Bihar Board Class 10 Mathematics Question Paper 2025 (Code 110 Set-I) Available- Download Here with Solution PDF

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (\(d\)). The nth term of an A.P. can be found using a standard formula.


Step 2: Key Formula or Approach:

The formula for the nth term (\(a_n\)) of an A.P. is:
\[ a_n = a + (n-1)d \]
where \(a\) is the first term, \(n\) is the term number, and \(d\) is the common difference.


Step 3: Detailed Explanation:

We are given the following information:

The 5th term, \(a_5 = 11\).

The common difference, \(d = 2\).

The term number, \(n = 5\).

We need to find the first term, \(a\).

Substitute the given values into the formula:
\[ 11 = a + (5-1) \times 2 \] \[ 11 = a + (4) \times 2 \] \[ 11 = a + 8 \]
To find \(a\), subtract 8 from both sides of the equation:
\[ a = 11 - 8 \] \[ a = 3 \]
So, the first term of the A.P. is 3.


Step 4: Final Answer:

The first term of the A.P. is 3.
Quick Tip: To verify the answer, you can construct the A.P. with \(a=3\) and \(d=2\). The sequence is 3, 5, 7, 9, 11... The 5th term is indeed 11, which confirms the calculation.

Step 1: Understanding the Concept:

The nth term (\(a_n\)) of a sequence can be found from the formula for the sum of its first n terms (\(S_n\)). The nth term is the difference between the sum of the first n terms and the sum of the first (n-1) terms. Note that for a sequence to be an A.P., its sum \(S_n\) must be a quadratic expression of the form \(An^2 + Bn\). The given \(S_n\) has a constant term (+1), which means the sequence generated is not a true A.P. However, we proceed by applying the standard method to find the term.


Step 2: Key Formula or Approach:

The formula to find the nth term (\(a_n\)) from the sum of n terms (\(S_n\)) is:
\[ a_n = S_n - S_{n-1} \]
We will use this formula to find the 6th term, \(a_6\).


Step 3: Detailed Explanation:

The given formula for the sum of n terms is:
\[ S_n = n^2 + 2n + 1 \]
First, calculate the sum of the first 6 terms (\(S_6\)):
\[ S_6 = (6)^2 + 2(6) + 1 = 36 + 12 + 1 = 49 \]
Next, calculate the sum of the first 5 terms (\(S_5\)):
\[ S_5 = (5)^2 + 2(5) + 1 = 25 + 10 + 1 = 36 \]
Now, find the 6th term (\(a_6\)) by subtracting \(S_5\) from \(S_6\):
\[ a_6 = S_6 - S_5 = 49 - 36 = 13 \]
The calculated 6th term is 13. This value is not present in options (A), (B), or (C).


Step 4: Final Answer:

Since the calculated 6th term is 13 and this is not among the given options, the correct choice is (D) none of these.
Quick Tip: A quick way to find \(a_n\) from a quadratic \(S_n = An^2+Bn+C\) is to use the formula \(a_n = 2An + B-A\). Here, \(S_n = 1n^2+2n+1\), so \(A=1, B=2\), which gives \(a_n = 2(1)n + 2-1 = 2n+1\). For \(n=6\), \(a_6=2(6)+1=13\).

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is a sequence where the difference between any two consecutive terms is constant. This constant value is called the common difference.


Step 2: Key Formula or Approach:

To check if a sequence is an A.P., we calculate the difference between consecutive terms. If \(a_2-a_1 = a_3-a_2 = a_4-a_3 = \dots\), then the sequence is an A.P.


Step 3: Detailed Explanation:

Let's check each option:

(A) 1, 7, 9, 16, ...
\(a_2 - a_1 = 7 - 1 = 6\)
\(a_3 - a_2 = 9 - 7 = 2\)

Since the differences are not constant (6 \(\neq\) 2), this is not an A.P.


(B) \(x^2, x^3, x^4, x^5, ...\)
\(a_2 - a_1 = x^3 - x^2 = x^2(x - 1)\)
\(a_3 - a_2 = x^4 - x^3 = x^3(x - 1)\)

The differences are not constant (unless x=0 or x=1), so this is not an A.P. in general. It is a Geometric Progression.


(C) x, 2x, 3x, 4x, ...
\(a_2 - a_1 = 2x - x = x\)
\(a_3 - a_2 = 3x - 2x = x\)
\(a_4 - a_3 = 4x - 3x = x\)

Since the difference is constant (\(x\)), this sequence is an A.P.


(D) \(2^2, 4^2, 6^2, 8^2, ...\) which is 4, 16, 36, 64, ...
\(a_2 - a_1 = 16 - 4 = 12\)
\(a_3 - a_2 = 36 - 16 = 20\)

Since the differences are not constant (12 \(\neq\) 20), this is not an A.P.


Step 4: Final Answer:

The sequence that is in an A.P. is x, 2x, 3x, 4x, ....
Quick Tip: To quickly identify an A.P., simply subtract the first term from the second, and the second from the third. If these two results are not equal, you can immediately rule it out.

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is a sequence of numbers where the difference between consecutive terms is constant. We need to find the sequence that does not have a constant difference.


Step 2: Key Formula or Approach:

We will check the difference between consecutive terms for each sequence. If the difference is not constant, the sequence is not an A.P.


Step 3: Detailed Explanation:

Let's analyze each option:

(A) 1, 2, 3, 4, ...
\(2 - 1 = 1\), \(3 - 2 = 1\), \(4 - 3 = 1\). The common difference is 1. This is an A.P.


(B) 3, 6, 9, 12, ...
\(6 - 3 = 3\), \(9 - 6 = 3\), \(12 - 9 = 3\). The common difference is 3. This is an A.P.


(C) 2, 4, 6, 8, ...
\(4 - 2 = 2\), \(6 - 4 = 2\), \(8 - 6 = 2\). The common difference is 2. This is an A.P.


(D) \(2^2, 4^2, 6^2, 8^2, ...\) which is the sequence 4, 16, 36, 64, ...
\(16 - 4 = 12\)
\(36 - 16 = 20\)

The difference between consecutive terms is not constant (12 \(\neq\) 20). Therefore, this is not an A.P.


Step 4: Final Answer:

The sequence that is not in an A.P. is \(2^2, 4^2, 6^2, 8^2, ...\).
Quick Tip: Be careful with sequences involving squares or powers. They are rarely Arithmetic Progressions. Always write out the first few terms of the sequence to make the pattern clear before calculating differences.

Step 1: Understanding the Concept:

We need to find the sum of the first 20 terms of a given Arithmetic Progression. We can use the standard formula for the sum of an A.P.


Step 2: Key Formula or Approach:

The formula for the sum of the first \(n\) terms of an A.P. is:
\[ S_n = \frac{n}{2}[2a + (n-1)d] \]
where \(S_n\) is the sum of \(n\) terms, \(a\) is the first term, and \(d\) is the common difference.


Step 3: Detailed Explanation:

From the given A.P. 1, 4, 7, 10, ...:

The first term, \(a = 1\).

The common difference, \(d = 4 - 1 = 3\).

The number of terms, \(n = 20\).

Now, substitute these values into the sum formula:
\[ S_{20} = \frac{20}{2}[2(1) + (20-1) \times 3] \] \[ S_{20} = 10[2 + (19) \times 3] \] \[ S_{20} = 10[2 + 57] \] \[ S_{20} = 10(59) \] \[ S_{20} = 590 \]

Step 4: Final Answer:

The sum of the first 20 terms of the A.P. is 590.
Quick Tip: Always identify the values of a, d, and n from the question first. Then, carefully substitute them into the formula. A common mistake is an error in the order of operations (BODMAS/PEMDAS) - calculate the bracket contents completely before multiplying by \(n/2\).

Step 1: Understanding the Concept:

The question requires evaluating four trigonometric expressions to determine which one equals 1. This involves knowing standard trigonometric values and identities.


Step 2: Key Formula or Approach:

We will use the standard values of trigonometric functions for angles 45°, 60°, and 90°.
Key values: \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\), \(\sin 45^\circ = \frac{1}{\sqrt{2}}\), \(\cos 45^\circ = \frac{1}{\sqrt{2}}\).
Key identity: \(\frac{\sin \theta}{\cos \theta} = \tan \theta\).


Step 3: Detailed Explanation:

Let's evaluate each option:

(A) \(\sin^2 60^\circ + \cos 60^\circ = (\frac{\sqrt{3}}{2})^2 + \frac{1}{2} = \frac{3}{4} + \frac{1}{2} = \frac{3}{4} + \frac{2}{4} = \frac{5}{4}\). This is not equal to 1.


(B) \(\sin 90^\circ \times \cos 90^\circ = 1 \times 0 = 0\). This is not equal to 1.


(C) \(\sin^2 60^\circ = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4}\). This is not equal to 1.


(D) \(\sin 45^\circ \times \frac{1}{\cos 45^\circ} = \frac{\sin 45^\circ}{\cos 45^\circ} = \tan 45^\circ\). Since \(\tan 45^\circ = 1\), this expression is equal to 1.


Step 4: Final Answer:

The expression that is equal to 1 is \(\sin 45^\circ \times \frac{1}{\cos 45^\circ}\).
Quick Tip: Recognizing identities can save time. The expression in option (D) is equivalent to \(\tan 45^\circ\). Knowing that \(\tan 45^\circ = 1\) allows you to find the answer without calculating with fractions.

Step 1: Understanding the Concept:

This problem requires simplifying a trigonometric expression using fundamental Pythagorean identities.


Step 2: Key Formula or Approach:

We will use the Pythagorean identity:
\[ 1 + \tan^2 A = \sec^2 A \]
And the reciprocal identity:
\[ \sec A = \frac{1}{\cos A} \]

Step 3: Detailed Explanation:

We start with the given expression:
\[ \cos^2 A (1 + \tan^2 A) \]
First, substitute the Pythagorean identity \(1 + \tan^2 A = \sec^2 A\) into the expression:
\[ = \cos^2 A (\sec^2 A) \]
Next, use the reciprocal identity \(\sec A = \frac{1}{\cos A}\), which means \(\sec^2 A = \frac{1}{\cos^2 A}\):
\[ = \cos^2 A \left( \frac{1}{\cos^2 A} \right) \]
Now, cancel out the \(\cos^2 A\) terms:
\[ = 1 \]

Step 4: Final Answer:

The value of the expression \(\cos^2 A (1 + \tan^2 A)\) is 1.
Quick Tip: Memorizing the three Pythagorean identities (\(\sin^2\theta + \cos^2\theta = 1\), \(1 + \tan^2\theta = \sec^2\theta\), and \(1 + \cot^2\theta = \csc^2\theta\)) is crucial for quickly solving trigonometric simplification problems.

Step 1: Understanding the Concept:

This question asks for the standard value of the tangent function for the angle 30 degrees.


Step 2: Key Formula or Approach:

The value of \(\tan 30^\circ\) is a fundamental trigonometric ratio that should be memorized. It can also be derived from the ratio \(\frac{\sin 30^\circ}{\cos 30^\circ}\).

We know \(\sin 30^\circ = \frac{1}{2}\) and \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).


Step 3: Detailed Explanation:

Using the ratio definition:
\[ \tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{1/2}{\sqrt{3}/2} \] \[ = \frac{1}{2} \times \frac{2}{\sqrt{3}} \] \[ = \frac{1}{\sqrt{3}} \]
This is a standard value.


Step 4: Final Answer:

The value of \(\tan 30^\circ\) is \(\frac{1}{\sqrt{3}}\).
Quick Tip: Memorizing the trigonometric values for 0°, 30°, 45°, 60°, and 90° is essential for speed and accuracy in exams. For tangent, remember the pattern: 0, \(1/\sqrt{3}\), 1, \(\sqrt{3}\), undefined.

Step 1: Understanding the Concept:

This question asks for the standard value of the cosine function for the angle 60 degrees.


Step 2: Key Formula or Approach:

This is a fundamental value in trigonometry. We can recall it from memory or use the complementary angle identity \(\cos \theta = \sin(90^\circ - \theta)\).


Step 3: Detailed Explanation:

The value of \(\cos 60^\circ\) is a standard result from the properties of a 30-60-90 triangle.
\[ \cos 60^\circ = \frac{1}{2} \]
Alternatively, using the complementary angle identity:
\[ \cos 60^\circ = \sin(90^\circ - 60^\circ) = \sin 30^\circ = \frac{1}{2} \]

Step 4: Final Answer:

The value of \(\cos 60^\circ\) is \(\frac{1}{2}\).
Quick Tip: Remember the relationship \(\cos 60^\circ = \sin 30^\circ = 1/2\) and \(\sin 60^\circ = \cos 30^\circ = \sqrt{3}/2\). This pairing of complementary angles can help you memorize the values more easily.

Step 1: Understanding the Concept:

We need to evaluate the given trigonometric expression by substituting the known values of \(\sin 90^\circ\) and \(\tan 45^\circ\).


Step 2: Key Formula or Approach:

We will use the standard trigonometric values:
\[ \sin 90^\circ = 1 \] \[ \tan 45^\circ = 1 \]

Step 3: Detailed Explanation:

The expression is \(\sin^2 90^\circ - \tan^2 45^\circ\).

This can be written as \((\sin 90^\circ)^2 - (\tan 45^\circ)^2\).

Substitute the known values:
\[ = (1)^2 - (1)^2 \] \[ = 1 - 1 \] \[ = 0 \]

Step 4: Final Answer:

The value of the expression is 0.
Quick Tip: Be careful with the notation \(\sin^2 \theta\), which means \((\sin \theta)^2\). First find the value of the trigonometric function, then apply the exponent.

Step 1: Understanding the Concept:

This problem requires finding the distance between two points in a Cartesian coordinate system. First, we need to evaluate the coordinates using trigonometric values, and then apply the distance formula.


Step 2: Key Formula or Approach:

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We also need the standard values: \(\sin 60^\circ = \frac{\sqrt{3}}{2}\) and \(\cos 60^\circ = \frac{1}{2}\).


Step 3: Detailed Explanation:

First, let's find the coordinates of the two points.

Point 1: \((x_1, y_1) = (8 \sin 60^\circ, 0) = (8 \times \frac{\sqrt{3}}{2}, 0) = (4\sqrt{3}, 0)\).

Point 2: \((x_2, y_2) = (0, 8 \cos 60^\circ) = (0, 8 \times \frac{1}{2}) = (0, 4)\).

Now, apply the distance formula:
\[ d = \sqrt{(0 - 4\sqrt{3})^2 + (4 - 0)^2} \] \[ d = \sqrt{(-4\sqrt{3})^2 + (4)^2} \] \[ d = \sqrt{(16 \times 3) + 16} \] \[ d = \sqrt{48 + 16} \] \[ d = \sqrt{64} \] \[ d = 8 \]

Step 4: Final Answer:

The distance between the two points is 8.
Quick Tip: Before applying the distance formula, always simplify the coordinates as much as possible. This makes the calculation steps cleaner and reduces the chance of errors.

Step 1: Understanding the Concept:

This question asks for the formula for the distance of a point from the origin in a Cartesian coordinate system. This is a direct application of the distance formula.


Step 2: Key Formula or Approach:

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
We will apply this formula for the origin O(0, 0) and the point P(x, y).


Step 3: Detailed Explanation:

Let the two points be:

Point 1 (Origin O): \((x_1, y_1) = (0, 0)\).

Point 2 (Point P): \((x_2, y_2) = (x, y)\).

Substitute these coordinates into the distance formula:
\[ Distance OP = \sqrt{(x - 0)^2 + (y - 0)^2} \] \[ Distance OP = \sqrt{(x)^2 + (y)^2} \] \[ Distance OP = \sqrt{x^2 + y^2} \]
This is the standard formula for the distance of a point from the origin.


Step 4: Final Answer:

The distance OP is \(\sqrt{x^2 + y^2}\).
Quick Tip: The distance of any point (x, y) from the origin (0, 0) is simply the square root of the sum of the squares of its coordinates. This is a direct consequence of the Pythagorean theorem.

Step 1: Understanding the Concept:

The distance of any point \((x, y)\) from the y-axis is the perpendicular distance to the axis. This distance is always equal to the absolute value of the x-coordinate of the point.


Step 2: Key Formula or Approach:

Distance from y-axis = \(|x|\)

For the given point \((12, 14)\), the x-coordinate is 12.


Step 3: Detailed Explanation:

The point \((12, 14)\) is located 12 units to the right of the y-axis and 14 units above the x-axis.

The perpendicular distance from the point to the y-axis is determined solely by its horizontal position, which is its x-coordinate.

Therefore, the distance is \(|12| = 12\).


Step 4: Final Answer:

The distance of the point (12, 14) from the y-axis is 12.
Quick Tip: A simple way to remember this is: the distance from the y-axis is the x-coordinate, and the distance from the x-axis is the y-coordinate (always take the positive value).

Step 1: Understanding the Concept:

In a Cartesian coordinate system, a point is represented by an ordered pair \((x, y)\). The x-coordinate is called the abscissa, and the y-coordinate is called the ordinate.


Step 2: Detailed Explanation:

The given point is \((-6, -8)\).

Here, the x-coordinate (abscissa) is -6.

The y-coordinate (ordinate) is -8.

The question asks for the ordinate of the point.


Step 3: Final Answer:

The ordinate of the point (-6, -8) is -8.
Quick Tip: Remember the alphabetical order: Abscissa (A) comes before Ordinate (O), just as x comes before y. Abscissa = x, Ordinate = y.

Step 1: Understanding the Concept:

The Cartesian plane is divided into four quadrants by the x-axis and y-axis. The sign of the x and y coordinates determines the quadrant.

- Quadrant I: x is positive (+), y is positive (+)

- Quadrant II: x is negative (-), y is positive (+)

- Quadrant III: x is negative (-), y is negative (-)

- Quadrant IV: x is positive (+), y is negative (-)


Step 2: Detailed Explanation:

The given point is \((3, -4)\).

The x-coordinate is 3, which is positive.

The y-coordinate is -4, which is negative.

A point with a positive x-coordinate and a negative y-coordinate \((+, -)\) lies in the Fourth Quadrant.


Step 3: Final Answer:

The point (3, -4) lies in the Fourth quadrant.
Quick Tip: You can visualize the quadrants by starting from the top-right and moving counter-clockwise. Top-right is I (+,+), top-left is II (-,+), bottom-left is III (-,-), and bottom-right is IV (+,-).

Step 1: Understanding the Concept:

A point lies in the second quadrant if its x-coordinate is negative and its y-coordinate is positive. The sign pattern for the second quadrant is \((-, +)\).


Step 2: Detailed Explanation:

Let's check the coordinates of each option:

(A) (3, 2): x is positive, y is positive \((+, +)\). This is in the First Quadrant.

(B) (-3, 2): x is negative, y is positive \((-, +)\). This is in the Second Quadrant.

(C) (3, -2): x is positive, y is negative \((+, -)\). This is in the Fourth Quadrant.

(D) (-3, -2): x is negative, y is negative \((-, -)\). This is in the Third Quadrant.


Step 3: Final Answer:

The point that lies in the second quadrant is (-3, 2).
Quick Tip: To quickly find a point in the second quadrant, look for the option that has a negative number first and a positive number second.

Step 1: Understanding the Concept:

The midpoint of a line segment is the point that lies exactly halfway between the two endpoints. Its coordinates are the average of the x-coordinates and the average of the y-coordinates of the endpoints.


Step 2: Key Formula or Approach:

The formula for the midpoint \((x_m, y_m)\) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Step 3: Detailed Explanation:

The given endpoints are \((4, -4)\) and \((-4, 4)\).

Let \((x_1, y_1) = (4, -4)\) and \((x_2, y_2) = (-4, 4)\).

Calculate the x-coordinate of the midpoint:
\[ x_m = \frac{4 + (-4)}{2} = \frac{0}{2} = 0 \]
Calculate the y-coordinate of the midpoint:
\[ y_m = \frac{-4 + 4}{2} = \frac{0}{2} = 0 \]
So, the coordinates of the midpoint are \((0, 0)\).


Step 4: Final Answer:

The co-ordinates of the mid-point are (0, 0).
Quick Tip: The midpoint (0,0) is also known as the origin. If you notice that the coordinates of the two points are opposites of each other (like (a, -b) and (-a, b)), their midpoint will always be the origin.

Step 1: Understanding the Concept:

This problem requires using the midpoint formula to find the coordinates of one endpoint when the other endpoint and the midpoint are known.


Step 2: Key Formula or Approach:

Let the coordinates of A be \((x_1, y_1)\), B be \((x_2, y_2)\), and the midpoint M be \((x_m, y_m)\). The midpoint formula is:
\[ x_m = \frac{x_1 + x_2}{2} \quad and \quad y_m = \frac{y_1 + y_2}{2} \]
We can rearrange this to solve for the unknown endpoint coordinates:
\[ x_2 = 2x_m - x_1 \quad and \quad y_2 = 2y_m - y_1 \]

Step 3: Detailed Explanation:

We are given:

Midpoint M \((x_m, y_m) = (2, 4)\)

Point A \((x_1, y_1) = (5, 7)\)

We need to find Point B \((x_2, y_2)\).

Using the rearranged formulas:

For the x-coordinate of B:
\[ x_2 = 2(2) - 5 = 4 - 5 = -1 \]
For the y-coordinate of B:
\[ y_2 = 2(4) - 7 = 8 - 7 = 1 \]
So, the coordinates of point B are \((-1, 1)\).


Step 4: Final Answer:

The co-ordinates of point B are (-1, 1).
Quick Tip: Think of the midpoint as the average. To find the other end, you can think: "How did I get from A to M?" From x=5 to x=2, you subtract 3. Do it again: 2 - 3 = -1. From y=7 to y=4, you subtract 3. Do it again: 4 - 3 = 1. So, B is (-1, 1).

Step 1: Understanding the Concept:

The center of a circle is the midpoint of any of its diameters. Therefore, we can find the center by calculating the midpoint of the given endpoints of the diameter.


Step 2: Key Formula or Approach:

The midpoint formula for endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ Center = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Step 3: Detailed Explanation:

The given endpoints of the diameter are \((10, -6)\) and \((-6, 10)\).

Let \((x_1, y_1) = (10, -6)\) and \((x_2, y_2) = (-6, 10)\).

Calculate the x-coordinate of the center:
\[ x_c = \frac{10 + (-6)}{2} = \frac{4}{2} = 2 \]
Calculate the y-coordinate of the center:
\[ y_c = \frac{-6 + 10}{2} = \frac{4}{2} = 2 \]
The coordinates of the center are \((2, 2)\).


Step 4: Final Answer:

The co-ordinates of the centre of the circle are (2, 2).
Quick Tip: This is a direct application of the midpoint formula. Remember that the center of a circle is always the midpoint of its diameter.

