CUET Mathematics Question Paper 2025 with Answer Key

Step 1: Understanding the Concept:

This question tests the knowledge of basic definitions related to matrices, specifically symmetric, skew-symmetric, singular, and non-singular matrices.


Step 3: Detailed Explanation:

Let's analyze each item in List-I and match it with the correct definition in List-II.


(A) \(A^{T}\) = A: This is the definition of a symmetric matrix. A matrix is symmetric if it is equal to its transpose. This matches with (IV).

(B) \(A^{T}\) = -A: This is the definition of a skew-symmetric matrix. A matrix is skew-symmetric if its transpose is equal to its negative. This matches with (III).

(C) |A| = 0: The determinant of a matrix being zero is the condition for the matrix to be a singular matrix. This matches with (I).

(D) |A| \(\neq\) 0: The determinant of a matrix being non-zero is the condition for the matrix to be a non-singular matrix. Such matrices have an inverse. This matches with (II).



Step 4: Final Answer:

Combining the matches, we get:

(A) \(\rightarrow\) (IV)

(B) \(\rightarrow\) (III)

(C) \(\rightarrow\) (I)

(D) \(\rightarrow\) (II)

This combination corresponds to option (2).
Quick Tip: Memorize the fundamental definitions of matrix types. The relationship between the determinant and singularity is a crucial concept in linear algebra. \(|A| = 0 \Leftrightarrow\) Singular, \(|A| \neq 0 \Leftrightarrow\) Non-singular.

Step 1: Understanding the Concept:
This question requires the multiplication of two 2x2 matrices.

Step 2: Key Formula or Approach:

Step 3: Detailed Explanation:

Let's calculate the product of the given matrices A and B.

If B = I, then the product AB would be:

This result matches option (B).

Step 4: Final Answer:

B should be the identity matrix, the product AB equals A, which is option (B).
Quick Tip: In multiple-choice questions, if your calculated answer is not among the options, double-check your calculations. If the calculation is correct, consider common typos in the question (e.g., a matrix intended to be an identity or zero matrix).

Step 1: Understanding the Concept:

This problem involves simplifying a matrix polynomial expression using the properties of matrix algebra and a given condition \( (A^{2} = I)\). Since a matrix A and the identity matrix I commute (AI = IA = A), we can use standard binomial expansion formulas.


Step 2: Key Formula or Approach:

We use the binomial expansion formulas:
\((x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3\)
\((x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\)


Step 3: Detailed Explanation:

Let's expand the terms \((A - I)^3\) and \((A + I)^3\).
\[ (A - I)^3 = A^3 - 3A^2I + 3AI^2 - I^3 \]
Since \(I^n = I\) and \(A^2 = I\), this becomes:
\[ (A - I)^3 = A^3 - 3A^2 + 3A - I \]
Now, let's expand the second term:
\[ (A + I)^3 = A^3 + 3A^2I + 3AI^2 + I^3 = A^3 + 3A^2 + 3A + I \]
Now add the two expansions:
\[ (A - I)^3 + (A + I)^3 = (A^3 - 3A^2 + 3A - I) + (A^3 + 3A^2 + 3A + I) \] \[ = 2A^3 + 6A \]
We are given that \(A^2 = I\). Let's find an expression for \(A^3\).
\[ A^3 = A^2 \cdot A = I \cdot A = A \]
Substitute \(A^3 = A\) back into the expression:
\[ (A - I)^3 + (A + I)^3 = 2(A) + 6A = 8A \]
Finally, subtract the last term from the original question:
\[ (A - I)^3 + (A + I)^3 - 3A = 8A - 3A = 5A \]

Step 4: Final Answer:

The value of the expression is 5A.
Quick Tip: When dealing with matrix polynomials involving the identity matrix I, remember that I behaves like the number 1 in scalar algebra. It commutes with any matrix, and \(I^n = I\). This allows the use of standard algebraic formulas like the binomial theorem.