Step 1: Understanding the Concept:

The centroid of a triangle is the point where its three medians intersect. The coordinates of the centroid are the average of the x-coordinates and the average of the y-coordinates of the three vertices.


Step 2: Key Formula or Approach:

The formula for the centroid \((x_c, y_c)\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
\[ (x_c, y_c) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

Step 3: Detailed Explanation:

The given vertices are \((4, 6)\), \((0, 4)\), and \((5, 5)\).

Calculate the x-coordinate of the centroid:
\[ x_c = \frac{4 + 0 + 5}{3} = \frac{9}{3} = 3 \]
Calculate the y-coordinate of the centroid:
\[ y_c = \frac{6 + 4 + 5}{3} = \frac{15}{3} = 5 \]
The coordinates of the centroid are \((3, 5)\).


Step 4: Final Answer:

The co-ordinates of the centroid of the triangle are (3, 5).
Quick Tip: The centroid is essentially the "average point" of the triangle's vertices. The formula is a straightforward extension of the midpoint formula from two points to three.

Step 1: Understanding the Concept:

This question tests the knowledge of fundamental trigonometric identities, specifically the complementary angle identities.


Step 2: Key Formula or Approach:

The complementary angle identities state the relationship between trigonometric functions of an angle and its complement (90° minus the angle). The key identity here is:
\[ \sin(90^\circ - A) = \cos A \]

Step 3: Detailed Explanation:

This is a direct application of the identity. The sine of an angle is equal to the cosine of its complementary angle. Therefore, \(\sin(90^\circ - A)\) is equal to \(\cos A\).


Step 4: Final Answer:

The value of \(\sin(90^\circ - A)\) is \(\cos A\).
Quick Tip: Remember the main pairs of complementary identities: \(\sin(90^\circ - A) = \cos A\), \(\cos(90^\circ - A) = \sin A\), and \(\tan(90^\circ - A) = \cot A\).

Step 1: Understanding the Concept:

We need to evaluate a trigonometric function by first substituting the given values for the angles.


Step 2: Key Formula or Approach:

First, calculate the value of the angle inside the cosine function, which is \((\alpha - \beta)\). Then, find the cosine of that resulting angle. We need to know the value of \(\cos 0^\circ\).


Step 3: Detailed Explanation:

We are given \(\alpha = 60^\circ\) and \(\beta = 60^\circ\).

Substitute these values into the expression \(\cos(\alpha - \beta)\):
\[ \cos(60^\circ - 60^\circ) \] \[ = \cos(0^\circ) \]
The value of \(\cos 0^\circ\) is a standard trigonometric value.
\[ \cos 0^\circ = 1 \]

Step 4: Final Answer:

The value of \(\cos(\alpha - \beta)\) is 1.
Quick Tip: This problem is simpler than it looks. It's a basic substitution and evaluation question. Don't be confused by the angle subtraction formula for cosine; it's not needed here since the angles are equal.

Step 1: Understanding the Concept:

We need to evaluate a trigonometric expression by substituting the given angle and using standard trigonometric values.


Step 2: Key Formula or Approach:

We will use the standard values for \(\sin 45^\circ\) and \(\cos 45^\circ\).
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \] \[ \cos 45^\circ = \frac{1}{\sqrt{2}} \]

Step 3: Detailed Explanation:

Substitute \(\theta = 45^\circ\) into the expression \(\sin \theta + \cos \theta\):
\[ \sin 45^\circ + \cos 45^\circ = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \] \[ = \frac{2}{\sqrt{2}} \]
To simplify, we can rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{2}\):
\[ = \frac{2 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{2}}{2} \] \[ = \sqrt{2} \]
Alternatively, we can write \(2\) as \((\sqrt{2})^2\): \[ \frac{(\sqrt{2})^2}{\sqrt{2}} = \sqrt{2} \]

Step 4: Final Answer:

The value of \(\sin 45^\circ + \cos 45^\circ\) is \(\sqrt{2}\).
Quick Tip: Remember that \(2/\sqrt{2} = \sqrt{2}\). This is a common simplification that can save you a step in calculations.

Step 1: Understanding the Concept:

The given expression is the formula for the tangent of a double angle. Recognizing this identity is the quickest way to solve the problem.


Step 2: Key Formula or Approach:

The double angle identity for tangent is:
\[ \tan(2A) = \frac{2 \tan A}{1 - \tan^2 A} \]

Step 3: Detailed Explanation:

The expression given is \(\frac{2 \tan A}{1 - \tan^2 A}\).

Using the double angle identity, this is equal to \(\tan(2A)\).

We are given that \(A = 30^\circ\).

Substitute this value into \(\tan(2A)\):
\[ \tan(2 \times 30^\circ) = \tan(60^\circ) \]
Alternatively, without using the identity:

We know \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).
\[ \frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} = \frac{2 \left(\frac{1}{\sqrt{3}}\right)}{1 - \left(\frac{1}{\sqrt{3}}\right)^2} = \frac{\frac{2}{\sqrt{3}}}{1 - \frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}} = \frac{2}{\sqrt{3}} \times \frac{3}{2} = \frac{3}{\sqrt{3}} = \sqrt{3} \]
And we know that \(\tan 60^\circ = \sqrt{3}\). Both methods give the same result.


Step 4: Final Answer:

The value of the expression is equal to \(\tan 60^\circ\).
Quick Tip: Recognizing the double angle formulas for \(\sin(2A)\), \(\cos(2A)\), and \(\tan(2A)\) can turn a multi-step calculation into a single-step problem.

Step 1: Understanding the Concept:

Given one trigonometric ratio, we can find any other trigonometric ratio by constructing a right-angled triangle or by using trigonometric identities.


Step 2: Key Formula or Approach:

We use the definitions of trigonometric ratios in a right-angled triangle:
\[ \tan \theta = \frac{Opposite}{Adjacent}, \quad \sin \theta = \frac{Opposite}{Hypotenuse} \]
We can find the hypotenuse using the Pythagorean theorem: \(h^2 = p^2 + b^2\).


Step 3: Detailed Explanation:

We are given \(\tan \theta = \frac{12}{5}\).

From this, we can consider a right-angled triangle where:

Opposite side (\(p\)) = 12

Adjacent side (\(b\)) = 5

Now, we find the hypotenuse (\(h\)) using the Pythagorean theorem:
\[ h = \sqrt{p^2 + b^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
Now we can find \(\sin \theta\):
\[ \sin \theta = \frac{Opposite}{Hypotenuse} = \frac{12}{13} \]

Step 4: Final Answer:

The value of \(\sin \theta\) is \(\frac{12}{13}\).
Quick Tip: Recognize common Pythagorean triples like (3, 4, 5), (5, 12, 13), and (8, 15, 17). If you see two of the numbers, you can instantly know the third without calculation. Here, we have 5 and 12, so the hypotenuse is 13.

Step 1: Understanding the Concept:

This problem can be simplified using complementary angle identities, which relate a trigonometric function of an angle to the co-function of its complement.


Step 2: Key Formula or Approach:

The key identities are:
\[ \cos \theta = \sin(90^\circ - \theta) \] \[ \tan \theta = \cot(90^\circ - \theta) \]

Step 3: Detailed Explanation:

Let's simplify each fraction separately.

First fraction: \(\frac{\cos 59^\circ}{\sin 31^\circ}\)

Using the identity, we can rewrite the numerator: \(\cos 59^\circ = \sin(90^\circ - 59^\circ) = \sin 31^\circ\).

So, the fraction becomes \(\frac{\sin 31^\circ}{\sin 31^\circ} = 1\).

Second fraction: \(\frac{\tan 80^\circ}{\cot 10^\circ}\)

Using the identity, we can rewrite the numerator: \(\tan 80^\circ = \cot(90^\circ - 80^\circ) = \cot 10^\circ\).

So, the fraction becomes \(\frac{\cot 10^\circ}{\cot 10^\circ} = 1\).

The entire expression is the product of the two simplified fractions:
\[ 1 \times 1 = 1 \]

Step 4: Final Answer:

The value of the expression is 1.
Quick Tip: When you see a fraction with a trigonometric function and its co-function (sin/cos, tan/cot, sec/csc), check if the angles add up to 90°. If they do, the value of the fraction is 1.

Step 1: Understanding the Concept:

This problem requires simplifying the product of two tangent functions using complementary angle identities, and then solving the resulting trigonometric equation.


Step 2: Key Formula or Approach:

The key identities are:
\[ \tan(90^\circ - \theta) = \cot \theta \] \[ \tan \theta \times \cot \theta = 1 \]

Step 3: Detailed Explanation:

First, let's simplify the left side of the equation: \(\tan 25^\circ \times \tan 65^\circ\).

Notice that \(25^\circ + 65^\circ = 90^\circ\), so the angles are complementary.

We can rewrite \(\tan 65^\circ\) using the identity:
\[ \tan 65^\circ = \tan(90^\circ - 25^\circ) = \cot 25^\circ \]
Now substitute this back into the expression:
\[ \tan 25^\circ \times \cot 25^\circ \]
Since \(\cot \theta = \frac{1}{\tan \theta}\), the product is:
\[ \tan 25^\circ \times \frac{1}{\tan 25^\circ} = 1 \]
So, the original equation becomes:
\[ 1 = \sin A \]
We need to find the angle A for which \(\sin A = 1\).

From standard trigonometric values, we know that \(\sin 90^\circ = 1\).

Therefore, \(A = 90^\circ\).


Step 4: Final Answer:

The value of A is 90°.
Quick Tip: A useful property to remember is: if A + B = 90°, then \(\tan A \times \tan B = 1\). This allows you to simplify the left side of the equation in a single step.

Step 1: Understanding the Concept:

We need to express \(\tan \theta\) in terms of \(x\), given that \(\cos \theta = x\). We can do this using trigonometric identities or by constructing a right-angled triangle.


Step 2: Key Formula or Approach:

Using identities:

1. The Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\), which gives \(\sin \theta = \sqrt{1 - \cos^2 \theta}\) (assuming \(\theta\) is in the first quadrant).
2. The ratio identity: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).


Step 3: Detailed Explanation:

We are given \(\cos \theta = x\).

First, find \(\sin \theta\) using the Pythagorean identity:
\[ \sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - x^2} \]
Now, use the ratio identity to find \(\tan \theta\):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\sqrt{1 - x^2}}{x} \]
Alternatively, using a triangle:

Since \(\cos \theta = \frac{Adjacent}{Hypotenuse} = \frac{x}{1}\), we can set Adjacent = \(x\) and Hypotenuse = 1.

By Pythagorean theorem, Opposite = \(\sqrt{Hypotenuse^2 - Adjacent^2} = \sqrt{1^2 - x^2} = \sqrt{1 - x^2}\).

Then, \(\tan \theta = \frac{Opposite}{Adjacent} = \frac{\sqrt{1 - x^2}}{x}\).


Step 4: Final Answer:

The value of \(\tan \theta\) is \(\frac{\sqrt{1-x^2}}{x}\).
Quick Tip: When given one trigonometric ratio and asked for another, drawing a simple right-angled triangle is often the most intuitive and error-free method.

Step 1: Understanding the Concept:

This problem involves simplifying a trigonometric expression by using algebraic factorization and fundamental trigonometric identities.


Step 2: Key Formula or Approach:

We will use the algebraic identity for the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).

We will also use the Pythagorean identity: \(\sin^2 \theta + \cos^2 \theta = 1\), which can be rearranged to \(\sin^2 \theta = 1 - \cos^2 \theta\).


Step 3: Detailed Explanation:

The given expression is \(1 - \cos^4 \theta\).

We can write this as \(1^2 - (\cos^2 \theta)^2\).

This is a difference of squares, where \(a=1\) and \(b=\cos^2 \theta\). Factoring it gives:
\[ (1 - \cos^2 \theta)(1 + \cos^2 \theta) \]
Now, we use the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\) to substitute for the first term:
\[ (\sin^2 \theta)(1 + \cos^2 \theta) \]
This matches option (B).


Step 4: Final Answer:

The expression \(1 - \cos^4 \theta\) is equal to \(\sin^2 \theta (1 + \cos^2 \theta)\).
Quick Tip: Whenever you see trigonometric functions with even powers (like 2, 4, 6), be on the lookout for opportunities to use the difference of squares factorization or the Pythagorean identities.

Step 1: Understanding the Concept:

The y-axis is the vertical line on the Cartesian plane where the value of the x-coordinate is always zero. A point lying on this axis will have no horizontal displacement from the origin.


Step 2: Detailed Explanation:

For any point to be on the y-axis, its x-coordinate must be 0. The y-coordinate can be any real number.

Thus, the general form of a point on the y-axis is \((0, y)\), where \(y\) can be any value.

Let's check the options:

(A) (y, 0): This represents a point on the x-axis, as its y-coordinate is 0.

(B) (2, y): This represents a point on the vertical line \(x=2\), which is parallel to the y-axis but not the y-axis itself.

(C) (0, x): This represents a point where the x-coordinate is 0 and the y-coordinate is some variable value (represented here by the letter 'x'). This matches the required form \((0, any value)\).


Step 3: Final Answer:

The form of a point lying on the y-axis is (0, any number). Option (C) represents this form, using 'x' as the variable for the y-coordinate. Therefore, (0, x) is the correct answer.
Quick Tip: Remember: On the y-axis, x is always zero. On the x-axis, y is always zero. This is a fundamental concept in coordinate geometry.

Step 1: Understanding the Concept:

A quadratic polynomial can be constructed if its zeroes (roots) are known. Let the zeroes be \(\alpha\) and \(\beta\). The polynomial can be expressed in the form \(k(x^2 - (\alpha + \beta)x + \alpha\beta)\), where \(k\) is a non-zero constant.


Step 2: Key Formula or Approach:

1. Find the sum of the zeroes: Sum = \(\alpha + \beta\).

2. Find the product of the zeroes: Product = \(\alpha \beta\).

3. Substitute these values into the formula: \(x^2 - (Sum)x + (Product)\). (Assuming k=1, which is standard for multiple-choice questions).


Step 3: Detailed Explanation:

The given zeroes are \(\alpha = 3\) and \(\beta = -10\).

Calculate the sum of the zeroes:
\[ Sum = 3 + (-10) = -7 \]
Calculate the product of the zeroes:
\[ Product = 3 \times (-10) = -30 \]
Now, construct the polynomial:
\[ x^2 - (Sum)x + (Product) = x^2 - (-7)x + (-30) \] \[ = x^2 + 7x - 30 \]
This matches option (A).


Step 4: Final Answer:

The required quadratic polynomial is \(x^2 + 7x - 30\).
Quick Tip: Pay close attention to the signs when substituting the sum and product into the formula. The formula has a minus sign before the sum, so `x² - (-7)x` becomes `x² + 7x`.

Step 1: Understanding the Concept:

We are given the sum and product of the zeroes of a quadratic polynomial and need to find the polynomial itself.


Step 2: Key Formula or Approach:

The general form of a quadratic polynomial with a given sum and product of zeroes is:
\[ P(x) = k(x^2 - (sum of zeroes)x + (product of zeroes)) \]
We can assume the constant \(k=1\) for simplicity, as it is standard for such problems.


Step 3: Detailed Explanation:

We are given:

Sum of zeroes = 3

Product of zeroes = -2

Substitute these values into the formula:
\[ P(x) = x^2 - (3)x + (-2) \] \[ P(x) = x^2 - 3x - 2 \]
This expression matches option (A).


Step 4: Final Answer:

The quadratic polynomial is \(x^2 - 3x - 2\).
Quick Tip: This is a direct application of the formula. Memorize the form \(x^2 - (Sum)x + (Product)\). Be careful with the signs during substitution.

Step 1: Understanding the Concept:

When a polynomial is divided by another polynomial, the degree of the quotient (the result of the division) can be determined by subtracting the degree of the divisor from the degree of the dividend.


Step 2: Key Formula or Approach:

Degree of Quotient = Degree of Dividend - Degree of Divisor


Step 3: Detailed Explanation:

The dividend is \(p(x) = x^4 - 2x^3 + 17x^2 - 4x + 30\).

The degree of a polynomial is the highest power of the variable.

The degree of the dividend \(p(x)\) is 4.

The divisor is \(q(x) = x + 2\).

The degree of the divisor \(q(x)\) is 1.

Now, apply the formula:
\[ Degree of Quotient = 4 - 1 = 3 \]
We do not need to perform the actual division.


Step 4: Final Answer:

The degree of the quotient is 3.
Quick Tip: This rule is very useful for quickly determining the nature of the result of polynomial division without carrying out the lengthy process.

Step 1: Understanding the Concept:

For a pair of linear equations in two variables, \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\), the number of solutions depends on the ratio of their coefficients.

1. If \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), the lines are intersecting and there is one unique solution.

2. If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\), the lines are parallel and there is no solution.

3. If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), the lines are coincident and there are infinitely many solutions.


Step 2: Detailed Explanation:

For the given equations:

Equation 1: \(x + 2y + 3 = 0\), so \(a_1 = 1, b_1 = 2, c_1 = 3\).

Equation 2: \(3x + 6y + 9 = 0\), so \(a_2 = 3, b_2 = 6, c_2 = 9\).

Now, let's find the ratios:
\[ \frac{a_1}{a_2} = \frac{1}{3} \] \[ \frac{b_1}{b_2} = \frac{2}{6} = \frac{1}{3} \] \[ \frac{c_1}{c_2} = \frac{3}{9} = \frac{1}{3} \]
Since \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\), the two lines are coincident (they are the same line).


Step 3: Final Answer:

The system of equations will have infinitely many solutions.
Quick Tip: Notice that the second equation, \(3x + 6y + 9 = 0\), is just the first equation, \(x + 2y + 3 = 0\), multiplied by 3. When one equation is a multiple of the other, the lines are coincident.

Step 1: Understanding the Concept:

The solutions to a system of linear equations correspond to the points of intersection of their graphs.


Step 2: Detailed Explanation:

Parallel lines are lines in a plane that never intersect, no matter how far they are extended.

Since they never meet, there are no common points that satisfy both equations simultaneously.

Therefore, a system of linear equations whose graphs are parallel has zero solutions (or no solution).

Let's analyze the given options:

(A) 1: This corresponds to intersecting lines.

(B) 2: A system of two linear equations cannot have exactly two solutions.

(C) infinitely many: This corresponds to coincident lines.

(D) none of these: Since the correct answer is 0 (no solution), and 0 is not listed in options A, B, or C, this option is the correct choice.


Step 3: Final Answer:

The number of solutions is zero. As this is not an option, the correct answer is none of these.
Quick Tip: Geometrically, solving a system of two linear equations means finding where the two lines cross. - Intersecting lines = 1 solution - Parallel lines = 0 solutions - Coincident (same) line = Infinite solutions

Step 1: Understanding the Concept:

A pair of linear equations can be classified based on its number of solutions:

- Consistent: The system has at least one solution (one or infinitely many).

- Inconsistent: The system has no solution.

- Dependent: The system has infinitely many solutions (a sub-category of consistent).

We can determine this by comparing the ratios of the coefficients \(a\), \(b\), and \(c\).


Step 2: Detailed Explanation:

The given equations are:

1. \(5x - 4y + 8 = 0\), where \(a_1 = 5, b_1 = -4, c_1 = 8\).

2. \(7x + 6y - 9 = 0\), where \(a_2 = 7, b_2 = 6, c_2 = -9\).

Let's compare the ratios of the coefficients of x and y:
\[ \frac{a_1}{a_2} = \frac{5}{7} \] \[ \frac{b_1}{b_2} = \frac{-4}{6} = -\frac{2}{3} \]
Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) (\(\frac{5}{7} \neq -\frac{2}{3}\)), the lines representing these equations will intersect at a single, unique point.

A system with one solution is defined as being consistent.


Step 3: Final Answer:

The pair of linear equations is consistent.
Quick Tip: The quickest way to check is to compare \(\frac{a_1}{a_2}\) and \(\frac{b_1}{b_2}\). If they are not equal, the system is consistent with a unique solution. You don't need to check the 'c' ratio.

Step 1: Understanding the Concept:

For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots (\(\alpha + \beta\)) is \(-b/a\) and the product of the roots (\(\alpha\beta\)) is \(c/a\). We can use these relationships to find the value of expressions involving the roots, like \(\alpha^2 + \beta^2\).


Step 2: Key Formula or Approach:

1. Find the sum and product of the roots.

2. Use the algebraic identity: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\).


Step 3: Detailed Explanation:

For the equation \(3x^2 - 5x + 2 = 0\), we have \(a=3, b=-5, c=2\).

Sum of roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-5}{3} = \frac{5}{3} \]
Product of roots:
\[ \alpha\beta = \frac{c}{a} = \frac{2}{3} \]
Now, use the identity to find \(\alpha^2 + \beta^2\):
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \] \[ = \left(\frac{5}{3}\right)^2 - 2\left(\frac{2}{3}\right) \] \[ = \frac{25}{9} - \frac{4}{3} \]
To subtract, find a common denominator (9):
\[ = \frac{25}{9} - \frac{12}{9} = \frac{13}{9} \]

Step 4: Final Answer:

The value of \(\alpha^2 + \beta^2\) is \(\frac{13}{9}\).
Quick Tip: Remember the key identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\). It is frequently used in problems involving the sum of the squares of roots.

Step 1: Understanding the Concept:

If a number is a root (or zero) of an equation, it means that substituting this number for the variable will make the equation true.


Step 2: Key Formula or Approach:

Substitute the value of the given root, \(x = 2\), into the quadratic equation and solve for the unknown parameter, \(p\).


Step 3: Detailed Explanation:

The given equation is \(2x^2 - 7x - p = 0\).

We are given that one root is \(x = 2\).

Substitute \(x = 2\) into the equation:
\[ 2(2)^2 - 7(2) - p = 0 \] \[ 2(4) - 14 - p = 0 \] \[ 8 - 14 - p = 0 \] \[ -6 - p = 0 \]
Now, solve for \(p\):
\[ p = -6 \]

Step 4: Final Answer:

The value of p is -6.
Quick Tip: This is a standard problem type. Whenever you are given a root and an unknown coefficient, the first step is always to substitute the root's value into the equation.

Step 1: Understanding the Concept:

For a quadratic equation \(ax^2 + bx + c = 0\) with roots \(\alpha\) and \(\beta\), we know the sum of roots is \(\alpha + \beta = -b/a\) and the product is \(\alpha\beta = c/a\). If one root is known, we can use either of these relations to find the other.


Step 2: Key Formula or Approach:

Using the sum of roots is often the easiest. Let the known root be \(\alpha\) and the unknown root be \(\beta\).
\[ \alpha + \beta = -\frac{b}{a} \]

Step 3: Detailed Explanation:

The equation is \(2x^2 - x - 6 = 0\), so \(a=2, b=-1, c=-6\).

One root is given as \(\alpha = -\frac{3}{2}\).

Using the sum of roots formula:
\[ -\frac{3}{2} + \beta = -\frac{-1}{2} \] \[ -\frac{3}{2} + \beta = \frac{1}{2} \] \[ \beta = \frac{1}{2} + \frac{3}{2} \] \[ \beta = \frac{4}{2} = 2 \]
To verify, let's use the product of roots: \(\alpha\beta = \frac{c}{a} = \frac{-6}{2} = -3\).
\[ \left(-\frac{3}{2}\right) \times 2 = -3 \]
The product matches, so the calculation is correct.


Step 4: Final Answer:

The another root is 2.
Quick Tip: Using the sum of roots is generally simpler than the product when finding the second root, as it involves addition/subtraction rather than division. However, using the product is a great way to double-check your answer.

Step 1: Understanding the Concept:

The nature of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is determined by its discriminant, \(D = b^2 - 4ac\).

- If \(D > 0\), the roots are real and unequal.

- If \(D = 0\), the roots are real and equal.

- If \(D < 0\), the roots are not real (they are complex).


Step 2: Key Formula or Approach:

Calculate the discriminant \(D = b^2 - 4ac\) and analyze its value.


Step 3: Detailed Explanation:

For the equation \(2x^2 - 6x + 3 = 0\), we have:
\(a = 2\), \(b = -6\), \(c = 3\).

Now, calculate the discriminant:
\[ D = (-6)^2 - 4(2)(3) \] \[ D = 36 - 24 \] \[ D = 12 \]
Since \(D = 12\), which is greater than 0, the roots of the equation are real and unequal.


Step 4: Final Answer:

The nature of the roots is real and unequal.
Quick Tip: The discriminant is a powerful tool. You don't need to solve the equation to know the type of roots it has. Just calculating \(b^2 - 4ac\) is enough.

Step 1: Understanding the Concept:

We need to find the area of a triangle given the coordinates of its three vertices. We can use the standard coordinate geometry formula for the area of a triangle, or a simpler method if the triangle has a vertical or horizontal base.


Step 2: Key Formula or Approach:

Method 1: Area Formula
\[ Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]
Method 2: Base and Height

Notice that points A(0, 1) and B(0, 5) both have an x-coordinate of 0. This means they lie on the y-axis, and the segment AB is a vertical line. We can use this as the base of the triangle.
\[ Area = \frac{1}{2} \times base \times height \]

Step 3: Detailed Explanation:

Using Method 2 (Base and Height), which is simpler here:

The base of the triangle is the distance between A(0, 1) and B(0, 5).
\[ Base = |5 - 1| = 4 units \]
The height of the triangle is the perpendicular distance from the third vertex C(3, 4) to the line containing the base (the y-axis). This distance is simply the absolute value of the x-coordinate of C.
\[ Height = |3| = 3 units \]
Now, calculate the area:
\[ Area = \frac{1}{2} \times 4 \times 3 = 6 square units \]

Step 4: Final Answer:

The area of \(\triangle ABC\) is 6 square units.
Quick Tip: Before jumping into the full area formula, always check if two of the vertices share the same x or y coordinate. If so, you have a vertical or horizontal base, and the `1/2 * base * height` method is much faster.

Step 1: Understanding the Concept:

This problem involves simplifying a product of tangent functions by using the complementary angle identity.


Step 2: Key Formula or Approach:

The key identities are:

1. \(\tan(90^\circ - \theta) = \cot \theta\)

2. \(\tan \theta \times \cot \theta = 1\)


Step 3: Detailed Explanation:

The given expression is \(\tan 10^\circ \tan 23^\circ \tan 80^\circ \tan 67^\circ\).

Let's pair the angles that add up to 90°: (10°, 80°) and (23°, 67°).

Now, convert one angle from each pair using the identity \(\tan(90^\circ - \theta) = \cot \theta\).
\[ \tan 80^\circ = \tan(90^\circ - 10^\circ) = \cot 10^\circ \] \[ \tan 67^\circ = \tan(90^\circ - 23^\circ) = \cot 23^\circ \]
Substitute these back into the expression:
\[ \tan 10^\circ \tan 23^\circ \cot 10^\circ \cot 23^\circ \]
Group the terms with the same angle:
\[ (\tan 10^\circ \cot 10^\circ) \times (\tan 23^\circ \cot 23^\circ) \]
Since \(\tan \theta \cot \theta = 1\), the expression becomes:
\[ 1 \times 1 = 1 \]

Step 4: Final Answer:

The value of the expression is 1.
Quick Tip: When you see a product of tangent functions, immediately check for pairs of angles that add up to 90°. Each such pair \(\tan A \tan B\) will simplify to 1.

Step 1: Understanding the Concept:

There is a fundamental theorem in geometry that relates the areas of similar triangles to their corresponding sides.


Step 2: Key Formula or Approach:

If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides.
\[ \frac{Area_1}{Area_2} = \left(\frac{Side_1}{Side_2}\right)^2 \]
Therefore, the ratio of the sides is the square root of the ratio of the areas.
\[ \frac{Side_1}{Side_2} = \sqrt{\frac{Area_1}{Area_2}} \]

Step 3: Detailed Explanation:

We are given the ratio of the areas:
\[ \frac{Area_1}{Area_2} = \frac{100}{144} \]
To find the ratio of the corresponding sides, we take the square root:
\[ \frac{Side_1}{Side_2} = \sqrt{\frac{100}{144}} = \frac{\sqrt{100}}{\sqrt{144}} = \frac{10}{12} \]
So, the ratio of the corresponding sides is 10 : 12.


Step 4: Final Answer:

The ratio of their corresponding sides is 10 : 12.
Quick Tip: Remember the relationship: Ratio of Sides = \(r\), Ratio of Areas = \(r^2\), Ratio of Volumes (for similar solids) = \(r^3\). To go from area to side, you need to take the square root.

Step 1: Understanding the Definitions:

This question tests the basic definitions of lines related to a circle.

- A Chord is a line \textit{segment whose two endpoints lie on the circle.

- A Tangent is a line that touches the circle at exactly one point.

- A Secant is a line that passes through the circle, intersecting it at two distinct points.


Step 2: Detailed Explanation:

The question asks for the name of a \textit{line that intersects a circle in two distinct points. Based on the standard definitions in geometry, this line is called a secant. The part of the secant that lies inside the circle is a chord.


Step 3: Final Answer:

The correct term is Secant.
Quick Tip: Remember the key difference: a secant is a full line that goes through the circle, while a chord is just the segment inside the circle. A tangent stays completely outside, touching at only one point.

Step 1: Understanding the Concept:

This question uses the theorem that relates the ratio of the sides of similar triangles to the ratio of their areas.


Step 2: Key Formula or Approach:

If the ratio of the corresponding sides of two similar triangles is \(a : b\), then the ratio of their areas is \(a^2 : b^2\).
\[ \frac{Area_1}{Area_2} = \left(\frac{Side_1}{Side_2}\right)^2 \]

Step 3: Detailed Explanation:

We are given the ratio of the corresponding sides:
\[ \frac{Side_1}{Side_2} = \frac{4}{9} \]
To find the ratio of the areas, we square this ratio:
\[ \frac{Area_1}{Area_2} = \left(\frac{4}{9}\right)^2 = \frac{4^2}{9^2} = \frac{16}{81} \]
So, the ratio of the areas is 16 : 81.


Step 4: Final Answer:

The ratio of the areas of the triangles will be 16 : 81.
Quick Tip: To go from the ratio of sides to the ratio of areas, you square the numbers. To go from the ratio of areas to the ratio of sides, you take the square root. Don't mix them up.

Step 1: Understanding the Concept:

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.


Step 2: Key Formula or Approach:
\[ \frac{Area(\triangle ABC)}{Area(\triangle DEF)} = \left(\frac{BC}{EF}\right)^2 \]

Step 3: Detailed Explanation:

We are given:

Area(\(\triangle ABC\)) = 54 cm\(^2\)

BC = 3 cm

EF = 4 cm

Let the Area(\(\triangle DEF\)) be \(x\).

Substitute the values into the formula:
\[ \frac{54}{x} = \left(\frac{3}{4}\right)^2 \] \[ \frac{54}{x} = \frac{9}{16} \]
Now, solve for \(x\):
\[ 9x = 54 \times 16 \] \[ x = \frac{54 \times 16}{9} \]
Since \(54/9 = 6\):
\[ x = 6 \times 16 = 96 \]
The area of \(\triangle DEF\) is 96 cm\(^2\).


Step 4: Final Answer:

The area of \(\triangle DEF\) is 96 cm\(^2\).
Quick Tip: When setting up the ratio, make sure to be consistent. If you put the area of triangle ABC in the numerator, you must also put its corresponding side (BC) in the numerator of the side ratio.

Step 1: Understanding the Concept:

The problem describes a right-angled triangle, since \(\angle A = 90^\circ\). We can use the Pythagorean theorem to find the length of the unknown side.


Step 2: Key Formula or Approach:

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
\[ (Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2 \]
In \(\triangle ABC\) with \(\angle A = 90^\circ\), the hypotenuse is BC. So, \(BC^2 = AB^2 + AC^2\).


Step 3: Detailed Explanation:

We are given:

Hypotenuse, BC = 13 cm

One side, AB = 12 cm

We need to find the other side, AC.

Substitute the values into the theorem:
\[ 13^2 = 12^2 + AC^2 \] \[ 169 = 144 + AC^2 \] \[ AC^2 = 169 - 144 \] \[ AC^2 = 25 \] \[ AC = \sqrt{25} = 5 \]
The length of AC is 5 cm.


Step 4: Final Answer:

The value of AC is 5 cm.
Quick Tip: This is a classic example of a Pythagorean triple (5, 12, 13). If you recognize this triple, you can find the answer instantly without any calculation.

Step 1: Understanding the Concept:

If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar by the Angle-Angle (AA) similarity criterion. A consequence of this is that their third angles must also be equal.


Step 2: Key Formula or Approach:

The sum of angles in any triangle is 180°.

In \(\triangle DEF\), \(\angle D + \angle E + \angle F = 180^\circ\).

In \(\triangle PQR\), \(\angle P + \angle Q + \angle R = 180^\circ\).


Step 3: Detailed Explanation:

We are given:
\(\angle D = \angle Q\) (Equation 1)
\(\angle E = \angle R\) (Equation 2)

From the sum of angles in \(\triangle DEF\), we can write \(\angle F = 180^\circ - \angle D - \angle E\).

From the sum of angles in \(\triangle PQR\), we can write \(\angle P = 180^\circ - \angle Q - \angle R\).

Now substitute the given equalities (Equations 1 and 2) into the equation for \(\angle P\):
\[ \angle P = 180^\circ - (\angle D) - (\angle E) \]
Comparing this with the equation for \(\angle F\), we see that:
\[ \angle F = \angle P \]

Step 4: Final Answer:

The correct statement is \(\angle F = \angle P\).
Quick Tip: When two pairs of angles are given as equal, the third pair of angles must also be equal. Simply match the remaining angle from the first triangle (\(\angle F\)) with the remaining angle from the second triangle (\(\angle P\)).

Step 1: Understanding the Concept:

The condition \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{DF}\) means that the sides of \(\triangle ABC\) are proportional to the corresponding sides of \(\triangle DEF\). By the Side-Side-Side (SSS) similarity criterion, this implies that \(\triangle ABC \sim \triangle DEF\).


Step 2: Key Formula or Approach:

In similar triangles, corresponding angles are equal. The correspondence is given by the order of vertices in the ratio.

A corresponds to D, B corresponds to E, and C corresponds to F.

Therefore, \(\angle A = \angle D\), \(\angle B = \angle E\), and \(\angle C = \angle F\).

The sum of angles in a triangle is 180°.


Step 3: Detailed Explanation:

We need to find the measure of \(\angle F\). Since \(\triangle ABC \sim \triangle DEF\), we have \(\angle F = \angle C\).

We can find \(\angle C\) using the angle sum property in \(\triangle ABC\).

We are given \(\angle A = 40^\circ\) and \(\angle B = 80^\circ\).
\[ \angle A + \angle B + \angle C = 180^\circ \] \[ 40^\circ + 80^\circ + \angle C = 180^\circ \] \[ 120^\circ + \angle C = 180^\circ \] \[ \angle C = 180^\circ - 120^\circ = 60^\circ \]
Since \(\angle F = \angle C\), the measure of \(\angle F\) is 60°.


Step 4: Final Answer:

The measure of \(\angle F\) is 60°.
Quick Tip: First, establish the similarity and identify the corresponding angles correctly. Then, use the angle sum property in the triangle for which you have two angles to find the third one.

Step 1: Understanding the Concept:

This question asks for the number of common tangents that can be drawn to two circles that intersect each other at two distinct points.


Step 2: Detailed Explanation:

Let's visualize the situation: When two circles intersect at two different points, they overlap partially.

- We can draw two tangents that are external to both circles. These are called direct common tangents.

- It is not possible to draw any transverse (or indirect) common tangents that would cross the space between the circles, because the circles themselves occupy that space.

Therefore, there are exactly two common tangents.


Step 3: Final Answer:

The number of common tangents of two intersecting circles is 2.
Quick Tip: Remember the number of common tangents for different circle positions: - Circles are separate: 4 common tangents (2 direct, 2 transverse) - Circles touch externally: 3 common tangents (2 direct, 1 transverse) - Circles intersect: 2 common tangents (both direct) - Circles touch internally: 1 common tangent - One circle inside another: 0 common tangents

Step 1: Understanding the Concept:

The volumes and surface areas of spheres are related to their radii. If we can find the ratio of the radii from the ratio of the volumes, we can then find the ratio of the surface areas.


Step 2: Key Formula or Approach:

Let the radii of the two spheres be \(r_1\) and \(r_2\).

Ratio of Volumes: \(\frac{V_1}{V_2} = \frac{\frac{4}{3}\pi r_1^3}{\frac{4}{3}\pi r_2^3} = \left(\frac{r_1}{r_2}\right)^3\)

Ratio of Surface Areas: \(\frac{A_1}{A_2} = \frac{4\pi r_1^2}{4\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2\)


Step 3: Detailed Explanation:

We are given the ratio of volumes:
\[ \frac{V_1}{V_2} = \frac{64}{125} \]
From this, we find the ratio of the radii:
\[ \left(\frac{r_1}{r_2}\right)^3 = \frac{64}{125} \] \[ \frac{r_1}{r_2} = \sqrt[3]{\frac{64}{125}} = \frac{4}{5} \]
Now, we can find the ratio of the surface areas:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25} \]
The ratio of their surface areas is 16 : 25.


Step 4: Final Answer:

The ratio of their surface areas is 16 : 25.
Quick Tip: The relationship is straightforward: to get from the volume ratio to the side/radius ratio, take the cube root. To get from the side/radius ratio to the area ratio, square it. So, cube root then square: \((\sqrt[3]{64})^2 : (\sqrt[3]{125})^2 = 4^2 : 5^2 = 16:25\).

Step 1: Understanding the Concept:

We need to find the ratio of the volumes of two cylinders given the ratios of their radii and heights.


Step 2: Key Formula or Approach:

The formula for the volume of a cylinder is \(V = \pi r^2 h\).

The ratio of the volumes of two cylinders will be:
\[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right) \]

Step 3: Detailed Explanation:

We are given the ratios:

Ratio of radii: \(\frac{r_1}{r_2} = \frac{4}{5}\)

Ratio of heights: \(\frac{h_1}{h_2} = \frac{6}{7}\)

Now, substitute these ratios into the volume ratio formula:
\[ \frac{V_1}{V_2} = \left(\frac{4}{5}\right)^2 \times \left(\frac{6}{7}\right) \] \[ \frac{V_1}{V_2} = \frac{16}{25} \times \frac{6}{7} \] \[ \frac{V_1}{V_2} = \frac{16 \times 6}{25 \times 7} = \frac{96}{175} \]
The ratio of the volumes is 96 : 175.


Step 4: Final Answer:

The ratio of their volumes is 96 : 175.
Quick Tip: Remember that the volume of a cylinder depends on the square of the radius but only on the first power of the height. So, you need to square the ratio of the radii before multiplying by the ratio of the heights.

Step 1: Understanding the Concept:

The total surface area (TSA) of a hemisphere is the sum of its curved surface area (CSA) and the area of its flat circular base.


Step 2: Key Formula or Approach:

1. The curved surface area of a hemisphere is half the surface area of a full sphere: \(CSA_{hemisphere} = \frac{1}{2} (4\pi R^2) = 2\pi R^2\).

2. The area of the circular base is given by the formula for the area of a circle: \(A_{base} = \pi R^2\).

3. The total surface area is the sum of these two areas: \(TSA = CSA + A_{base}\).


Step 3: Detailed Explanation:
\[ TSA_{hemisphere} = (2\pi R^2) + (\pi R^2) \] \[ TSA_{hemisphere} = 3\pi R^2 \]

Step 4: Final Answer:

The total surface area of a hemisphere of radius R is \(3\pi R^2\).
Quick Tip: Don't confuse the total surface area (\(3\pi R^2\)) with the curved surface area (\(2\pi R^2\)). The "total" area includes the flat circular top.

Step 1: Understanding the Concept:

The curved surface area (CSA) of a cone is related to its radius (\(r\)) and slant height (\(l\)) by a standard formula. We can use this formula to find the slant height when the area and radius are known.


Step 2: Key Formula or Approach:

The formula for the curved surface area of a cone is:
\[ CSA = \pi r l \]
We need to solve for \(l\).


Step 3: Detailed Explanation:

We are given:

CSA = 880 cm\(^2\)

Radius, \(r = 14\) cm

Use the value \(\pi \approx \frac{22}{7}\).

Substitute the given values into the formula:
\[ 880 = \frac{22}{7} \times 14 \times l \]
Simplify the right side:
\[ 880 = 22 \times 2 \times l \] \[ 880 = 44 \times l \]
Now, solve for \(l\):
\[ l = \frac{880}{44} \] \[ l = 20 \]
The slant height is 20 cm.


Step 4: Final Answer:

The slant height is 20 cm.
Quick Tip: When the radius or diameter is a multiple of 7, it's almost always easier to use \(\pi = 22/7\) as it will lead to cancellations and simpler arithmetic.

Step 1: Understanding the Concept:

The diagonal of a cube is the line segment connecting two opposite vertices. Its length is related to the length of the cube's edge by a specific formula.


Step 2: Key Formula or Approach:

The formula for the length of the diagonal (\(d\)) of a cube with edge length (\(a\)) is:
\[ d = a\sqrt{3} \]
We are given \(d\) and need to solve for \(a\).


Step 3: Detailed Explanation:

We are given that the diagonal, \(d = 2\sqrt{3}\) cm.

Substitute this into the formula:
\[ 2\sqrt{3} = a\sqrt{3} \]
To find \(a\), divide both sides by \(\sqrt{3}\):
\[ a = \frac{2\sqrt{3}}{\sqrt{3}} \] \[ a = 2 \]
The length of the edge is 2 cm.


Step 4: Final Answer:

The length of its edge is 2 cm.
Quick Tip: Remember the formula for the diagonal of a cube: \(d = a\sqrt{3}\). Don't confuse it with the diagonal of a face of the cube, which is \(a\sqrt{2}\).

Step 1: Understanding the Concept:

The total surface area (TSA) of a cube depends on the square of its edge length. We need to see how the TSA changes when the edge length is doubled.


Step 2: Key Formula or Approach:

The formula for the total surface area of a cube with edge length \(a\) is:
\[ TSA = 6a^2 \]

Step 3: Detailed Explanation:

Let the original edge length be \(a\). The original TSA is \(TSA_{original} = 6a^2\).

Now, the edge is doubled. The new edge length is \(a_{new} = 2a\).

Let's calculate the new total surface area with this new edge:
\[ TSA_{new} = 6(a_{new})^2 = 6(2a)^2 \] \[ TSA_{new} = 6(4a^2) = 4 \times (6a^2) \]
Since \(6a^2 = TSA_{original}\), we have:
\[ TSA_{new} = 4 \times TSA_{original} \]
This means the new total surface area is four times the original total surface area.


Step 4: Final Answer:

The total surface area will become Four times the previous total surface area.
Quick Tip: When a linear dimension (like edge, radius) is scaled by a factor of \(k\), the area is scaled by a factor of \(k^2\), and the volume is scaled by a factor of \(k^3\). Here, the edge is doubled (\(k=2\)), so the area becomes \(2^2 = 4\) times larger.

Step 1: Understanding the Concept:

We need to find the ratio of the total surface area (TSA) of a sphere to the TSA of a hemisphere, given that they both have the same radius, \(R\).


Step 2: Key Formula or Approach:

1. The total surface area of a sphere of radius \(R\) is \(TSA_{sphere} = 4\pi R^2\).

2. The total surface area of a hemisphere of radius \(R\) is the sum of its curved area (\(2\pi R^2\)) and its circular base area (\(\pi R^2\)), which is \(TSA_{hemisphere} = 3\pi R^2\).

3. We need to find the ratio \(\frac{TSA_{sphere}}{TSA_{hemisphere}}\).


Step 3: Detailed Explanation:
\[ Ratio = \frac{TSA_{sphere}}{TSA_{hemisphere}} = \frac{4\pi R^2}{3\pi R^2} \]
Cancel the common terms \(\pi R^2\) from the numerator and the denominator:
\[ Ratio = \frac{4}{3} \]
So, the ratio is 4 : 3.


Step 4: Final Answer:

The ratio is 4 : 3.
Quick Tip: This is a direct comparison of formulas. Make sure you use the total surface area of the hemisphere (\(3\pi R^2\)), not just its curved surface area (\(2\pi R^2\)).

Step 1: Understanding the Concept:

We are given the curved surface area (CSA) of a hemisphere and need to find its radius. We can do this by using the formula for the CSA of a hemisphere and solving for the radius, \(r\).


Step 2: Key Formula or Approach:

The formula for the curved surface area of a hemisphere is:
\[ CSA = 2\pi r^2 \]
We will substitute the given values and solve for \(r\).


Step 3: Detailed Explanation:

We are given CSA = 1232 cm\(^2\). Use \(\pi \approx \frac{22}{7}\).
\[ 1232 = 2 \times \frac{22}{7} \times r^2 \] \[ 1232 = \frac{44}{7} \times r^2 \]
To isolate \(r^2\), multiply both sides by \(\frac{7}{44}\):
\[ r^2 = 1232 \times \frac{7}{44} \]
Let's simplify the division. We can divide 1232 by 44. (Hint: 1232 / 44 = 1232 / (4 * 11) = 308 / 11 = 28).
\[ r^2 = 28 \times 7 \] \[ r^2 = (4 \times 7) \times 7 = 4 \times 49 \]
Now, take the square root to find \(r\):
\[ r = \sqrt{4 \times 49} = \sqrt{4} \times \sqrt{49} = 2 \times 7 = 14 \]
The radius is 14 cm.