Step 1: Understanding the Concept:
This question tests the property relating the determinant of the adjugate of a matrix to the determinant of the matrix itself.

Step 2: Key Formula or Approach:

For any square matrix A of order n, the determinant of its adjugate is given by the formula:
\[ |adj(A)| = |A|^{n-1} \]

Step 3: Detailed Explanation:

The given matrix A is a 3x3 matrix, so n = 3. The formula becomes:
\[ |adj(A)| = |A|^{3-1} = |A|^2 \]
First, we need to calculate the determinant of A, |A|.

We can expand the determinant along the first row:

Now, we can find \(|adj(A)|\) using the formula:
\[ |adj(A)| = |A|^2 = (-3\sqrt{3})^2 \] \[ |adj(A)| = (-3)^2 \cdot (\sqrt{3})^2 = 9 \cdot 3 = 27 \]

Step 4: Final Answer:

The value of \(|adj(A)|\) is 27.
Quick Tip: For determinants of matrices with many zeros, always expand along the row or column containing the most zeros to simplify the calculation. Also, remember the key property \(|adj(A)| = |A|^{n-1}\), which is frequently tested.

Step 1: Understanding the Concept:

This question involves finding the first and second derivatives of a function and then substituting them into a given expression to simplify it.


Step 2: Key Formula or Approach:

The key differentiation rule needed is for the exponential function:
\[ \frac{d}{dx}(e^{ax}) = a e^{ax} \]

Step 3: Detailed Explanation:

The given function is \(y = 3e^{2x} + 2e^{3x}\).

First, let's find the first derivative, \(\frac{dy}{dx}\).
\[ \frac{dy}{dx} = \frac{d}{dx}(3e^{2x} + 2e^{3x}) = 3(2e^{2x}) + 2(3e^{3x}) \] \[ \frac{dy}{dx} = 6e^{2x} + 6e^{3x} \]
Next, let's find the second derivative, \(\frac{d^2y}{dx^2}\).
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(6e^{2x} + 6e^{3x}) = 6(2e^{2x}) + 6(3e^{3x}) \] \[ \frac{d^2y}{dx^2} = 12e^{2x} + 18e^{3x} \]
Now, we need to evaluate the expression \(\frac{d^2y}{dx^2} + 6y\).

Substitute the expressions for \(\frac{d^2y}{dx^2}\) and \(y\):
\[ \frac{d^2y}{dx^2} + 6y = (12e^{2x} + 18e^{3x}) + 6(3e^{2x} + 2e^{3x}) \] \[ = 12e^{2x} + 18e^{3x} + 18e^{2x} + 12e^{3x} \]
Group the like terms:
\[ = (12 + 18)e^{2x} + (18 + 12)e^{3x} \] \[ = 30e^{2x} + 30e^{3x} \]
Now, we compare this result with the options, which are in terms of \(\frac{dy}{dx}\).

Let's factor out 5 from our result:
\[ 30e^{2x} + 30e^{3x} = 5(6e^{2x} + 6e^{3x}) \]
We recognize that the term in the parenthesis is exactly \(\frac{dy}{dx}\).
\[ \frac{d^2y}{dx^2} + 6y = 5\left(\frac{dy}{dx}\right) \]

Step 4: Final Answer:

The expression \(\frac{d^2y}{dx^2} + 6y\) is equal to \(5\frac{dy}{dx}\).
Quick Tip: For problems of this type, which relate a function to its derivatives, recognize the structure of a linear homogeneous differential equation. The function \(y = C_1e^{r_1x} + C_2e^{r_2x}\) is the solution to \(y'' - (r_1+r_2)y' + r_1r_2y = 0\). Here, \(r_1=2\) and \(r_2=3\), so it's a solution to \(y'' - 5y' + 6y = 0\). From this, we can directly see that \(y'' + 6y = 5y'\).

Step 1: Understanding the Concept:

A function is increasing on an interval where its first derivative is positive, i.e., f'(x) > 0.


Step 2: Key Formula or Approach:

We need to find the first derivative of f(x) using the product rule: \((uv)' = u'v + uv'\).