Step 4: Final Answer:

The radius is 14 cm.
Quick Tip: When solving for \(r^2\), try to break down the numbers into factors that are perfect squares before multiplying them out. This makes taking the square root much easier.

Step 1: Understanding the Concept:

This problem requires manipulating a given trigonometric equation to find the value of another trigonometric expression. The key is to use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).


Step 2: Key Formula or Approach:

1. Start with the given equation: \(\cos \theta + \cos^2 \theta = 1\).

2. Rearrange it to express one function in terms of another.

3. Substitute this relationship into the expression we need to evaluate: \(\sin^2 \theta + \sin^4 \theta\).


Step 3: Detailed Explanation:

From the given equation:
\[ \cos \theta + \cos^2 \theta = 1 \]
Rearrange it by subtracting \(\cos^2 \theta\) from both sides:
\[ \cos \theta = 1 - \cos^2 \theta \]
Using the Pythagorean identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can substitute \(\sin^2 \theta\) into the equation:
\[ \cos \theta = \sin^2 \theta \quad (*) \]
Now, let's look at the expression we need to find:
\[ \sin^2 \theta + \sin^4 \theta \]
This can be written as:
\[ \sin^2 \theta + (\sin^2 \theta)^2 \]
From our derived relationship (*), we know that \(\sin^2 \theta = \cos \theta\). Let's substitute this into the expression:
\[ = (\cos \theta) + (\cos \theta)^2 \] \[ = \cos \theta + \cos^2 \theta \]
From the original given equation, we know that \(\cos \theta + \cos^2 \theta = 1\).

Therefore, \(\sin^2 \theta + \sin^4 \theta = 1\).


Step 4: Final Answer:

The value of \(\sin^2 \theta + \sin^4 \theta\) is 1.
Quick Tip: The key to this type of problem is to use the given equation to find a substitution. Rearranging the given equation using the Pythagorean identity is the crucial first step.

Step 1: Understanding the Concept:

This problem involves simplifying a trigonometric expression using the Pythagorean identities.


Step 2: Key Formula or Approach:

We will use the two Pythagorean identities:

1. \(1 + \tan^2 A = \sec^2 A\)

2. \(1 + \cot^2 A = \csc^2 A\)

And the reciprocal identities \(\sec A = 1/\cos A\) and \(\csc A = 1/\sin A\).


Step 3: Detailed Explanation:

Start with the given expression:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} \]
Substitute the Pythagorean identities in the numerator and the denominator:
\[ = \frac{\sec^2 A}{\csc^2 A} \]
Now, use the reciprocal identities to express secant and cosecant in terms of sine and cosine:
\[ = \frac{1/\cos^2 A}{1/\sin^2 A} \]
To simplify the complex fraction, multiply the numerator by the reciprocal of the denominator:
\[ = \frac{1}{\cos^2 A} \times \frac{\sin^2 A}{1} = \frac{\sin^2 A}{\cos^2 A} \]
Using the identity \(\tan A = \sin A / \cos A\), we get:
\[ = \tan^2 A \]

Step 4: Final Answer:

The value of the expression is \(\tan^2 A\).
Quick Tip: A useful shortcut to remember is that \(\frac{\sec^2 A}{\csc^2 A} = \tan^2 A\). This can save a few steps in the simplification process.

Step 1: Understanding the Concept:

The roots of a quadratic equation \(ax^2 + bx + c = 0\) are real and equal if and only if its discriminant, \(D\), is equal to zero.


Step 2: Key Formula or Approach:

The discriminant is given by \(D = b^2 - 4ac\). We need to set \(D = 0\) and solve for \(k\).


Step 3: Detailed Explanation:

For the given equation \(kx^2 - 6x + 1 = 0\), we have:
\(a = k\), \(b = -6\), \(c = 1\).

Set the discriminant to zero for real and equal roots:
\[ D = b^2 - 4ac = 0 \] \[ (-6)^2 - 4(k)(1) = 0 \] \[ 36 - 4k = 0 \]
Now, solve for \(k\):
\[ 36 = 4k \] \[ k = \frac{36}{4} \] \[ k = 9 \]

Step 4: Final Answer:

The value of k for which the roots are real and equal is 9.
Quick Tip: Remember the conditions for the nature of roots based on the discriminant: - \(D > 0\): real and distinct roots - \(D = 0\): real and equal roots - \(D < 0\): no real roots This question specifically asks for the "real and equal" case, so \(D=0\) is the key.

Step 1: Understanding the Concept:

This question is based on the Factor Theorem. The Factor Theorem states that if \(x = a\) is a zero (or root) of a polynomial \(p(x)\), then \((x - a)\) is a factor of that polynomial.


Step 2: Key Formula or Approach:

According to the Factor Theorem, if \(a\) is a zero, then \((x-a)\) is a factor.


Step 3: Detailed Explanation:

We are given that one of the zeros of the polynomial \(p(x)\) is 2.

This means that when we set \(x = 2\), \(p(2) = 0\).

Using the Factor Theorem with \(a = 2\), the corresponding factor is \((x - a)\), which is \((x - 2)\).


Step 4: Final Answer:

The factor of \(p(x)\) is x - 2.
Quick Tip: The relationship is simple: if the zero is a positive number `a`, the factor is `(x - a)`. If the zero is a negative number `-a`, the factor is `(x - (-a))` which is `(x + a)`.

Step 1: Understanding the Concept:

For a general quadratic polynomial of the form \(Px^2 + Qx + R\), the product of its zeros is given by the ratio of the constant term to the coefficient of the \(x^2\) term.


Step 2: Key Formula or Approach:

For a polynomial \(Px^2 + Qx + R\), the product of zeros \(\alpha \beta = \frac{R}{P}\).


Step 3: Detailed Explanation:

The given polynomial is \(cx^2 + ax + b\).

Here, the coefficient of the \(x^2\) term is \(P = c\).

The coefficient of the \(x\) term is \(Q = a\).

The constant term is \(R = b\).

Applying the formula for the product of zeros:
\[ \alpha \beta = \frac{constant term}{coefficient of x^2} = \frac{b}{c} \]

Step 4: Final Answer:

The value of \(\alpha . \beta\) is \(\frac{b}{c}\).
Quick Tip: Be careful with the standard form. The question uses \(cx^2+ax+b\) instead of the usual \(ax^2+bx+c\). Always identify the coefficients based on the powers of x, not on the letters used.

Step 1: Understanding the Concept:

A quadratic equation is a polynomial equation of the second degree, meaning the highest exponent of the variable is 2. The standard form is \(ax^2 + bx + c = 0\), where \(a \neq 0\). We need to simplify each option to see which one fits this form.


Step 2: Detailed Explanation:

(A) \((x + 3)(x - 3) = x^2 - 4x^3 \Rightarrow x^2 - 9 = x^2 - 4x^3 \Rightarrow 4x^3 - 9 = 0\). The highest power is 3, so this is a cubic equation.


(B) \((x+3)^2 = 4(x+4) \Rightarrow x^2 + 6x + 9 = 4x + 16 \Rightarrow x^2 + 2x - 7 = 0\). The highest power is 2, so this is a quadratic equation.


(C) \((2x-2)^2 = 4x^2 + 7 \Rightarrow 4x^2 - 8x + 4 = 4x^2 + 7 \Rightarrow -8x - 3 = 0\). The \(x^2\) terms cancel out, leaving a linear equation.


(D) \(4x + \frac{1}{4x} = 4x \Rightarrow \frac{1}{4x} = 0\). This equation has no solution and is not a polynomial equation. If we multiply by \(4x\), we get \(1 = 0\), which is a contradiction.


Step 3: Final Answer:

The equation that simplifies to a quadratic form is \((x+3)^2 = 4(x+4)\).
Quick Tip: To check if an equation is quadratic, expand all terms and move them to one side. If the highest power of the variable that remains is 2, it is a quadratic equation.

Step 1: Understanding the Concept:

A quadratic equation has the standard form \(ax^2 + bx + c = 0\) with \(a \neq 0\). We need to simplify each equation and find the one where the highest power of \(x\) is not 2.


Step 2: Detailed Explanation:

(A) \(5x - x^2 = x^2 + 3 \Rightarrow 2x^2 - 5x + 3 = 0\). This is a quadratic equation.


(B) \(x^3 - x^2 = (x-1)^3 \Rightarrow x^3 - x^2 = x^3 - 3x^2(1) + 3x(1)^2 - 1^3 \Rightarrow x^3 - x^2 = x^3 - 3x^2 + 3x - 1\). The \(x^3\) terms cancel. \(\Rightarrow 2x^2 - 3x + 1 = 0\). This is a quadratic equation.


(C) \((x+3)^2 = 3(x^2 - 5) \Rightarrow x^2 + 6x + 9 = 3x^2 - 15 \Rightarrow 2x^2 - 6x - 24 = 0\). This is a quadratic equation.


(D) \((\sqrt{2}x + 3)^2 = 2x^2 + 5 \Rightarrow (\sqrt{2}x)^2 + 2(\sqrt{2}x)(3) + 3^2 = 2x^2 + 5 \Rightarrow 2x^2 + 6\sqrt{2}x + 9 = 2x^2 + 5\). The \(2x^2\) terms cancel out, leaving \(6\sqrt{2}x + 4 = 0\). This is a linear equation, not a quadratic equation.


Step 3: Final Answer:

The equation that is not a quadratic equation is \((\sqrt{2}x + 3)^2 = 2x^2 + 5\).
Quick Tip: Always fully expand and simplify the expressions on both sides of the equation. An equation might look quadratic at first, but the \(x^2\) terms might cancel out.

Step 1: Understanding the Concept:

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is a value that determines the nature of its roots.


Step 2: Key Formula or Approach:

The formula for the discriminant (\(D\)) is:
\[ D = b^2 - 4ac \]

Step 3: Detailed Explanation:

For the given equation \(2x^2 - 7x + 6 = 0\), we identify the coefficients:
\(a = 2\), \(b = -7\), \(c = 6\).

Now, substitute these values into the discriminant formula:
\[ D = (-7)^2 - 4(2)(6) \] \[ D = 49 - 48 \] \[ D = 1 \]

Step 4: Final Answer:

The discriminant of the quadratic equation is 1.
Quick Tip: Be very careful with signs, especially when the coefficient 'b' is negative. Remember that \((-b)^2\) is always positive.

Step 1: Understanding the Concept:

The equation \(x = 2\) represents a vertical line on the Cartesian plane. Every point on this line has an x-coordinate of exactly 2, while the y-coordinate can be any real number.


Step 2: Detailed Explanation:

We need to check which of the given points satisfy the condition that its x-coordinate is 2.

(A) (2, 0): The x-coordinate is 2. This point lies on the line.

(B) (2, 1): The x-coordinate is 2. This point lies on the line.

(C) (2, 2): The x-coordinate is 2. This point lies on the line.

Since all the points (A), (B), and (C) have an x-coordinate of 2, they all lie on the graph of \(x = 2\).


Step 3: Final Answer:

The correct option is all of these.
Quick Tip: Remember that \(x = k\) is the equation of a vertical line passing through the x-axis at \(k\), and \(y = k\) is the equation of a horizontal line passing through the y-axis at \(k\).

Step 1: Understanding the Concept:

If three terms \(a, b, c\) are in an Arithmetic Progression (A.P.), then the difference between consecutive terms is constant. This means \(b - a = c - b\), which can be rearranged to \(2b = a + c\). The middle term is the arithmetic mean of the first and third terms.


Step 2: Key Formula or Approach:

We will use the property \(2 \times (middle term) = (first term) + (third term)\).


Step 3: Detailed Explanation:

The given terms are:

First term: \(a = P + 1\)

Middle term: \(b = 2P + 1\)

Third term: \(c = 4P - 1\)

Apply the A.P. property:
\[ 2(2P + 1) = (P + 1) + (4P - 1) \]
Expand both sides:
\[ 4P + 2 = P + 4P + 1 - 1 \] \[ 4P + 2 = 5P \]
Now, solve for P:
\[ 2 = 5P - 4P \] \[ P = 2 \]

Step 4: Final Answer:

The value of P is 2.
Quick Tip: To verify your answer, substitute P=2 back into the terms: (2+1), 2(2)+1, 4(2)-1 -> 3, 5, 7. The sequence is 3, 5, 7, which is an A.P. with a common difference of 2. This confirms the answer is correct.

Step 1: Understanding the Concept:

The common difference (\(d\)) of an Arithmetic Progression (A.P.) is the constant difference between any two consecutive terms.


Step 2: Key Formula or Approach:
\(d = a_2 - a_1\) or \(d = a_3 - a_2\), etc.


Step 3: Detailed Explanation:

The given A.P. is 1, 5, 9, ...

The first term (\(a_1\)) is 1.

The second term (\(a_2\)) is 5.

The third term (\(a_3\)) is 9.

Calculate the difference:
\[ d = a_2 - a_1 = 5 - 1 = 4 \]
To be sure, let's check the next pair:
\[ d = a_3 - a_2 = 9 - 5 = 4 \]
The difference is constant.


Step 4: Final Answer:

The common difference is 4.
Quick Tip: Finding the common difference is the most fundamental step in solving problems related to Arithmetic Progressions. Simply subtract any term from the term that follows it.

Step 1: Understanding the Concept:

We need to find the position (\(n\)) of a specific term in a given Arithmetic Progression.


Step 2: Key Formula or Approach:

We will use the formula for the nth term of an A.P.:
\[ a_n = a + (n-1)d \]
Here, we are given \(a_n\), \(a\), and we can find \(d\). We need to solve for \(n\).


Step 3: Detailed Explanation:

From the A.P. 5, 8, 11, 14, ...:

The first term, \(a = 5\).

The common difference, \(d = 8 - 5 = 3\).

The term we are looking for, \(a_n = 38\).

Substitute these values into the formula:
\[ 38 = 5 + (n-1)3 \]
Now, solve for \(n\):
\[ 38 - 5 = (n-1)3 \] \[ 33 = (n-1)3 \] \[ \frac{33}{3} = n-1 \] \[ 11 = n - 1 \] \[ n = 11 + 1 = 12 \]
So, 38 is the 12th term of the A.P.


Step 4: Final Answer:

38 is the 12th term.
Quick Tip: You can also use a rearranged version of the formula to find 'n' directly: \(n = \frac{a_n - a}{d} + 1\). In this case, \(n = \frac{38 - 5}{3} + 1 = \frac{33}{3} + 1 = 11 + 1 = 12\).

Step 1: Understanding the Concept:

The length (or size or width) of a class interval is the difference between the upper class limit and the lower class limit.


Step 2: Key Formula or Approach:

Class Length = Upper Limit - Lower Limit


Step 3: Detailed Explanation:

Let's calculate the length for each given class interval:

For the class 2 - 5:

Length = 5 - 2 = 3

For the class 5 - 8:

Length = 8 - 5 = 3

For the class 8 - 11:

Length = 11 - 8 = 3

The length is constant for all the given class intervals.


Step 4: Final Answer:

The length of the class intervals is 3.
Quick Tip: The length of a class interval is simply the size of the range it covers. Be careful not to confuse it with the midpoint of the class.

Step 1: Understanding the Concept:

Consecutive odd numbers differ by 2. We can represent them algebraically, set up an equation for their mean, solve for the variable, and then find the largest number.


Step 2: Key Formula or Approach:

Let the four consecutive odd numbers be \(x\), \(x+2\), \(x+4\), and \(x+6\).

The mean (average) is the sum of the numbers divided by the count of the numbers.
\[ Mean = \frac{x + (x+2) + (x+4) + (x+6)}{4} \]

Step 3: Detailed Explanation:

We are given that the mean is 6.
\[ 6 = \frac{x + x+2 + x+4 + x+6}{4} \] \[ 6 = \frac{4x + 12}{4} \]
Multiply both sides by 4:
\[ 24 = 4x + 12 \]
Subtract 12 from both sides:
\[ 12 = 4x \]
Solve for \(x\):
\[ x = 3 \]
The numbers are:

First number: \(x = 3\)

Second number: \(x+2 = 5\)

Third number: \(x+4 = 7\)

Fourth (largest) number: \(x+6 = 9\)


Step 4: Final Answer:

The largest number is 9.
Quick Tip: For an even number of consecutive terms in an arithmetic progression, the mean is the average of the two middle terms. The four numbers are 3, 5, 7, 9. The middle two are 5 and 7. Their mean is \((5+7)/2 = 6\), which matches the given information. The largest is 9.

Step 1: Understanding the Concept:

We need to find the mean (average) of the first 6 even natural numbers. First, we must identify these numbers.


Step 2: Key Formula or Approach:

1. List the first 6 even natural numbers.

2. Calculate their sum.

3. Divide the sum by the count of the numbers (which is 6).
\[ Mean = \frac{Sum of numbers}{Count of numbers} \]

Step 3: Detailed Explanation:

The first 6 even natural numbers are: 2, 4, 6, 8, 10, 12.

Calculate their sum:
\[ Sum = 2 + 4 + 6 + 8 + 10 + 12 = 42 \]
Calculate the mean:
\[ Mean = \frac{42}{6} = 7 \]

Step 4: Final Answer:

The mean of the first 6 even natural numbers is 7.
Quick Tip: The mean of the first \(n\) even natural numbers is simply \(n+1\). Here, \(n=6\), so the mean is \(6+1=7\). This shortcut works because these numbers form an arithmetic progression.

Step 1: Understanding the Concept:

This question asks for one of the fundamental Pythagorean identities in trigonometry.


Step 2: Key Formula or Approach:

The three Pythagorean identities are:

1. \(\sin^2 \theta + \cos^2 \theta = 1\)

2. \(1 + \tan^2 \theta = \sec^2 \theta\)

3. \(1 + \cot^2 \theta = \csc^2 \theta\)


Step 3: Detailed Explanation:

The expression given is \(1 + \cot^2 \theta\). By direct application of the third Pythagorean identity, this is equal to \(\csc^2 \theta\).


Step 4: Final Answer:

The expression \(1 + \cot^2 \theta\) is equal to \(\csc^2 \theta\).
Quick Tip: Memorizing the three Pythagorean identities is absolutely essential for trigonometry. They are used frequently for simplifying expressions and proving other identities.

Step 1: Understanding the Concept:

The mode of a set of data is the value that appears most frequently.


Step 2: Key Formula or Approach:

1. Tally the frequency of each number in the dataset.

2. Identify the number with the highest frequency.


Step 3: Detailed Explanation:

The given dataset is: 8, 7, 9, 3, 9, 5, 4, 5, 7, 5.

Let's count the occurrences of each number:

- 3: appears 1 time

- 4: appears 1 time

- 5: appears 3 times

- 6: appears 0 times

- 7: appears 2 times

- 8: appears 1 time

- 9: appears 2 times

The number 5 appears most often (3 times).


Step 4: Final Answer:

The mode of the dataset is 5.
Quick Tip: To avoid errors, it's helpful to first write the numbers in ascending order: 3, 4, 5, 5, 5, 7, 7, 8, 9, 9. This makes it much easier to see which number occurs most frequently.

Step 1: Understanding the Concept:

This question deals with complementary events in probability. \(E'\) (or \(\bar{E}\)) represents the complement of event E, meaning the event that E does not occur. The sum of the probabilities of an event and its complement is always 1.


Step 2: Key Formula or Approach:

The formula for the probability of a complementary event is:
\[ P(E') = 1 - P(E) \]

Step 3: Detailed Explanation:

We are given the probability of event E:
\[ P(E) = 0.02 \]
Using the formula, we can find the probability of its complement, \(E'\):
\[ P(E') = 1 - 0.02 \] \[ P(E') = 0.98 \]

Step 4: Final Answer:

The value of \(P(E')\) is 0.98.
Quick Tip: An event and its complement cover all possible outcomes. Think of it as a percentage: if there is a 2% chance of something happening, there is a 98% (100% - 2%) chance of it not happening.

Step 1: Understanding the Concept:

We need to find the probability of a specific event when two dice are thrown. The total number of possible outcomes is the product of the number of faces on each die. The number of favorable outcomes is the count of outcomes that satisfy the given condition.


Step 2: Key Formula or Approach:
\[ Probability = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]
- Total outcomes when throwing two dice = \(6 \times 6 = 36\).

- We need to find the outcomes where the difference of the numbers is zero.


Step 3: Detailed Explanation:

The difference between the numbers on the two dice is zero only if the numbers are the same. These outcomes are called doublets.

The favorable outcomes are:

(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)

The number of favorable outcomes is 6.

The total number of possible outcomes is 36.

Now, calculate the probability:
\[ P(difference is zero) = \frac{6}{36} = \frac{1}{6} \]

Step 4: Final Answer:

The probability is \(\frac{1}{6}\).
Quick Tip: When working with two dice, the total number of outcomes is always 36. Memorizing the number of outcomes for common events (like getting a specific sum, a doublet, etc.) can save time.

Step 1: Understanding the Concept:

This problem asks for the probability of a specific outcome when two coins are tossed simultaneously. We need to identify all possible outcomes and then find the ones that match the desired event.


Step 2: Key Formula or Approach:
\[ Probability = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]

Step 3: Detailed Explanation:

When two coins are tossed, the set of all possible outcomes (the sample space) is:

\{HH, HT, TH, TT\

where H = Heads and T = Tails.

The total number of possible outcomes is 4.

The event we are interested in is "getting heads on both the coins."

The favorable outcome for this event is just one: \{HH\.

The number of favorable outcomes is 1.

Now, calculate the probability:
\[ P(both heads) = \frac{1}{4} \]

Step 4: Final Answer:

The probability is \(\frac{1}{4}\).
Quick Tip: For independent events like coin tosses, you can also multiply their individual probabilities. The probability of getting a head on one coin is 1/2. The probability of getting heads on both is \(1/2 \times 1/2 = 1/4\).

Step 1: Understanding the Concept:

We need to find the probability of selecting one of two specific months from a total of 12 months in a year.


Step 2: Key Formula or Approach:
\[ Probability = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]

Step 3: Detailed Explanation:

The total number of possible outcomes is the total number of months in a year, which is 12.

The favorable outcomes are the months "June" or "September".

The number of favorable outcomes is 2.

Now, calculate the probability:
\[ P(June or September) = \frac{2}{12} = \frac{1}{6} \]

Step 4: Final Answer:

The probability is \(\frac{1}{6}\).
Quick Tip: In "or" probability problems where the events are mutually exclusive (like selecting June or September - you can't select both at once), you can find the probability of each and add them: P(June) + P(September) = 1/12 + 1/12 = 2/12 = 1/6.

Step 1: Understanding the Concept:

This problem asks for the probability of getting one of two possible outcomes when a single standard six-sided die is thrown.


Step 2: Key Formula or Approach:
\[ Probability = \frac{Number of Favorable Outcomes}{Total Number of Possible Outcomes} \]

Step 3: Detailed Explanation:

When a single die is thrown, the total number of possible outcomes is 6. The sample space is \{1, 2, 3, 4, 5, 6\.

The favorable outcomes are "getting a number 4 or 5".

The numbers in the sample space that satisfy this condition are 4 and 5.

The number of favorable outcomes is 2.

Now, calculate the probability:
\[ P(4 or 5) = \frac{2}{6} = \frac{1}{3} \]

Step 4: Final Answer:

The probability is \(\frac{1}{3}\).
Quick Tip: This is another "or" probability problem with mutually exclusive events. P(4 or 5) = P(4) + P(5) = 1/6 + 1/6 = 2/6 = 1/3.

Step 1: Understanding the Concept:

A rational number (a fraction) has a terminating decimal expansion if and only if its denominator, in the simplest form, can be expressed as a product of powers of 2 and 5 only (i.e., in the form \(2^n 5^m\), where n and m are non-negative integers).


Step 2: Detailed Explanation:

Let's analyze each option after simplifying the fraction:

(A) \(\frac{14}{2^0 \times 3^2} = \frac{14}{1 \times 9} = \frac{14}{9}\). The denominator has a factor of 3. Not terminating.


(B) \(\frac{9}{5^1 \times 7^2} = \frac{9}{5 \times 49}\). The denominator has a factor of 7. Not terminating.


(C) \(\frac{8}{2^2 \times 3^2} = \frac{8}{4 \times 9} = \frac{2}{9}\). The denominator has a factor of 3. Not terminating.


(D) \(\frac{15}{2^2 \times 5^3} = \frac{3 \times 5}{2^2 \times 5^3} = \frac{3}{2^2 \times 5^2}\). The denominator is in the form \(2^n 5^m\). Therefore, this fraction has a terminating decimal expansion.


Step 3: Final Answer:

The fraction with a terminating decimal expansion is \(\frac{15}{2^2 \times 5^3}\).
Quick Tip: Always simplify the fraction first before checking the prime factors of the denominator. Sometimes, prime factors other than 2 or 5 in the denominator might cancel out with factors in the numerator.

Step 1: Understanding the Concept:

To express a decimal number in the form of a rational number \(\frac{p}{q}\), where the denominator \(q\) is in the form of \(2^n \times 5^m\), we first convert the decimal to a fraction and then simplify it by finding the prime factorization of the numerator and the denominator.


Step 2: Detailed Explanation:

First, convert the decimal 0.505 into a fraction.
\[ 0.505 = \frac{505}{1000} \]
Now, find the prime factorization of the numerator and the denominator.

The numerator is 505.
\[ 505 = 5 \times 101 \]
The denominator is 1000.
\[ 1000 = 10^3 = (2 \times 5)^3 = 2^3 \times 5^3 \]
Now, write the fraction with its prime factors:
\[ \frac{505}{1000} = \frac{5 \times 101}{2^3 \times 5^3} \]
Cancel out the common factor of 5 from the numerator and the denominator.
\[ = \frac{101}{2^3 \times 5^{(3-1)}} = \frac{101}{2^3 \times 5^2} \]

Step 3: Final Answer:

Thus, 0.505 can be written as \(\frac{101}{2^3 \times 5^2}\). This matches option (D).
Quick Tip: For terminating decimals, the number of digits after the decimal point gives the power of 10 in the denominator. For example, 0.505 has 3 digits after the decimal, so the denominator is \(10^3 = 1000\).

Step 1: Understanding the Concept:

This question is based on Euclid's Division Algorithm. The algorithm states that for any two positive integers 'a' (dividend) and 'b' (divisor), there exist unique integers 'q' (quotient) and 'r' (remainder) such that a = bq + r, where 0 \(\le\) r \(<\) b.


Step 2: Key Formula or Approach:

The formula to use is the division algorithm itself:
\[ a = bq + r \]

Step 3: Detailed Explanation:

We are given the values:

Divisor, \(b = 4\)

Quotient, \(q = 5\)

Remainder, \(r = 1\)

We need to find the dividend, 'a'.

Substitute the given values into the formula:
\[ a = (4 \times 5) + 1 \] \[ a = 20 + 1 \] \[ a = 21 \]

Step 4: Final Answer:

The value of 'a' is 21. This corresponds to option (B).
Quick Tip: Always remember the components of the division algorithm: Dividend = (Divisor \(\times\) Quotient) + Remainder. This is a fundamental concept in number theory.

Step 1: Understanding the Concept:

The zeroes of a polynomial P(x) are the values of x for which P(x) = 0. For a quadratic polynomial, we can find the zeroes by factoring the polynomial, using the quadratic formula, or by splitting the middle term.


Step 2: Key Formula or Approach:

We will find the zeroes by setting the polynomial equal to zero and solving for x. Let's use the factorization method by splitting the middle term.

The polynomial is \(P(x) = 2x^2 - 4x - 6\).

Set \(P(x) = 0\):
\[ 2x^2 - 4x - 6 = 0 \]

Step 3: Detailed Explanation:

First, we can simplify the equation by dividing the entire equation by 2, as it is a common factor.
\[ x^2 - 2x - 3 = 0 \]
Now, we need to find two numbers that multiply to -3 and add up to -2. These numbers are -3 and 1.

Split the middle term:
\[ x^2 - 3x + 1x - 3 = 0 \]
Factor by grouping:
\[ x(x - 3) + 1(x - 3) = 0 \] \[ (x + 1)(x - 3) = 0 \]
Now, set each factor to zero to find the values of x.
\[ x + 1 = 0 \implies x = -1 \] \[ x - 3 = 0 \implies x = 3 \]

Step 4: Final Answer:

The zeroes of the polynomial are -1 and 3. This matches option (B).
Quick Tip: Always look for a common factor in the polynomial terms before factoring. It simplifies the numbers and makes the factorization process easier.

Step 1: Understanding the Concept:

The degree of a polynomial is the highest exponent of its variable. When two polynomials are multiplied, the degree of the resulting polynomial is the sum of the degrees of the individual polynomials.


Step 2: Key Formula or Approach:

Let \(P(x)\) and \(Q(x)\) be two polynomials.

Degree of \((P(x) \times Q(x))\) = Degree of \(P(x)\) + Degree of \(Q(x)\).


Step 3: Detailed Explanation:

Let's identify the two polynomials:

First polynomial: \(P(x) = x^3 + x^2 + 2x + 1\).

The highest power of x in \(P(x)\) is 3. So, the degree of \(P(x)\) is 3.


Second polynomial: \(Q(x) = x^2 + 2x + 1\).

The highest power of x in \(Q(x)\) is 2. So, the degree of \(Q(x)\) is 2.


The degree of the product polynomial \((x^3 + x^2 + 2x + 1)(x^2 + 2x + 1)\) is the sum of their degrees.
\[ Degree = Degree of P(x) + Degree of Q(x) = 3 + 2 = 5 \]

Step 4: Final Answer:

The degree of the resulting polynomial is 5. This corresponds to option (C).

Alternatively, we can see that the term with the highest degree in the product will be obtained by multiplying the terms with the highest degrees from each polynomial: \(x^3 \times x^2 = x^{3+2} = x^5\). The degree is 5.
Quick Tip: To find the degree of a product of polynomials, you don't need to perform the full multiplication. Simply add the degrees of the highest power terms of each polynomial.

Step 1: Understanding the Concept:

A polynomial is an algebraic expression in which the variables involved have only non-negative integer powers. The coefficients can be any real number.


Step 2: Detailed Explanation:

Let's analyze each option:

(A) \(x^2 - 7\): The powers of x are 2 and 0 (since \(7 = 7x^0\)). Both are non-negative integers. So, this is a polynomial.


(B) \(2x^2 + 7x + 6\): The powers of x are 2, 1, and 0. All are non-negative integers. So, this is a polynomial.


(C) \(\frac{1}{2}x^2 + \frac{1}{2}x + 4\): The powers of x are 2, 1, and 0. All are non-negative integers. The coefficients (\(\frac{1}{2}, \frac{1}{2}, 4\)) are real numbers. So, this is a polynomial.


(D) \(x + \frac{4}{x}\): This expression can be rewritten using exponents as \(x^1 + 4x^{-1}\). The power of x in the second term is -1, which is a negative integer. According to the definition of a polynomial, the powers of the variable must be non-negative. Therefore, this expression is not a polynomial.


Step 3: Final Answer:

The expression \(x + \frac{4}{x}\) is not a polynomial because it contains a term with a negative exponent. This matches option (D).
Quick Tip: A quick way to spot a non-polynomial is to look for variables in the denominator of a fraction or under a radical sign (e.g., \(\sqrt{x}\) which is \(x^{1/2}\)).

Step 1: Understanding the Concept:

If \(\alpha\) and \(\beta\) are the zeroes of a quadratic polynomial, then the polynomial can be written in the form \(k(x - \alpha)(x - \beta)\), where k is a non-zero constant. Another way is to use the sum and product of zeroes: the polynomial is \(k(x^2 - (sum of zeroes)x + (product of zeroes))\).


Step 2: Key Formula or Approach:

Using the sum and product of zeroes:

Polynomial = \(x^2 - (\alpha + \beta)x + (\alpha \beta)\) (assuming k=1).


Step 3: Detailed Explanation:

We are given the zeroes \(\alpha = 2\) and \(\beta = -2\).

Method 1: Using factors

The factors of the polynomial are \((x - \alpha)\) and \((x - \beta)\).

So the factors are \((x - 2)\) and \((x - (-2)) = (x + 2)\).

The polynomial is the product of these factors:
\[ (x - 2)(x + 2) \]
This is in the form of \((a - b)(a + b) = a^2 - b^2\).
\[ = x^2 - 2^2 = x^2 - 4 \]
Method 2: Using sum and product of zeroes

Sum of zeroes: \(\alpha + \beta = 2 + (-2) = 0\).

Product of zeroes: \(\alpha \beta = 2 \times (-2) = -4\).

The polynomial is \(x^2 - (sum)x + (product)\).
\[ = x^2 - (0)x + (-4) = x^2 - 4 \]

Step 4: Final Answer:

Both methods give the polynomial \(x^2 - 4\). This matches option (B).
Quick Tip: Remember the difference of squares identity: \((a-b)(a+b) = a^2 - b^2\). This is very useful for finding a polynomial when the zeroes are opposites of each other (like a and -a).

Step 1: Understanding the Concept:

For a general quadratic polynomial of the form \(ax^2 + bx + c\), the sum of its zeroes (\(\alpha + \beta\)) is given by the formula \(-\frac{b}{a}\), and the product of its zeroes (\(\alpha\beta\)) is given by \(\frac{c}{a}\).


Step 2: Key Formula or Approach:

Sum of zeroes, \(\alpha + \beta = -\frac{Coefficient of t}{Coefficient of t^2} = -\frac{b}{a}\).


Step 3: Detailed Explanation:

The given polynomial is \(t^2 + 7t + 10\).

Comparing this with the standard form \(at^2 + bt + c\), we have:
\(a = 1\) (coefficient of \(t^2\))
\(b = 7\) (coefficient of \(t\))
\(c = 10\) (constant term)

Now, we can find the sum of the zeroes using the formula:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{7}{1} = -7 \]

Step 4: Final Answer:

The value of \(\alpha + \beta\) is -7. This corresponds to option (C).
Quick Tip: Be careful with the signs. The formula for the sum of roots has a negative sign \((-\frac{b}{a})\), while the product of roots does not \((\frac{c}{a})\). This is a common point of error.

Step 1: Understanding the Concept:

This problem requires knowledge of the standard trigonometric ratios for specific angles, namely 30° and 60°.


Step 2: Key Formula or Approach:

We need the following standard values:
\(\sin 30^\circ = \frac{1}{2}\)
\(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
\(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
\(\cos 60^\circ = \frac{1}{2}\)


Step 3: Detailed Explanation:

Substitute these values into the given expression:
\[ (\sin 30^\circ + \cos 30^\circ) - (\sin 60^\circ + \cos 60^\circ) \] \[ = \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{3}}{2} + \frac{1}{2}\right) \]
Since the terms inside both parentheses are identical, subtracting them will result in zero.
\[ = \frac{1 + \sqrt{3}}{2} - \frac{\sqrt{3} + 1}{2} \] \[ = 0 \]

Step 4: Final Answer:

The value of the expression is 0. This matches option (B).
Quick Tip: Recognize that \(\sin \theta = \cos(90^\circ - \theta)\) and \(\cos \theta = \sin(90^\circ - \theta)\). Here, \(\sin 30^\circ = \cos 60^\circ\) and \(\cos 30^\circ = \sin 60^\circ\). The expression is \((A+B)-(B+A)\), which is always zero.

Step 1: Understanding the Concept:

If a certain value is a "zero" of a polynomial, it means that when you substitute this value for the variable (x), the entire polynomial evaluates to zero.


Step 2: Key Formula or Approach:

Let the polynomial be \(P(x) = (k-1)x^2 + kx + 1\).

Since -4 is a zero, we have \(P(-4) = 0\). We will substitute \(x = -4\) into the polynomial and solve the resulting equation for k.


Step 3: Detailed Explanation:

Substitute \(x = -4\) into the polynomial:
\[ P(-4) = (k-1)(-4)^2 + k(-4) + 1 = 0 \]
Now, simplify the equation:
\[ (k-1)(16) - 4k + 1 = 0 \]
Distribute the 16:
\[ 16k - 16 - 4k + 1 = 0 \]
Combine like terms (terms with k and constant terms):
\[ (16k - 4k) + (-16 + 1) = 0 \] \[ 12k - 15 = 0 \]
Now, solve for k:
\[ 12k = 15 \] \[ k = \frac{15}{12} \]
Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 3.
\[ k = \frac{15 \div 3}{12 \div 3} = \frac{5}{4} \]

Step 4: Final Answer:

The value of k is \(\frac{5}{4}\). This corresponds to option (B).
Quick Tip: When substituting a negative number into an expression with exponents, always use parentheses to avoid calculation errors. For example, write \((-4)^2\) which is 16, not \(-4^2\) which is -16.

Step 1: Understanding the Concept:

This question is based on a fundamental theorem in circle geometry. The theorem states that the lengths of the two tangents drawn from an external point to a circle are equal.


Step 2: Key Formula or Approach:

Theorem: If PA and PB are two tangents to a circle from an external point P, then PA = PB.


Step 3: Detailed Explanation:

According to the theorem, the lengths of the tangents from the external point P to the circle are equal.

We are given that PA and PB are the two tangents from point P.

The length of one tangent, PA, is given as 8 cm.

Therefore, the length of the other tangent, PB, must be equal to PA.
\[ PB = PA = 8 cm \]

Step 4: Final Answer:

The length of PB is 8 cm. This corresponds to option (B).
Quick Tip: Always remember this key property of circles: tangents from the same external point are equal in length. This is a frequently tested concept.

Step 1: Understanding the Concept:

This problem uses properties of tangents from an external point to a circle. Specifically, we use the fact that the line joining the center of the circle to the external point bisects the angle between the tangents, and the radius is perpendicular to the tangent at the point of contact.


Step 2: Key Formula or Approach:

1. The line segment PO bisects \(\angle APB\). So, \(\angle OPA = \frac{1}{2} \angle APB\).

2. The radius is perpendicular to the tangent at the point of contact. So, \(\angle OAP = 90^\circ\).

3. The sum of angles in a triangle is \(180^\circ\). In \(\triangle OAP\), \(\angle POA + \angle OAP + \angle APO = 180^\circ\).


Step 3: Detailed Explanation:

We are given that \(\angle APB = 80^\circ\).

The line PO bisects this angle, so:
\[ \angle APO = \frac{1}{2} \angle APB = \frac{1}{2} \times 80^\circ = 40^\circ \]
Now, consider the triangle \(\triangle OAP\). We know that the radius OA is perpendicular to the tangent PA at the point of contact A.

Therefore, \(\angle OAP = 90^\circ\).

The sum of angles in \(\triangle OAP\) is \(180^\circ\).
\[ \angle POA + \angle OAP + \angle APO = 180^\circ \]
Substitute the known values:
\[ \angle POA + 90^\circ + 40^\circ = 180^\circ \] \[ \angle POA + 130^\circ = 180^\circ \] \[ \angle POA = 180^\circ - 130^\circ = 50^\circ \]

Step 4: Final Answer:

The measure of \(\angle POA\) is \(50^\circ\). This matches option (B).
Quick Tip: In problems involving tangents from an external point P to a circle with center O, the triangles \(\triangle OAP\) and \(\triangle OBP\) are always congruent right-angled triangles.

Step 1: Understanding the Concept:

This question asks for a direct statement of a fundamental theorem related to tangents and radii of a circle.


Step 2: Detailed Explanation:

The theorem states that the tangent at any point on a circle is perpendicular to the radius that passes through that point of contact.

"Perpendicular" means that the angle formed between the two lines (the tangent and the radius) is \(90^\circ\).


Step 3: Final Answer:

Therefore, the angle between the tangent and the radius at the point of contact is \(90^\circ\). This corresponds to option (D).
Quick Tip: This is one of the most important theorems in the study of circles. It's the basis for solving many geometry problems. Memorize it: Radius \(\perp\) Tangent at the point of contact.

Step 1: Understanding the Concept:

The area of a circle depends on the square of its radius. This question asks for the ratio of the areas of two circles given the ratio of their radii.


Step 2: Key Formula or Approach:

Area of a circle, \(A = \pi r^2\).

Let the radii of the two circles be \(r_1\) and \(r_2\), and their areas be \(A_1\) and \(A_2\).

The ratio of their areas is \(\frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2} = \left(\frac{r_1}{r_2}\right)^2\).


Step 3: Detailed Explanation:

We are given the ratio of the radii:
\[ \frac{r_1}{r_2} = \frac{3}{4} \]
Now, we find the ratio of their areas using the formula derived in Step 2:
\[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{3}{4}\right)^2 \] \[ \frac{A_1}{A_2} = \frac{3^2}{4^2} = \frac{9}{16} \]
So, the ratio of the areas is 9:16.


Step 4: Final Answer:

The ratio of the areas of the two circles is 9:16. This matches option (C).
Quick Tip: For any two similar 2D shapes, if the ratio of their corresponding lengths (like radius, side, etc.) is a:b, then the ratio of their areas will be \(a^2:b^2\).

Step 1: Understanding the Concept:

A sector of a circle is the portion of the circle enclosed by two radii and the arc between them. Its area is a fraction of the total area of the circle, determined by the central angle.


Step 2: Key Formula or Approach:

The area of a sector with central angle \(\theta\) (in degrees) and radius \(r\) is given by the formula:
\[ Area of Sector = \frac{\theta}{360^\circ} \times \pi r^2 \]

Step 3: Detailed Explanation:

We are given:

Radius, \(r = 42\) cm

Central angle, \(\theta = 30^\circ\)

Substitute these values into the formula. We will use the approximation \(\pi = \frac{22}{7}\).
\[ Area = \frac{30}{360} \times \frac{22}{7} \times (42)^2 \]
Simplify the fraction \(\frac{30}{360}\):
\[ \frac{30}{360} = \frac{3}{36} = \frac{1}{12} \]
Now the calculation is:
\[ Area = \frac{1}{12} \times \frac{22}{7} \times 42 \times 42 \]
We can simplify by canceling terms:
\[ Area = \frac{1}{12} \times 22 \times (6 \times 42) \] \[ Area = \frac{1}{2 \times 6} \times 22 \times 6 \times 42 \]
Cancel the 6:
\[ Area = \frac{1}{2} \times 22 \times 42 \] \[ Area = 11 \times 42 \] \[ Area = 462 cm^2 \]

Step 4: Final Answer:

The area of the sector is 462 cm\(^2\). This corresponds to option (C).
Quick Tip: When performing calculations with fractions, always look for opportunities to cancel common factors before multiplying. This simplifies the arithmetic and reduces the chance of errors.

Step 1: Understanding the Concept:

The circumference of a circle is directly proportional to its radius. This question tests the relationship between the ratio of circumferences and the ratio of radii.


Step 2: Key Formula or Approach:

The circumference of a circle, \(C = 2\pi r\).

Let the radii of the two circles be \(r_1\) and \(r_2\), and their circumferences be \(C_1\) and \(C_2\).

The ratio of their circumferences is \(\frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2}\).


Step 3: Detailed Explanation:

We are given the ratio of the circumferences:
\[ \frac{C_1}{C_2} = \frac{5}{7} \]
Using the formula, we have:
\[ \frac{2\pi r_1}{2\pi r_2} = \frac{5}{7} \]
The term \(2\pi\) is a common factor in the numerator and the denominator, so it cancels out.
\[ \frac{r_1}{r_2} = \frac{5}{7} \]
So, the ratio of the radii is 5:7.


Step 4: Final Answer:

The ratio of the radii of the two circles is 5:7. This matches option (B).
Quick Tip: For circles, the ratio of their radii, the ratio of their diameters, and the ratio of their circumferences are all the same. If the ratio of radii is a:b, the ratio of circumferences is also a:b.

Step 1: Understanding the Concept:

This question uses one of the fundamental Pythagorean identities in trigonometry.


Step 2: Key Formula or Approach:

The relevant trigonometric identity is:
\[ \sec^2 A - \tan^2 A = 1 \]

Step 3: Detailed Explanation:

The given expression is \(7 \sec^2 A - 7 \tan^2 A\).

We can see that 7 is a common factor in both terms. Let's factor it out:
\[ 7 (\sec^2 A - \tan^2 A) \]
Now, we substitute the value of the identity \(\sec^2 A - \tan^2 A = 1\) into the expression:
\[ = 7 (1) \] \[ = 7 \]

Step 4: Final Answer:

The value of the expression is 7. This corresponds to option (B).
Quick Tip: Memorize the three Pythagorean identities: \(\sin^2 \theta + \cos^2 \theta = 1\), \(1 + \tan^2 \theta = \sec^2 \theta\), and \(1 + \cot^2 \theta = \csc^2 \theta\). They are essential for simplifying trigonometric expressions.

Step 1: Understanding the Concept:

This problem involves substituting parametric equations into an algebraic expression and simplifying it using a trigonometric identity. The goal is to eliminate the parameter \(\theta\).


Step 2: Key Formula or Approach:

We will substitute the given expressions for \(x\) and \(y\) into \(b^2x^2 + a^2y^2\) and use the Pythagorean identity \(\cos^2 \theta + \sin^2 \theta = 1\).


Step 3: Detailed Explanation:

We are given:
\(x = a \cos \theta\)
\(y = b \sin \theta\)

The expression to evaluate is \(b^2x^2 + a^2y^2\).

Substitute the values of \(x\) and \(y\) into the expression:
\[ b^2x^2 + a^2y^2 = b^2(a \cos \theta)^2 + a^2(b \sin \theta)^2 \]
Square the terms inside the parentheses:
\[ = b^2(a^2 \cos^2 \theta) + a^2(b^2 \sin^2 \theta) \] \[ = a^2b^2 \cos^2 \theta + a^2b^2 \sin^2 \theta \]
Factor out the common term \(a^2b^2\):
\[ = a^2b^2 (\cos^2 \theta + \sin^2 \theta) \]
Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\):
\[ = a^2b^2 (1) \] \[ = a^2b^2 \]

Step 4: Final Answer:

The value of the expression is \(a^2b^2\). This matches option (A).
Quick Tip: When you see expressions like \(x = a \cos \theta\) and \(y = b \sin \theta\), think about how to isolate \(\cos \theta\) and \(\sin \theta\) (\(\cos \theta = x/a\), \(\sin \theta = y/b\)) and then use \(\cos^2\theta + \sin^2\theta = 1\). This gives \((x/a)^2 + (y/b)^2 = 1\), which is the equation of an ellipse.

Step 1: Understanding the Concept:

This is a problem of heights and distances, which is an application of trigonometry. We can model the situation with a right-angled triangle where the tower is the perpendicular side, the distance from the base is the base side, and the angle of elevation is the angle between the base and the hypotenuse.


Step 2: Key Formula or Approach:

In a right-angled triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
\[ \tan \theta = \frac{Opposite}{Adjacent} = \frac{Height}{Base} \]

Step 3: Detailed Explanation:

Let \(h\) be the height of the tower (opposite side).

The distance from the base is 10 m (adjacent side).

The angle of elevation, \(\theta\), is \(60^\circ\).

Using the tangent ratio:
\[ \tan 60^\circ = \frac{h}{10} \]
We know the standard value \(\tan 60^\circ = \sqrt{3}\).
\[ \sqrt{3} = \frac{h}{10} \]
To find \(h\), multiply both sides by 10:
\[ h = 10\sqrt{3} m \]

Step 4: Final Answer:

The height of the tower is \(10\sqrt{3}\) m. This corresponds to option (B).
Quick Tip: Remember the SOH CAH TOA mnemonic to choose the correct trigonometric ratio: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Step 1: Understanding the Concept:

This problem can also be modeled with a right-angled triangle. The height of the kite is the opposite side, the length of the string is the hypotenuse, and the angle the string makes with the ground is the angle of elevation.


Step 2: Key Formula or Approach:

In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
\[ \sin \theta = \frac{Opposite}{Hypotenuse} \]

Step 3: Detailed Explanation:

The height of the kite (opposite side) = 30 m.

Let the length of the string be \(l\) (hypotenuse).

The angle with the earth, \(\theta\), is \(60^\circ\).

Using the sine ratio:
\[ \sin 60^\circ = \frac{30}{l} \]
We know the standard value \(\sin 60^\circ = \frac{\sqrt{3}}{2}\).
\[ \frac{\sqrt{3}}{2} = \frac{30}{l} \]
To solve for \(l\), we can cross-multiply:
\[ l \times \sqrt{3} = 30 \times 2 \] \[ l\sqrt{3} = 60 \] \[ l = \frac{60}{\sqrt{3}} \]
To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{3}\):
\[ l = \frac{60 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{60\sqrt{3}}{3} \] \[ l = 20\sqrt{3} m \]

Step 4: Final Answer:

The length of the string is \(20\sqrt{3}\) m. This matches option (C).
Quick Tip: It's good practice to rationalize the denominator in your final answer. This involves removing any square roots from the bottom of the fraction.

Step 1: Understanding the Concept:

To prove that two triangles are similar, we can use one of the similarity criteria: Angle-Angle (AA), Side-Angle-Side (SAS), or Side-Side-Side (SSS). In this problem, we will use the AA similarity criterion by showing that two pairs of corresponding angles in \(\triangle ABD\) and \(\triangle ECF\) are equal.


Step 2: Detailed Explanation:

We are given the following information:

\(\triangle ABC\) is an isosceles triangle with \(AB = AC\).
E is a point on the side CB produced.
\(AD \perp BC\), which means \(\angle ADB = 90^\circ\).
\(EF \perp AC\), which means \(\angle EFC = 90^\circ\).

Our goal is to prove \(\triangle ABD \sim \triangle ECF\).


Proof:

In \(\triangle ABC\), since \(AB = AC\), the angles opposite to these sides are equal.

Therefore, \(\angle ABC = \angle ACB\).


Let's consider \(\triangle ABD\) and \(\triangle ECF\).


\(\angle ADB = \angle EFC = 90^\circ\) (Given that AD \(\perp\) BC and EF \(\perp\) AC).

\(\angle ABD = \angle ABC\). Also, since E lies on CB produced, C is between B and E. The angle \(\angle ECF\) is the same as \(\angle ACB\). So, \(\angle ECF = \angle ACB\).

As established earlier, \(\angle ABC = \angle ACB\).

Therefore, \(\angle ABD = \angle ECF\).



Now we have two pairs of equal corresponding angles in \(\triangle ABD\) and \(\triangle ECF\).

By the Angle-Angle (AA) similarity criterion, we can conclude that \(\triangle ABD \sim \triangle ECF\).


Step 3: Final Answer:

Hence, it is proved that \(\triangle ABD\) is similar to \(\triangle ECF\).
Quick Tip: In geometry proofs involving isosceles triangles, always start by using the property that angles opposite to equal sides are equal. This often provides the key to solving the problem.

Step 1: Understanding the Concept:

To prove that \(\triangle ABC \sim \triangle PQR\), we can use the SAS (Side-Angle-Side) similarity criterion. The strategy is to first prove that two smaller triangles formed by the medians are similar (\(\triangle ABD \sim \triangle PQM\)) to establish the equality of \(\angle B\) and \(\angle Q\). A common typo in this problem is listing PR instead of QR; we will assume the proportion is with QR.


Step 2: Detailed Explanation:

We are given: \[ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AD}{PM} \]
where AD and PM are medians. This means D is the midpoint of BC and M is the midpoint of QR.

So, \(BC = 2BD\) and \(QR = 2QM\).


Substitute this into the given proportion: \[ \frac{AB}{PQ} = \frac{2BD}{2QM} = \frac{AD}{PM} \] \[ \implies \frac{AB}{PQ} = \frac{BD}{QM} = \frac{AD}{PM} \]
Now consider \(\triangle ABD\) and \(\triangle PQM\). Since all three corresponding sides are in proportion, by the SSS (Side-Side-Side) similarity criterion, we have: \[ \triangle ABD \sim \triangle PQM \]
Because the triangles are similar, their corresponding angles are equal. \[ \therefore \angle B = \angle Q \]
Now, let's consider the original triangles, \(\triangle ABC\) and \(\triangle PQR\). We have:

\(\frac{AB}{PQ} = \frac{BC}{QR}\) (Given).
\(\angle B = \angle Q\) (Proved above).

By the SAS (Side-Angle-Side) similarity criterion, we can conclude that: \[ \triangle ABC \sim \triangle PQR \]

Step 3: Final Answer:

Hence, it is proved that \(\triangle ABC\) is similar to \(\triangle PQR\).
Quick Tip: When a problem involves medians and proportionality, try to use the fact that a median divides a side into two equal halves. This often allows you to establish similarity for smaller triangles, which then helps prove the similarity of the larger triangles.

Step 1: Understanding the Concept:

This problem uses the theorem that states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.


Step 2: Key Formula or Approach:

If \(\triangle ABC \sim \triangle DEF\), then: \[ \frac{Area(\triangle ABC)}{Area(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{BC}{EF}\right)^2 = \left(\frac{AC}{DF}\right)^2 \]

Step 3: Detailed Explanation:

We are given:

Area(\(\triangle ABC\)) = 9 cm\(^2\)

Area(\(\triangle DEF\)) = 64 cm\(^2\)

DE = 5.1 cm

Since \(\triangle ABC \sim \triangle DEF\), AB and DE are corresponding sides.

Using the formula: \[ \frac{9}{64} = \left(\frac{AB}{5.1}\right)^2 \]
To find the ratio of the sides, take the square root of both sides: \[ \sqrt{\frac{9}{64}} = \frac{AB}{5.1} \] \[ \frac{3}{8} = \frac{AB}{5.1} \]
Now, solve for AB: \[ AB = \frac{3 \times 5.1}{8} \] \[ AB = \frac{15.3}{8} \] \[ AB = 1.9125 cm \]

Step 4: Final Answer:

The length of side AB is 1.9125 cm.
Quick Tip: A common mistake is to forget to take the square root of the area ratio. Remember: Area ratio is the square of the side ratio, so the side ratio is the square root of the area ratio.

Step 1: Understanding the Concept:

To prove this trigonometric identity, we will work with the Left Hand Side (LHS). The standard technique for expressions involving square roots of fractions like this is to rationalize the denominator inside the square root.


Step 2: Key Formula or Approach:

We will use the algebraic identity \((a-b)(a+b) = a^2 - b^2\) and the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\), which implies \(1 - \cos^2\theta = \sin^2\theta\).


Step 3: Detailed Explanation:

Let's start with the LHS: \[ LHS = \sqrt{\frac{1+\cos\theta}{1-\cos\theta}} \]
Multiply the numerator and the denominator inside the square root by \((1+\cos\theta)\) to rationalize the denominator: \[ LHS = \sqrt{\frac{(1+\cos\theta)(1+\cos\theta)}{(1-\cos\theta)(1+\cos\theta)}} \] \[ LHS = \sqrt{\frac{(1+\cos\theta)^2}{1^2 - \cos^2\theta}} \]
Using the identity \(1 - \cos^2\theta = \sin^2\theta\): \[ LHS = \sqrt{\frac{(1+\cos\theta)^2}{\sin^2\theta}} \]
Now, we can take the square root of the numerator and the denominator: \[ LHS = \frac{1+\cos\theta}{\sin\theta} \]
This is equal to the Right Hand Side (RHS).


Step 4: Final Answer:

Since LHS = RHS, the identity is proved.
Quick Tip: Rationalization is a powerful tool for simplifying expressions with square roots in the denominator. Multiplying by the conjugate of the denominator often helps eliminate the square root by creating a difference of squares.

Step 1: Understanding the Concept:

This problem involves proving a trigonometric equality by using the relationship between trigonometric functions of complementary angles (angles that add up to \(90^\circ\)).


Step 2: Key Formula or Approach:

The key complementary angle identity we will use is: \[ \cot(90^\circ - \theta) = \tan\theta \]

Step 3: Detailed Explanation:

We can start from the Right Hand Side (RHS) and show that it is equal to the Left Hand Side (LHS).
\[ RHS = \cot 63^\circ \cdot \cot 81^\circ \]
Notice that \(63^\circ = 90^\circ - 27^\circ\) and \(81^\circ = 90^\circ - 9^\circ\).

Apply the complementary angle identity to each term:

For the first term: \[ \cot 63^\circ = \cot(90^\circ - 27^\circ) = \tan 27^\circ \]
For the second term: \[ \cot 81^\circ = \cot(90^\circ - 9^\circ) = \tan 9^\circ \]
Now substitute these back into the RHS expression: \[ RHS = (\tan 27^\circ) \cdot (\tan 9^\circ) \]
Rearranging the terms, we get: \[ RHS = \tan 9^\circ \cdot \tan 27^\circ \]
This is exactly the expression on the LHS.


Step 4: Final Answer:

Since RHS = LHS, the identity is proved.
Quick Tip: Whenever you see a trigonometric expression with angles that seem unrelated, check if any pairs add up to \(90^\circ\). If they do, using complementary angle identities is almost always the correct approach.

Step 1: Understanding the Concept:

Given one trigonometric ratio, we can find the others by constructing a right-angled triangle and using the Pythagorean theorem, or by using trigonometric identities. The triangle method is often more intuitive.


Step 2: Key Formula or Approach:

We know that \(\cos A = \frac{Adjacent}{Hypotenuse}\).

From this, we can find the Opposite side using Pythagoras theorem: \((Opposite)^2 + (Adjacent)^2 = (Hypotenuse)^2\).

Then we use the definitions: \(\cot A = \frac{Adjacent}{Opposite}\) and \(\csc A = \frac{Hypotenuse}{Opposite}\).


Step 3: Detailed Explanation:

We are given \(\cos A = \frac{4}{5}\).

Let's consider a right-angled triangle where for angle A:

Adjacent side = 4

Hypotenuse = 5

Let the opposite side be 'p'. By the Pythagorean theorem: \[ p^2 + 4^2 = 5^2 \] \[ p^2 + 16 = 25 \] \[ p^2 = 25 - 16 = 9 \] \[ p = \sqrt{9} = 3 \]
So, the Opposite side = 3.


Now we can find the required trigonometric ratios:

1. To find \(\cot A\): \[ \cot A = \frac{Adjacent}{Opposite} = \frac{4}{3} \]
2. To find \(\csc A\): \[ \csc A = \frac{Hypotenuse}{Opposite} = \frac{5}{3} \]

Step 4: Final Answer:

The values are \(\cot A = \frac{4}{3}\) and \(\csc A = \frac{5}{3}\).
Quick Tip: Recognizing Pythagorean triplets like (3, 4, 5) can save you time. When you see two of the numbers, you can immediately identify the third without calculation.

Step 1: Understanding the Concept:

Given the tangent of an angle, we can find the sine and cosine of that angle by constructing a right-angled triangle, finding all its sides using the Pythagorean theorem, and then applying the definitions of sine and cosine.


Step 2: Key Formula or Approach:

We are given \(\tan\theta = \frac{Opposite}{Adjacent}\).

We will find the Hypotenuse using \((Hypotenuse)^2 = (Opposite)^2 + (Adjacent)^2\).

Then we calculate \(\sin\theta = \frac{Opposite}{Hypotenuse}\) and \(\cos\theta = \frac{Adjacent}{Hypotenuse}\).


Step 3: Detailed Explanation:

We are given \(\tan\theta = \frac{5}{12}\).

In a right-angled triangle, let:

Opposite side = 5

Adjacent side = 12

Let the hypotenuse be 'h'. By the Pythagorean theorem: \[ h^2 = 5^2 + 12^2 \] \[ h^2 = 25 + 144 = 169 \] \[ h = \sqrt{169} = 13 \]
So, the Hypotenuse = 13.


Now we find \(\sin\theta\) and \(\cos\theta\): \[ \sin\theta = \frac{Opposite}{Hypotenuse} = \frac{5}{13} \] \[ \cos\theta = \frac{Adjacent}{Hypotenuse} = \frac{12}{13} \]
Finally, we calculate the value of \(\sin\theta + \cos\theta\): \[ \sin\theta + \cos\theta = \frac{5}{13} + \frac{12}{13} = \frac{5+12}{13} = \frac{17}{13} \]

Step 4: Final Answer:

The value of \(\sin\theta + \cos\theta\) is \(\frac{17}{13}\).
Quick Tip: Memorizing common Pythagorean triplets like (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) will significantly speed up solving trigonometry problems.

Step 1: Understanding the Concept:

To solve this trigonometric equation, we need to use the complementary angle identities to express both sides of the equation with the same trigonometric function.


Step 2: Key Formula or Approach:

We will use the identity \(\sin\theta = \cos(90^\circ - \theta)\). By applying this, we can make both sides of the equation a cosine function and then equate the angles.


Step 3: Detailed Explanation:

The given equation is: \[ \sin 3A = \cos(A - 26^\circ) \]
Using the identity, we can rewrite the LHS as: \[ \sin 3A = \cos(90^\circ - 3A) \]
Now substitute this into the equation: \[ \cos(90^\circ - 3A) = \cos(A - 26^\circ) \]
Since both 3A and (A - 26\(^\circ\)) must represent angles for which the cosine function is defined and equal, we can equate the angles (assuming they are in the primary range): \[ 90^\circ - 3A = A - 26^\circ \]
Now, solve for A by rearranging the terms: \[ 90^\circ + 26^\circ = A + 3A \] \[ 116^\circ = 4A \] \[ A = \frac{116^\circ}{4} \] \[ A = 29^\circ \]
We should check if the condition that 3A is an acute angle is met. \(3A = 3 \times 29^\circ = 87^\circ\). Since \(87^\circ < 90^\circ\), 3A is indeed an acute angle.


Step 4: Final Answer:

The value of A is \(29^\circ\).
Quick Tip: When you see an equation with sine on one side and cosine on the other, like \(\sin(X) = \cos(Y)\), it implies that the angles are complementary, i.e., \(X + Y = 90^\circ\). In this case, \(3A + (A - 26^\circ) = 90^\circ\), which gives \(4A = 116^\circ\) and \(A=29^\circ\). This is a quick shortcut.

Step 1: Understanding the Concept:

This word problem can be solved by setting up a system of two linear equations with two variables based on the given information and then solving them simultaneously.


Step 2: Detailed Explanation:

Let the two numbers be \(x\) and \(y\).

From the problem statement, we can form two equations:

Equation 1: The sum of the two numbers is 50. \[ x + y = 50 \]
Equation 2: One number is \(\frac{7}{3}\) times the other. \[ x = \frac{7}{3}y \]
Now we can solve this system. We will use the substitution method by substituting the expression for \(x\) from Equation 2 into Equation 1.
\[ \left(\frac{7}{3}y\right) + y = 50 \]
To solve for \(y\), find a common denominator: \[ \frac{7y}{3} + \frac{3y}{3} = 50 \] \[ \frac{10y}{3} = 50 \]
Multiply both sides by 3: \[ 10y = 150 \]
Divide by 10: \[ y = 15 \]
Now substitute the value of \(y\) back into Equation 2 to find \(x\): \[ x = \frac{7}{3}(15) \] \[ x = 7 \times 5 \] \[ x = 35 \]

Step 3: Final Answer:

The two numbers are 35 and 15. We can check our answer: \(35 + 15 = 50\) and \(35 = \frac{7}{3} \times 15\). The conditions are met.
Quick Tip: When translating word problems into equations, define your variables clearly. For example, "Let x be the larger number and y be the smaller number." This helps in correctly setting up relationships like \(x = \frac{7}{3}y\).

Step 1: Understanding the Concept:

We can solve this problem using polynomial long division. It is important to write the dividend (\(x^3+1\)) with placeholders for the missing terms (\(x^2\) and \(x\)) to keep the columns aligned correctly.


Step 2: Detailed Explanation:

We set up the long division as follows, writing \(x^3 + 1\) as \(x^3 + 0x^2 + 0x + 1\).
\[ \begin{array}{c|cc cc} \multicolumn{2}{r}{x^2} & -x & +1
\cline{2-5} x+1 & x^3 & +0x^2 & +0x & +1
\multicolumn{2}{r}{- (x^3} & +x^2)
\cline{2-3} \multicolumn{2}{r}{0} & -x^2 & +0x
\multicolumn{2}{r}{} & -(-x^2 & -x)
\cline{3-4} \multicolumn{2}{r}{} & 0 & +x & +1
\multicolumn{2}{r}{} & & -(x & +1)
\cline{4-5} \multicolumn{2}{r}{} & & 0 & 0
\end{array} \]
Step-by-step process:


Divide the first term of the dividend (\(x^3\)) by the first term of the divisor (\(x\)). \(x^3/x = x^2\). Write \(x^2\) in the quotient.
Multiply the divisor (\(x+1\)) by \(x^2\). \(x^2(x+1) = x^3+x^2\). Write this below the dividend.
Subtract. \((x^3+0x^2) - (x^3+x^2) = -x^2\). Bring down the next term (\(0x\)).
Divide the new first term (\(-x^2\)) by \(x\). \(-x^2/x = -x\). Write \(-x\) in the quotient.
Multiply the divisor (\(x+1\)) by \(-x\). \(-x(x+1) = -x^2-x\). Write this below.
Subtract. \((-x^2+0x) - (-x^2-x) = x\). Bring down the next term (\(+1\)).
Divide the new first term (\(x\)) by \(x\). \(x/x = 1\). Write \(+1\) in the quotient.
Multiply the divisor (\(x+1\)) by \(1\). \(1(x+1) = x+1\). Write this below.
Subtract. \((x+1) - (x+1) = 0\). The remainder is 0.


Step 3: Final Answer:

The quotient is \(x^2 - x + 1\) and the remainder is 0.
Quick Tip: This problem can be solved instantly by using the sum of cubes factorization formula: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\). Here, \(x^3 + 1^3 = (x+1)(x^2 - x \cdot 1 + 1^2) = (x+1)(x^2-x+1)\). Therefore, \((x^3+1) \div (x+1) = x^2-x+1\).

Step 1: Understanding the Concept:

Euclid's division algorithm is a method for finding the Highest Common Factor (H.C.F.) of two integers. It is based on the principle that the H.C.F. of two numbers does not change if the larger number is replaced by its difference with the smaller number. The algorithm uses the division lemma \(a = bq + r\), where the H.C.F. of \(a\) and \(b\) is the same as the H.C.F. of \(b\) and \(r\).


Step 2: Detailed Explanation:

Let \(a = 1188\) and \(b = 504\). We apply the division lemma repeatedly until the remainder becomes 0.

Step 1: Divide 1188 by 504. \[ 1188 = 504 \times 2 + 180 \]
The remainder is 180. Now, the new dividend is 504 and the new divisor is 180.


Step 2: Divide 504 by 180. \[ 504 = 180 \times 2 + 144 \]
The remainder is 144. Now, the new dividend is 180 and the new divisor is 144.


Step 3: Divide 180 by 144. \[ 180 = 144 \times 1 + 36 \]
The remainder is 36. Now, the new dividend is 144 and the new divisor is 36.


Step 4: Divide 144 by 36. \[ 144 = 36 \times 4 + 0 \]
The remainder is 0. The algorithm stops. The H.C.F. is the last non-zero remainder.


Step 3: Final Answer:

The last non-zero remainder is 36. Therefore, the H.C.F. of 504 and 1188 is 36.
Quick Tip: To keep the process organized, always remember the pattern: the divisor of the current step becomes the dividend of the next step, and the remainder of the current step becomes the divisor of the next step.

Step 1: Understanding the Concept:

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is a value that determines the nature of its roots (solutions). The nature of the roots can be real and distinct, real and equal, or not real (complex).


Step 2: Key Formula or Approach:

The discriminant, denoted by \(D\), is calculated using the formula: \[ D = b^2 - 4ac \]
The nature of the roots is determined as follows:

If \(D > 0\), the roots are real and distinct (unequal).
If \(D = 0\), the roots are real and equal.
If \(D < 0\), the roots are not real (they are complex conjugates).


Step 3: Detailed Explanation:

The given quadratic equation is \(2x^2 + 5x - 3 = 0\).

Comparing it with the standard form \(ax^2 + bx + c = 0\), we have: \(a = 2\), \(b = 5\), \(c = -3\).


First, let's calculate the discriminant \(D\): \[ D = b^2 - 4ac \] \[ D = (5)^2 - 4(2)(-3) \] \[ D = 25 - (-24) \] \[ D = 25 + 24 = 49 \]
Now, let's determine the nature of the roots based on the value of \(D\).
Since \(D = 49\), which is greater than 0 (\(D > 0\)), the equation has two real and distinct roots.


Step 4: Final Answer:

The discriminant is 49. The nature of the roots is real and distinct.
Quick Tip: Be very careful with signs, especially when the coefficient 'c' is negative. A common error is calculating \(b^2 - 4ac\) as \(b^2 - 4(a)(c)\) instead of \(b^2 + 4(a)(|c|)\).

Step 1: Understanding the Concept:

This word problem needs to be translated into a quadratic equation, which can then be solved to find the unknown integers.


Step 2: Detailed Explanation:

Let the two consecutive positive integers be \(x\) and \((x+1)\).

According to the problem, the sum of their squares is 365. We can write this as an equation: \[ x^2 + (x+1)^2 = 365 \]
Expand the term \((x+1)^2\): \[ x^2 + (x^2 + 2x + 1) = 365 \]
Combine like terms and simplify the equation: \[ 2x^2 + 2x + 1 = 365 \] \[ 2x^2 + 2x - 364 = 0 \]
Divide the entire equation by 2 to simplify it: \[ x^2 + x - 182 = 0 \]
Now, we need to solve this quadratic equation. We can factor it by finding two numbers that multiply to -182 and add to +1. These numbers are 14 and -13. \[ (x + 14)(x - 13) = 0 \]
This gives two possible values for \(x\): \(x + 14 = 0 \implies x = -14\) \(x - 13 = 0 \implies x = 13\)


Since the problem asks for positive integers, we discard the solution \(x = -14\).
So, the first integer is \(x = 13\).

The second consecutive integer is \(x + 1 = 13 + 1 = 14\).


Step 3: Final Answer:

The two consecutive positive integers are 13 and 14. Let's check: \(13^2 + 14^2 = 169 + 196 = 365\). The answer is correct.
Quick Tip: When factoring a quadratic equation like \(x^2 + x - 182 = 0\), if the numbers are large, think about the approximate square root. \(\sqrt{182}\) is close to 13.5. So, check integers near 13.5, like 13 and 14, as potential factors.

Step 1: Understanding the Concept:

The task is to convert the two statements given in the word problem into a single algebraic equation in one variable.


Step 2: Detailed Explanation:

Let the larger number be \(x\) and the smaller number be \(y\).

Let's translate each sentence into an equation.

Sentence 1: "The difference of squares of two numbers is 180."
Since x is the larger number, this translates to: \[ x^2 - y^2 = 180 \quad --- (1) \]
Sentence 2: "The square of the smaller number is 8 times the larger number."
This translates to: \[ y^2 = 8x \quad --- (2) \]
The question asks to write the equation for this statement, which usually means a single equation in one variable that models the situation. We can achieve this by substituting the expression for \(y^2\) from Equation (2) into Equation (1). \[ x^2 - (8x) = 180 \]
Rearranging this into the standard form of a quadratic equation (\(ax^2+bx+c=0\)): \[ x^2 - 8x - 180 = 0 \]

Step 3: Final Answer:

This single equation, \(x^2 - 8x - 180 = 0\), represents the conditions given in the statement, where \(x\) is the larger of the two numbers.
Quick Tip: When a problem involves two variables and two conditions, the goal is often to use one equation to express one variable in terms of the other, and then substitute it into the second equation to get a final equation in a single variable.

Step 1: Understanding the Concept:

This proof involves two key theorems from triangle geometry. First, the Converse of the Basic Proportionality Theorem (also known as Thales's Theorem or BPT), which relates side ratios to parallel lines. Second, the property of isosceles triangles, which relates equal angles to equal opposite sides.


Step 2: Detailed Explanation:

We are given a \(\triangle PQR\) with points S on PQ and T on PR.

Given Condition 1: The sides are divided in the same ratio. \[ \frac{PS}{SQ} = \frac{PT}{TR} \]
By the Converse of the Basic Proportionality Theorem, if a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

Therefore, we can conclude that \(ST \parallel QR\).


Now, since \(ST \parallel QR\) and PQ is a transversal line, the corresponding angles are equal. \[ \angle PST = \angle PQR \quad --- (i) \]
Given Condition 2: We are also given an equality of angles. \[ \angle PST = \angle PRQ \quad --- (ii) \]
From equations (i) and (ii), we can see that both \(\angle PQR\) and \(\angle PRQ\) are equal to \(\angle PST\).
Therefore, \[ \angle PQR = \angle PRQ \]
In \(\triangle PQR\), we have now shown that two angles are equal. In any triangle, the sides opposite to equal angles are equal in length.

The side opposite to \(\angle PQR\) is PR.

The side opposite to \(\angle PRQ\) is PQ.

Therefore, \(PQ = PR\).


A triangle with two equal sides is an isosceles triangle.


Step 3: Final Answer:

Thus, \(\triangle PQR\) is an isosceles triangle. Hence Proved.
Quick Tip: The logical flow is crucial in such proofs. The ratio condition leads to parallel lines. Parallel lines lead to equal corresponding angles. Combining this with the given angle condition proves two angles of the large triangle are equal, which in turn proves it is isosceles.

Step 1: Understanding the Concept:

The curved surface area (CSA) of a cone is the area of its lateral surface, excluding the circular base. The formula for CSA involves the radius and the slant height. We must first calculate the slant height using the given radius and perpendicular height.


Step 2: Key Formula or Approach:

1. The slant height (\(l\)) of a cone is related to its radius (\(r\)) and height (\(h\)) by the Pythagorean theorem: \(l = \sqrt{r^2 + h^2}\).

2. The Curved Surface Area (CSA) of a cone is given by the formula: \(CSA = \pi r l\).


Step 3: Detailed Explanation:

We are given:

Radius, \(r = 7\) cm

Height, \(h = 24\) cm


First, we calculate the slant height (\(l\)): \[ l = \sqrt{r^2 + h^2} \] \[ l = \sqrt{7^2 + 24^2} \] \[ l = \sqrt{49 + 576} \] \[ l = \sqrt{625} \] \[ l = 25 cm \]
Now that we have the slant height, we can calculate the Curved Surface Area using \(\pi = \frac{22}{7}\). \[ CSA = \pi r l \] \[ CSA = \frac{22}{7} \times 7 \times 25 \]
The 7 in the numerator and denominator cancels out. \[ CSA = 22 \times 25 \] \[ CSA = 550 cm^2 \]

Step 4: Final Answer:

The curved surface area of the cone is 550 cm\(^2\).
Quick Tip: The triplet (7, 24, 25) is a Pythagorean triplet. Recognizing this allows you to find the slant height of 25 cm instantly without calculating \(\sqrt{625}\), saving valuable time in an exam.

Step 1: Understanding the Concept:

The area swept by the minute hand of a clock forms a sector of a circle. To find this area, we need to determine the central angle of the sector created in the given time and use the formula for the area of a sector. The length of the minute hand will be the radius of this circle.


Step 2: Key Formula or Approach:

1. Find the angle swept by the minute hand. The minute hand completes a full circle (360\(^\circ\)) in 60 minutes.

Angle swept in 1 minute = \(\frac{360^\circ}{60} = 6^\circ\).

2. The formula for the area of a sector is:
\[ Area = \frac{\theta}{360^\circ} \times \pi r^2 \]
where \(\theta\) is the central angle and \(r\) is the radius.


Step 3: Detailed Explanation:

Given:

Radius (length of minute hand), \(r = 7\) cm.

Time = 40 minutes.


First, calculate the angle \(\theta\) swept in 40 minutes: \[ \theta = 40 minutes \times 6^\circ/minute = 240^\circ \]
Now, calculate the area of the sector using the formula with \(\pi = \frac{22}{7}\): \[ Area = \frac{240}{360} \times \frac{22}{7} \times (7)^2 \]
Simplify the fraction: \[ \frac{240}{360} = \frac{24}{36} = \frac{2}{3} \]
Substitute this back into the area formula: \[ Area = \frac{2}{3} \times \frac{22}{7} \times 49 \] \[ Area = \frac{2}{3} \times 22 \times 7 \] \[ Area = \frac{308}{3} cm^2 \]
This can also be written as \(102.67\) cm\(^2\).


Step 4: Final Answer:

The area swept by the minute hand in 40 minutes is \(\frac{308}{3}\) cm\(^2\).
Quick Tip: Remember the speeds of the clock hands: the minute hand moves 6\(^\circ\) per minute, and the hour hand moves 0.5\(^\circ\) per minute. These are useful for various clock-related problems.

Step 1: Understanding the Concept:

This problem requires the use of trigonometric identities, specifically the relationship between trigonometric functions of complementary angles (angles that sum to 90\(^\circ\)).


Step 2: Key Formula or Approach:

The key identities are:
1. \(\tan(90^\circ - \theta) = \cot\theta\)
2. \(\tan\theta \cdot \cot\theta = 1\)
We also need the standard value of \(\tan 60^\circ = \sqrt{3}\).


Step 3: Detailed Explanation:

Let's start with the Left Hand Side (LHS) of the equation: \[ LHS = \tan 7^\circ \cdot \tan 60^\circ \cdot \tan 83^\circ \]
Notice that \(7^\circ + 83^\circ = 90^\circ\). This means they are complementary angles. We can rewrite \(\tan 83^\circ\) using the complementary angle identity: \[ \tan 83^\circ = \tan(90^\circ - 7^\circ) = \cot 7^\circ \]
Now substitute this back into the LHS expression: \[ LHS = \tan 7^\circ \cdot \tan 60^\circ \cdot \cot 7^\circ \]
Rearrange the terms to group the complementary functions together: \[ LHS = (\tan 7^\circ \cdot \cot 7^\circ) \cdot \tan 60^\circ \]
Using the identity \(\tan\theta \cdot \cot\theta = 1\): \[ LHS = (1) \cdot \tan 60^\circ \]
Now, substitute the known value of \(\tan 60^\circ\): \[ LHS = 1 \cdot \sqrt{3} = \sqrt{3} \]
This is equal to the Right Hand Side (RHS).


Step 4: Final Answer:

Since LHS = RHS, the identity is proved.
Quick Tip: In trigonometric products or sums, always check for pairs of angles that add up to 90\(^\circ\) or 180\(^\circ\). This is a strong hint to use complementary or supplementary angle identities.

Step 1: Understanding the Concept:

This is a proof by contradiction. We start by assuming the opposite of what we want to prove. If this assumption leads to a logical contradiction, then our original statement must be true. The proof relies on the known fact that \(\sqrt{3}\) is an irrational number.


Step 2: Detailed Explanation:

Assumption: Let us assume, to the contrary, that \(5 - \sqrt{3}\) is a rational number.

By the definition of a rational number, it can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). \[ 5 - \sqrt{3} = \frac{p}{q} \]
Now, we rearrange the equation to isolate the irrational part (\(\sqrt{3}\)): \[ 5 - \frac{p}{q} = \sqrt{3} \] \[ \sqrt{3} = \frac{5q - p}{q} \]
Contradiction:

Since \(p\) and \(q\) are integers, \(5q\) is an integer and \(5q - p\) is also an integer. Also, \(q \neq 0\).

Therefore, \(\frac{5q - p}{q}\) is a rational number.

This implies that \(\sqrt{3}\) is a rational number.

However, this contradicts the well-established fact that \(\sqrt{3}\) is an irrational number.

This contradiction arises because our initial assumption that \(5 - \sqrt{3}\) is rational was incorrect.


Step 3: Final Answer:

Therefore, we must conclude that \(5 - \sqrt{3}\) is an irrational number. Hence proved.
Quick Tip: The key to this type of proof is to isolate the known irrational root (like \(\sqrt{2}\), \(\sqrt{3}\), etc.) on one side of the equation. The other side will be an expression of rational numbers, which is itself rational, leading to the necessary contradiction.

Step 1: Understanding the Concept:

If three points are collinear, they lie on the same straight line. This means the slope between any two pairs of these points must be the same. Another method is to use the fact that the area of a triangle formed by three collinear points is zero. We will use the slope method.


Step 2: Key Formula or Approach:

Let the points be A(1, 1), B(3, k), and C(-1, 4).

If the points are collinear, then Slope of AB = Slope of BC.

The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).


Step 3: Detailed Explanation:

Calculate the slope of the line segment AB: \[ m_{AB} = \frac{k - 1}{3 - 1} = \frac{k - 1}{2} \]
Calculate the slope of the line segment BC: \[ m_{BC} = \frac{4 - k}{-1 - 3} = \frac{4 - k}{-4} \]
Set the slopes equal to each other: \[ \frac{k - 1}{2} = \frac{4 - k}{-4} \]
Cross-multiply to solve for k: \[ -4(k - 1) = 2(4 - k) \] \[ -4k + 4 = 8 - 2k \]
Move the terms with k to one side and the constant terms to the other: \[ 4 - 8 = -2k + 4k \] \[ -4 = 2k \] \[ k = \frac{-4}{2} = -2 \]

Step 4: Final Answer:

The value of k for which the points are collinear is -2.
Quick Tip: Using the area of a triangle formula for collinearity (\(Area = 0\)) is also a reliable method. The formula is \(\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| = 0\). It avoids division and can sometimes be quicker.

Step 1: Understanding the Concept:

Any point on the y-axis has its x-coordinate equal to zero. The problem requires us to find a point P(0, y) that is equidistant from two given points A(6, 5) and B(-4, 3). This means the distance PA must be equal to the distance PB.


Step 2: Key Formula or Approach:

We will use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\), which is \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
To simplify calculations, we can work with the squares of the distances: \(PA^2 = PB^2\).


Step 3: Detailed Explanation:

Let the point on the y-axis be P(0, y).

Let the given points be A(6, 5) and B(-4, 3).

The condition is PA = PB, which implies \(PA^2 = PB^2\).


Using the distance formula for \(PA^2\): \[ PA^2 = (6 - 0)^2 + (5 - y)^2 = 6^2 + (5 - y)^2 = 36 + 25 - 10y + y^2 = 61 - 10y + y^2 \]
Using the distance formula for \(PB^2\): \[ PB^2 = (-4 - 0)^2 + (3 - y)^2 = (-4)^2 + (3 - y)^2 = 16 + 9 - 6y + y^2 = 25 - 6y + y^2 \]
Now, set \(PA^2 = PB^2\): \[ 61 - 10y + y^2 = 25 - 6y + y^2 \]
The \(y^2\) terms on both sides cancel out. \[ 61 - 10y = 25 - 6y \]
Rearrange the equation to solve for y: \[ 61 - 25 = -6y + 10y \] \[ 36 = 4y \] \[ y = \frac{36}{4} = 9 \]
The point on the y-axis is (0, 9).


Step 4: Final Answer:

The required point on the y-axis is (0, 9).
Quick Tip: The locus of points equidistant from two fixed points (A and B) is the perpendicular bisector of the line segment AB. This question is asking for the intersection of the y-axis and the perpendicular bisector of the segment joining (6,5) and (-4,3).

Step 1: Understanding the Concept:

This problem involves a right-angled triangle formed by the ladder, the wall, and the ground. We need to use trigonometric ratios to find the height on the wall. It is crucial to correctly identify which angle is given.


Step 2: Key Formula or Approach:

Let's visualize the setup. The wall is vertical, the ground is horizontal, and the ladder leans against the wall, forming the hypotenuse.
Let \(h\) be the height on the wall.
The length of the ladder is the hypotenuse = 7 m.
The angle given is between the ladder and the wall, which is \(30^\circ\).
In the right-angled triangle, the height \(h\) is the side adjacent to the \(30^\circ\) angle.
The appropriate trigonometric ratio is cosine: \(\cos\theta = \frac{Adjacent}{Hypotenuse}\).


Step 3: Detailed Explanation:

Using the cosine ratio: \[ \cos(30^\circ) = \frac{h}{7} \]
We know the standard value \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\). \[ \frac{\sqrt{3}}{2} = \frac{h}{7} \]
Solve for \(h\): \[ h = 7 \times \frac{\sqrt{3}}{2} = \frac{7\sqrt{3}}{2} m \]

Alternative Method:
The angle between the ladder and the ground would be \(90^\circ - 30^\circ = 60^\circ\).
In this case, the height \(h\) is the side opposite to the \(60^\circ\) angle.
Using the sine ratio: \(\sin\theta = \frac{Opposite}{Hypotenuse}\). \[ \sin(60^\circ) = \frac{h}{7} \]
We know \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). \[ \frac{\sqrt{3}}{2} = \frac{h}{7} \implies h = \frac{7\sqrt{3}}{2} m \]
Both methods yield the same result.


Step 4: Final Answer:

The height of the point on the wall is \(\frac{7\sqrt{3}}{2}\) m.
Quick Tip: Always draw a diagram for problems involving heights and distances. Carefully label the sides and angles. Pay close attention to whether the angle is an angle of elevation (with the horizontal) or an angle given with a vertical object.

Step 1: Understanding the Concept:

To prove that two triangles are similar, we can use the Angle-Angle (AA) similarity criterion. This involves finding two pairs of corresponding angles that are equal. We will use the properties of parallelograms, such as opposite angles being equal and opposite sides being parallel.


Step 2: Detailed Explanation:

Given: ABCD is a parallelogram. E is a point on the extension of side AD. BE intersects CD at F.

To Prove: \(\triangle ABE \sim \triangle CFB\).


Proof:

In \(\triangle ABE\) and \(\triangle CFB\):

Angle A and Angle C:
In a parallelogram, opposite angles are equal.
Therefore, \(\angle DAB = \angle BCD\).
This means \(\angle EAB = \angle FCB\). So we have one pair of equal angles.


Alternate Interior Angles:
Since ABCD is a parallelogram, AD is parallel to BC.
As E lies on the extension of AD, the entire line AE is parallel to BC (\(AE \parallel BC\)).
Now, consider BE as a transversal line intersecting these two parallel lines.
The alternate interior angles formed are equal.
Therefore, \(\angle AEB = \angle CBF\). So we have a second pair of equal angles.


Since two pairs of corresponding angles of \(\triangle ABE\) and \(\triangle CFB\) are equal, the third pair must also be equal (\(\angle ABE = \angle CFB\)).

By the AA similarity criterion, \(\triangle ABE \sim \triangle CFB\).


Step 3: Final Answer:

Hence, it is proved that \(\triangle ABE\) is similar to \(\triangle CFB\).
Quick Tip: When dealing with geometry proofs involving parallelograms, always list the key properties: opposite sides are parallel, opposite sides are equal, opposite angles are equal, and diagonals bisect each other. These properties are the tools you will use to find equal angles or sides.

Step 1: Understanding the Concept:

This is a direct application of the Pythagorean theorem to an isosceles right-angled triangle. We need to use the properties of both types of triangles to establish the relationship.


Step 2: Detailed Explanation:

Given: \(\triangle ABC\) is an isosceles right triangle, with the right angle at C (\(\angle C = 90^\circ\)).

Since the triangle is isosceles and the right angle is at C, the sides adjacent to the right angle must be equal.
Therefore, \(AC = BC\).

AB is the hypotenuse.


By the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. \[ AB^2 = AC^2 + BC^2 \]
Now, substitute the condition for an isosceles triangle (\(BC = AC\)) into the Pythagorean theorem: \[ AB^2 = AC^2 + (AC)^2 \] \[ AB^2 = 2AC^2 \]

Step 3: Final Answer:

Hence, it is proved that \(AB^2 = 2AC^2\).
Quick Tip: This result is a standard property of isosceles right triangles (also known as 45-45-90 triangles). The ratio of the sides is \(x : x : x\sqrt{2}\). Squaring the hypotenuse gives \((x\sqrt{2})^2 = 2x^2\), which is twice the square of a leg.

Step 1: Understanding the Concept:

This problem requires the use of the section formula, which gives the coordinates of a point that divides a line segment into a given ratio.


Step 2: Key Formula or Approach:

The section formula for a point \(P(x, y)\) that divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m_1:m_2\) internally is: \[ x = \frac{m_1x_2 + m_2x_1}{m_1 + m_2} \quad and \quad y = \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \]

Step 3: Detailed Explanation:

Let the given points be \(A(x_1, y_1) = (-1, 7)\) and \(B(x_2, y_2) = (4, -3)\).

The ratio is \(m_1:m_2 = 2:3\).

So, \(m_1 = 2\) and \(m_2 = 3\).


Calculate the x-coordinate: \[ x = \frac{(2)(4) + (3)(-1)}{2 + 3} = \frac{8 - 3}{5} = \frac{5}{5} = 1 \]
Calculate the y-coordinate: \[ y = \frac{(2)(-3) + (3)(7)}{2 + 3} = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \]
The coordinates of the point are (1, 3).


Step 4: Final Answer:

The coordinates of the point that divides the line segment are (1, 3).
Quick Tip: To avoid confusion, always label your points as \((x_1, y_1)\) and \((x_2, y_2)\) and the ratio as \(m_1\) and \(m_2\). Be careful to match \(m_1\) with the coordinates of the second point \((x_2, y_2)\) and \(m_2\) with the coordinates of the first point \((x_1, y_1)\) in the formula.

Step 1: Understanding the Concept:

To find the area of a triangle when the coordinates of its three vertices are given, we use the coordinate geometry formula for the area of a triangle.


Step 2: Key Formula or Approach:

The area of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by the formula: \[ Area = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \]

Step 3: Detailed Explanation:

Let the given vertices be: \( (x_1, y_1) = (-5, -1) \)
\( (x_2, y_2) = (3, -5) \)
\( (x_3, y_3) = (5, 2) \)


Substitute these values into the area formula: \[ Area = \frac{1}{2} |(-5)(-5 - 2) + 3(2 - (-1)) + 5(-1 - (-5))| \]
Simplify the terms inside the absolute value: \[ Area = \frac{1}{2} |(-5)(-7) + 3(2 + 1) + 5(-1 + 5)| \] \[ Area = \frac{1}{2} |35 + 3(3) + 5(4)| \] \[ Area = \frac{1}{2} |35 + 9 + 20| \] \[ Area = \frac{1}{2} |64| \] \[ Area = \frac{1}{2} \times 64 \] \[ Area = 32 \]

Step 4: Final Answer:

The area of the triangle is 32 square units.
Quick Tip: To avoid mistakes with signs, calculate each part of the formula \(x_1(y_2 - y_3)\), \(x_2(y_3 - y_1)\), and \(x_3(y_1 - y_2)\) separately before adding them together. The absolute value at the end ensures that the area is always a positive quantity.

Step 1: Understanding the Concept:

This problem connects three properties of a cube: its side length (a), its space diagonal (d), and its total surface area (TSA). We need to use the formula for the diagonal to find the side length first, and then use the side length to find the TSA.


Step 2: Key Formula or Approach:

1. The length of the space diagonal of a cube with side length \(a\) is given by \(d = a\sqrt{3}\).

2. The Total Surface Area (TSA) of a cube is the sum of the areas of its six square faces, given by \(TSA = 6a^2\).


Step 3: Detailed Explanation:

Given:
Diagonal of the cube, \(d = 9\sqrt{3}\) cm.


First, find the side length \(a\) of the cube using the diagonal formula: \[ a\sqrt{3} = 9\sqrt{3} \]
Divide both sides by \(\sqrt{3}\): \[ a = 9 cm \]
Now that we have the side length, we can calculate the Total Surface Area: \[ TSA = 6a^2 \] \[ TSA = 6 \times (9)^2 \] \[ TSA = 6 \times 81 \] \[ TSA = 486 cm^2 \]

Step 4: Final Answer:

The total surface area of the cube is 486 cm\(^2\).
Quick Tip: Remember the difference between a face diagonal and a space diagonal of a cube. A face diagonal has length \(a\sqrt{2}\), while a space diagonal (connecting opposite vertices) has length \(a\sqrt{3}\). Read the question carefully to know which one is given.

Step 1: Understanding the Concept:

The quadratic formula is a general solution for finding the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\).


Step 2: Key Formula or Approach:

The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The term \(D = b^2 - 4ac\) is the discriminant, which tells us the nature of the roots.


Step 3: Detailed Explanation:

The given equation is \(2x^2 - 2\sqrt{2}x + 1 = 0\).

Comparing this to the standard form \(ax^2 + bx + c = 0\), we have: \(a = 2\), \(b = -2\sqrt{2}\), \(c = 1\).


First, let's calculate the discriminant \(D\): \[ D = b^2 - 4ac = (-2\sqrt{2})^2 - 4(2)(1) \] \[ D = (4 \times 2) - 8 = 8 - 8 = 0 \]
Since the discriminant is 0, the equation has two real and equal roots.


Now, apply the quadratic formula: \[ x = \frac{-(-2\sqrt{2}) \pm \sqrt{0}}{2(2)} \] \[ x = \frac{2\sqrt{2}}{4} \]
Simplify the fraction: \[ x = \frac{\sqrt{2}}{2} \]
Since the roots are equal, both roots are \(\frac{\sqrt{2}}{2}\).


Step 4: Final Answer:

The roots of the equation are \(x = \frac{\sqrt{2}}{2}\) and \(x = \frac{\sqrt{2}}{2}\).
Quick Tip: When the discriminant \(D=0\), the quadratic expression is a perfect square. In this case, \(2x^2 - 2\sqrt{2}x + 1 = 0\) can be written as \((\sqrt{2}x - 1)^2 = 0\), which directly gives \(\sqrt{2}x = 1\) or \(x = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\).

Step 1: Understanding the Concept:

The given series is an Arithmetic Progression (A.P.) because the difference between consecutive terms is constant. To find the sum, we first need to determine the number of terms in the series and then use the formula for the sum of an A.P.


Step 2: Key Formula or Approach:

1. To find the number of terms (\(n\)), use the formula for the n-th term of an A.P.: \(a_n = a + (n-1)d\).

2. To find the sum (\(S_n\)), use the formula: \(S_n = \frac{n}{2}(a + l)\), where \(a\) is the first term and \(l\) is the last term.


Step 3: Detailed Explanation:

From the series \(3, 11, 19, \dots, 67\):

First term, \(a = 3\).

Last term, \(l = a_n = 67\).

Common difference, \(d = 11 - 3 = 8\).


First, find the number of terms, \(n\): \[ 67 = 3 + (n-1)8 \] \[ 64 = (n-1)8 \] \[ 8 = n-1 \] \[ n = 9 \]
There are 9 terms in the series.


Now, find the sum of these 9 terms: \[ S_9 = \frac{9}{2}(3 + 67) \] \[ S_9 = \frac{9}{2}(70) \] \[ S_9 = 9 \times 35 \] \[ S_9 = 315 \]

Step 4: Final Answer:

The sum of the series is 315.
Quick Tip: The formula \(S_n = \frac{n}{2}(a + l)\) is generally faster than \(S_n = \frac{n}{2}[2a + (n-1)d]\) if the last term is known. Always check if the last term is given before starting calculations.

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) is determined by its first term (\(a\)) and common difference (\(d\)). We are given two terms of the A.P., which allows us to set up a system of two linear equations in \(a\) and \(d\). Solving this system will give us the required values.


Step 2: Key Formula or Approach:

The formula for the n-th term of an A.P. is \(a_n = a + (n-1)d\).


Step 3: Detailed Explanation:

Given:
The 5th term is 43: \(a_5 = a + (5-1)d = a + 4d = 43\) ---(Equation 1)

The 9th term is 79: \(a_9 = a + (9-1)d = a + 8d = 79\) ---(Equation 2)


To solve this system, we can subtract Equation 1 from Equation 2: \[ (a + 8d) - (a + 4d) = 79 - 43 \] \[ 4d = 36 \] \[ d = \frac{36}{4} = 9 \]
Now that we have the common difference (\(d=9\)), substitute it back into Equation 1 to find the first term (\(a\)): \[ a + 4(9) = 43 \] \[ a + 36 = 43 \] \[ a = 43 - 36 = 7 \]
The first term is \(a = 7\) and the common difference is \(d = 9\).

The A.P. is formed by starting with \(a\) and repeatedly adding \(d\).
The A.P. is: 7, (7+9), (16+9), ...

The A.P. is: 7, 16, 25, 34, ...


Step 4: Final Answer:

The required Arithmetic Progression is 7, 16, 25, 34, ...
Quick Tip: A quick way to find the common difference \(d\) when given two terms \(a_m\) and \(a_n\) is to use the formula \(d = \frac{a_m - a_n}{m - n}\). Here, \(d = \frac{79 - 43}{9 - 5} = \frac{36}{4} = 9\).

Step 1: Understanding the Concept:

To solve a system of linear equations graphically, we need to draw the graph for each equation. The point where the two lines intersect is the solution to the system. To draw a line, we need at least two points that satisfy the equation.


Step 2: Finding Points for Each Line:

For the first equation: \(x + 3y - 6 = 0\)

We can rewrite it as \(x = 6 - 3y\). Let's find two points:

If \(y = 0\), then \(x = 6 - 3(0) = 6\). So, one point is A(6, 0).
If \(y = 2\), then \(x = 6 - 3(2) = 0\). So, another point is B(0, 2).


For the second equation: \(2x - 3y - 12 = 0\)

We can rewrite it as \(2x = 12 + 3y\). Let's find two points:

If \(y = 0\), then \(2x = 12 + 3(0) \implies 2x=12 \implies x=6\). So, one point is C(6, 0).
If \(y = -2\), then \(2x = 12 + 3(-2) \implies 2x=6 \implies x=3\). So, another point is D(3, -2).


Step 3: Graphing and Solving:

To draw the graph:

Plot the points A(6, 0) and B(0, 2) on a coordinate plane and draw a straight line passing through them. This line represents the equation \(x + 3y - 6 = 0\).
Plot the points C(6, 0) and D(3, -2) on the same coordinate plane and draw a straight line passing through them. This line represents the equation \(2x - 3y - 12 = 0\).
Observe the point where the two lines intersect. Both lines pass through the point (6, 0).


Step 4: Final Answer:

The point of intersection of the two lines is (6, 0). Therefore, the solution to the pair of linear equations is \(x = 6\) and \(y = 0\).
Quick Tip: To easily find points for graphing, set x=0 to find the y-intercept and then set y=0 to find the x-intercept. These two points are often the simplest to calculate and plot.

Step 1: Understanding the Concept:

This is a proof of the Side-Angle-Side (SAS) similarity criterion for triangles. We need to prove that if two triangles have a pair of equal corresponding angles and the sides including these angles are in proportion, then the triangles are similar.


Step 2: Statement and Given Information:

Let the two triangles be \(\triangle ABC\) and \(\triangle DEF\).

Given: \(\angle A = \angle D\) and \(\frac{AB}{DE} = \frac{AC}{DF}\).

To Prove: \(\triangle ABC \sim \triangle DEF\).


Step 3: Construction and Proof:

Construction: On the side DE, cut a segment \(DP = AB\), and on the side DF, cut a segment \(DQ = AC\). Join PQ.


Proof:

In \(\triangle ABC\) and \(\triangle DPQ\):

\(AB = DP\) (By construction)
\(\angle A = \angle D\) (Given)
\(AC = DQ\) (By construction)

Therefore, by SAS congruence rule, \(\triangle ABC \cong \triangle DPQ\).

This implies that \(\angle B = \angle DPQ\) and \(\angle C = \angle DQP\).


Now, it is given that \(\frac{AB}{DE} = \frac{AC}{DF}\).

Substituting \(AB = DP\) and \(AC = DQ\), we get: \[ \frac{DP}{DE} = \frac{DQ}{DF} \]
By the converse of the Basic Proportionality Theorem (Thales's Theorem), this condition implies that \(PQ \parallel EF\).


Since \(PQ \parallel EF\), the corresponding angles are equal: \(\angle DPQ = \angle E\) and \(\angle DQP = \angle F\).


From our earlier findings, we know \(\angle B = \angle DPQ\) and \(\angle C = \angle DQP\).

Therefore, we can conclude that \(\angle B = \angle E\) and \(\angle C = \angle F\).


Now, in \(\triangle ABC\) and \(\triangle DEF\):

\(\angle A = \angle D\) (Given)
\(\angle B = \angle E\) (Proved above)
\(\angle C = \angle F\) (Proved above)

Since all three corresponding angles are equal, by AAA similarity criterion, \(\triangle ABC \sim \triangle DEF\).


Step 4: Final Answer:

Hence, it is proved that the two triangles are similar.
Quick Tip: The key to this proof is the construction step. By creating a triangle (\(\triangle DPQ\)) that is congruent to \(\triangle ABC\), you can transfer the properties of \(\triangle ABC\) into \(\triangle DEF\) and use theorems related to parallel lines to complete the proof.

Step 1: Understanding the Concept:

We need to translate the given word problem into a system of two algebraic equations with two variables, representing the tens and units digits of the number.


Step 2: Setting up the Equations:

Let the tens digit be \(t\) and the units digit be \(u\).

The value of the two-digit number can be represented as \(10t + u\).

The sum of the digits is \(t + u\).

The product of the digits is \(t \cdot u\).


From the problem statement, we get two conditions:

Condition 1: The number is four times the sum of its digits. \[ 10t + u = 4(t + u) \]
Condition 2: The number is twice the product of its digits. \[ 10t + u = 2(t \cdot u) \]

Step 3: Solving the System of Equations:

Let's first simplify Equation 1: \[ 10t + u = 4t + 4u \] \[ 6t = 3u \] \[ u = 2t \]
Now, substitute this expression for \(u\) into Equation 2: \[ 10t + (2t) = 2(t \cdot (2t)) \] \[ 12t = 2(2t^2) \] \[ 12t = 4t^2 \]
Rearrange into a quadratic form: \[ 4t^2 - 12t = 0 \] \[ 4t(t - 3) = 0 \]
This gives two possible solutions for \(t\): \(t=0\) or \(t=3\).

Since it is a two-digit number, the tens digit \(t\) cannot be 0. So, we must have \(t = 3\).

Now, find the units digit \(u\): \[ u = 2t = 2(3) = 6 \]
The digits are \(t=3\) and \(u=6\). The number is \(10(3) + 6 = 36\).


Step 4: Final Answer and Verification:

The number is 36.

Check:
Sum of digits = 3 + 6 = 9. Four times the sum = \(4 \times 9 = 36\). (Condition 1 is met)
Product of digits = 3 \(\times\) 6 = 18. Twice the product = \(2 \times 18 = 36\). (Condition 2 is met)
The answer is correct.
Quick Tip: Always remember to represent a two-digit number with digits 't' and 'u' as \(10t + u\). A common mistake is to write it as \(tu\), which implies multiplication.

Step 1: Understanding the Concept:

This is a geometric construction problem that uses the concept of similar triangles (based on the Basic Proportionality Theorem) to divide a line segment into a given ratio. We also need to calculate the theoretical lengths of the parts to verify the measurement.


Step 2: Steps of Construction:


Draw a line segment AB of length 7.6 cm using a ruler.
Draw a ray AX from point A, making any acute angle with AB (e.g., \(\angle BAX\)).
The given ratio is 5:8. The sum of the ratio parts is \(5+8=13\). Using a compass, mark 13 equidistant points on the ray AX, starting from A. Label them \(A_1, A_2, \dots, A_{13}\) such that \(AA_1 = A_1A_2 = \dots = A_{12}A_{13}\).
Join the last point, \(A_{13}\), to point B to form the line segment \(A_{13}B\).
From the 5th point on the ray, \(A_5\), draw a line parallel to \(A_{13}B\). This can be done by constructing an angle at \(A_5\) equal to \(\angle AA_{13}B\). This parallel line intersects the original line segment AB at a point C.
The point C divides the line segment AB in the ratio 5:8. That is, \(AC:CB = 5:8\).


Step 3: Measurement and Calculation:

By Measurement:

Using a ruler, measure the lengths of AC and CB. You will find that AC is approximately 2.9 cm and CB is approximately 4.7 cm.


By Calculation:

Total length = 7.6 cm.

Total number of parts in the ratio = 5 + 8 = 13.

Length of the first part (AC) = \(\left(\frac{5}{13}\right) \times 7.6 = \frac{38}{13} \approx 2.923\) cm.

Length of the second part (CB) = \(\left(\frac{8}{13}\right) \times 7.6 = \frac{60.8}{13} \approx 4.677\) cm.


Step 4: Final Answer:

The line segment is divided at point C. On measuring, the two parts AC and CB are found to be approximately 2.9 cm and 4.7 cm, which is consistent with the calculated values of 2.92 cm and 4.68 cm.
Quick Tip: The accuracy of the geometric construction depends on the precision with which the parallel line is drawn. Using a set square or constructing equal corresponding angles carefully will give a better result.

Step 1: Understanding the Concept:

To prove this trigonometric identity, we will simplify the Left Hand Side (LHS) by rationalizing the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.


Step 2: Key Formula or Approach:

We will use the algebraic identity \((a+b)(a-b) = a^2 - b^2\) and the Pythagorean trigonometric identities:

\(\sec^2\theta - \tan^2\theta = 1\)
\(\sec^2\theta = 1 + \tan^2\theta\)


Step 3: Detailed Explanation:

Starting with the LHS: \[ LHS = \frac{\sec\theta - \tan\theta}{\sec\theta + \tan\theta} \]
Multiply the numerator and the denominator by the conjugate of the denominator, which is \((\sec\theta - \tan\theta)\): \[ LHS = \frac{(\sec\theta - \tan\theta) \times (\sec\theta - \tan\theta)}{(\sec\theta + \tan\theta) \times (\sec\theta - \tan\theta)} \]
The numerator becomes \((\sec\theta - \tan\theta)^2\) and the denominator becomes \((\sec^2\theta - \tan^2\theta)\). \[ LHS = \frac{(\sec\theta - \tan\theta)^2}{\sec^2\theta - \tan^2\theta} \]
Using the identity \(\sec^2\theta - \tan^2\theta = 1\), the denominator simplifies to 1. \[ LHS = (\sec\theta - \tan\theta)^2 \]
Now, expand the square using \((a-b)^2 = a^2 - 2ab + b^2\): \[ LHS = \sec^2\theta - 2\sec\theta\tan\theta + \tan^2\theta \]
To match the RHS, we need to express everything in terms of \(\tan\theta\). Use the identity \(\sec^2\theta = 1 + \tan^2\theta\): \[ LHS = (1 + \tan^2\theta) - 2\sec\theta\tan\theta + \tan^2\theta \]
Combine the \(\tan^2\theta\) terms: \[ LHS = 1 + 2\tan^2\theta - 2\sec\theta\tan\theta \]
This is equal to the Right Hand Side (RHS).


Step 4: Final Answer:

Since LHS = RHS, the identity is proved.
Quick Tip: When you see an expression like \(\frac{A-B}{A+B}\) in a trigonometric proof, a good first step is often to multiply the numerator and denominator by the conjugate, either \(A-B\) or \(A+B\), to simplify the denominator using a Pythagorean identity.

Step 1: Understanding the Concept:

This problem involves using the formula for the circumference of a circle and setting up an equation based on the given condition.


Step 2: Key Formula or Approach:

The formula for the circumference (\(C\)) of a circle with radius (\(r\)) is: \[ C = 2\pi r \]

Step 3: Detailed Explanation:

Let the radii of the two given circles be \(r_1\) and \(r_2\).
\(r_1 = 19\) cm
\(r_2 = 9\) cm


Let their respective circumferences be \(C_1\) and \(C_2\).
\(C_1 = 2\pi r_1 = 2\pi(19) = 38\pi\) cm.
\(C_2 = 2\pi r_2 = 2\pi(9) = 18\pi\) cm.


Let the radius of the new circle be \(R\) and its circumference be \(C\).

The problem states that the circumference of the new circle is equal to the sum of the circumferences of the two smaller circles. \[ C = C_1 + C_2 \]
Substitute the formulas: \[ 2\pi R = 2\pi r_1 + 2\pi r_2 \]
We can factor out \(2\pi\) from the right side: \[ 2\pi R = 2\pi (r_1 + r_2) \]
Divide both sides by \(2\pi\): \[ R = r_1 + r_2 \]
Now, substitute the given values of the radii: \[ R = 19 + 9 = 28 cm \]

Step 4: Final Answer:

The radius of the new circle is 28 cm.
Quick Tip: Notice that when circumferences are added, the resulting radius is simply the sum of the individual radii. However, this is not true for areas. If areas are added (\( \pi R^2 = \pi r_1^2 + \pi r_2^2 \)), the relationship is \(R^2 = r_1^2 + r_2^2\).

Step 1: Understanding the Concept:

To find the mean of a grouped frequency distribution, we use the direct method. This involves finding the midpoint (or class mark) of each class interval, multiplying it by the corresponding frequency, summing up these products, and finally dividing by the total frequency.


Step 2: Key Formula or Approach:

The formula for the mean (\(\bar{x}\)) is: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \]
where \(f_i\) is the frequency of the i-th class and \(x_i\) is the class mark of the i-th class.
The class mark \(x_i\) is calculated as \(\frac{Upper limit + Lower limit}{2}\).


Step 3: Calculation Table:

We can organize the calculations in a table:

\begin{tabular{|c|c|c|c|
\hline
Class-interval & Frequency (\(f_i\)) & Class Mark (\(x_i\)) & \(f_i x_i\)

\hline
11-13 & 7 & \((11+13)/2 = 12\) & \(7 \times 12 = 84\)

13-15 & 6 & \((13+15)/2 = 14\) & \(6 \times 14 = 84\)

15-17 & 9 & \((15+17)/2 = 16\) & \(9 \times 16 = 144\)

17-19 & 13 & \((17+19)/2 = 18\) & \(13 \times 18 = 234\)

19-21 & 20 & \((19+21)/2 = 20\) & \(20 \times 20 = 400\)

21-23 & 5 & \((21+23)/2 = 22\) & \(5 \times 22 = 110\)

23-25 & 4 & \((23+25)/2 = 24\) & \(4 \times 24 = 96\)

\hline
Total & \(\sum f_i = 64\) & & \(\sum f_i x_i = 1152\)

\hline
\end{tabular


Step 4: Calculating the Mean:

Sum of all frequencies, \(\sum f_i = 64\).

Sum of the products, \(\sum f_i x_i = 1152\).

Now, apply the formula for the mean: \[ \bar{x} = \frac{1152}{64} \] \[ \bar{x} = 18 \]

Step 5: Final Answer:

The mean of the given distribution is 18.
Quick Tip: For distributions where the class marks and frequencies are large, the 'Assumed Mean Method' or 'Step-Deviation Method' can simplify calculations. However, for manageable numbers like in this problem, the direct method is efficient and straightforward.

Step 1: Understanding the Concept:

A frustum of a cone is the portion of a cone left after its top has been cut off by a plane parallel to the base. We need to find its curved surface area (CSA) using the given slant height and the circumferences of its circular top and bottom bases.


Step 2: Key Formula or Approach:

The standard formula for the Curved Surface Area of a frustum is \(CSA = \pi(r_1 + r_2)l\), where \(r_1\) and \(r_2\) are the radii of the two circular ends and \(l\) is the slant height.
A more direct formula using the circumferences (\(C_1\) and \(C_2\)) can be derived and used. Since \(C = 2\pi r\), we have \(r = C/(2\pi)\).
So, \(r_1+r_2 = \frac{C_1}{2\pi} + \frac{C_2}{2\pi} = \frac{C_1+C_2}{2\pi}\).
Substituting this into the CSA formula gives: \[ CSA = \pi \left( \frac{C_1 + C_2}{2\pi} \right) l = \frac{1}{2}(C_1 + C_2)l \]
This formula allows us to calculate the CSA directly from the given circumferences and slant height.


Step 3: Detailed Explanation:

We are given:
Slant height, \(l = 4\) cm.

Circumference of the larger end, \(C_1 = 18\) cm.

Circumference of the smaller end, \(C_2 = 6\) cm.


Using the direct formula for CSA: \[ CSA = \frac{1}{2}(C_1 + C_2)l \]
Substitute the given values: \[ CSA = \frac{1}{2}(18 + 6) \times 4 \] \[ CSA = \frac{1}{2}(24) \times 4 \] \[ CSA = 12 \times 4 \] \[ CSA = 48 cm^2 \]

Step 4: Final Answer:

The curved surface area of the frustum is 48 cm\(^2\).
Quick Tip: Whenever a problem gives circumferences instead of radii for a frustum, using the formula \(CSA = \frac{1}{2}(C_1 + C_2)l\) is much faster than calculating the radii first and then using the standard formula.