Then, we find the intervals where f'(x) > 0.


Step 3: Detailed Explanation:

The given function is \(f(x) = x^2e^{-x}\).

Differentiating f(x) with respect to x using the product rule, where \(u = x^2\) and \(v = e^{-x}\):
\[ f'(x) = \frac{d}{dx}(x^2) \cdot e^{-x} + x^2 \cdot \frac{d}{dx}(e^{-x}) \] \[ f'(x) = (2x)e^{-x} + x^2(-e^{-x}) \] \[ f'(x) = e^{-x}(2x - x^2) \] \[ f'(x) = x(2-x)e^{-x} \]
For the function to be increasing, we must have f'(x) > 0.
\[ x(2-x)e^{-x} > 0 \]
Since \(e^{-x}\) is always positive for all real x, the sign of f'(x) depends on the term \(x(2-x)\).

So we need to solve the inequality:
\[ x(2-x) > 0 \]
The critical points are x = 0 and x = 2. We can analyze the sign in the intervals determined by these points:


For x \(<\) 0: \(x\) is negative, \((2-x)\) is positive. Product is negative.
For 0 \(<\) x \(<\) 2: \(x\) is positive, \((2-x)\) is positive. Product is positive.
For x \(>\) 2: \(x\) is positive, \((2-x)\) is negative. Product is negative.

The derivative f'(x) is positive when 0 \(<\) x \(<\) 2.


Step 4: Final Answer:

The function f(x) is increasing on the interval (0, 2).
Quick Tip: To find intervals of increasing/decreasing, always find the first derivative, set it to zero to find critical points, and then test the sign of the derivative in the intervals between these points. Remember that \(e^z\) is always positive.

Step 1: Understanding the Concept:

To find the maximum of a function, we find its critical points by setting the first derivative to zero. The second derivative test can confirm if a critical point is a maximum. We then evaluate the required expression.


Step 2: Key Formula or Approach:

Use the quotient rule for differentiation: \((\frac{u}{v})' = \frac{u'v - uv'}{v^2}\).

For a maximum at x = a, we must have f'(a) = 0 and f''(a) < 0.


Step 3: Detailed Explanation:

Given function: \(f(x) = \frac{\ln x}{x}\).

First Derivative:

Using the quotient rule with \(u = \ln x\) and \(v = x\):
\[ f'(x) = \frac{(\frac{1}{x}) \cdot x - (\ln x) \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \]
To find the maximum, set f'(x) = 0:
\[ \frac{1 - \ln x}{x^2} = 0 \implies 1 - \ln x = 0 \implies \ln x = 1 \implies x = e \]
So, the maximum occurs at \(a = e\).

Second Derivative:

Now, differentiate f'(x) using the quotient rule with \(u = 1 - \ln x\) and \(v = x^2\):
\[ f''(x) = \frac{(-\frac{1}{x}) \cdot x^2 - (1 - \ln x) \cdot (2x)}{(x^2)^2} \] \[ f''(x) = \frac{-x - 2x + 2x \ln x}{x^4} = \frac{-3x + 2x \ln x}{x^4} = \frac{2 \ln x - 3}{x^3} \]
Now evaluate f''(a) = f''(e):
\[ f''(e) = \frac{2 \ln e - 3}{e^3} = \frac{2(1) - 3}{e^3} = \frac{-1}{e^3} \]
Calculate the final expression:

We need to find \(a^2 f''(a)\). With \(a = e\):
\[ a^2 f''(a) = e^2 \cdot \left(\frac{-1}{e^3}\right) = \frac{-e^2}{e^3} = -\frac{1}{e} \]

Step 4: Final Answer:

The value of \(a^{2}f''\)(a) is \(-\frac{1}{e}\).
Quick Tip: When finding maxima/minima, always find the first derivative and set it to zero. For complex expressions, be systematic with differentiation rules (product, quotient, chain). Double-check the final expression the question asks for.

Step 1: Understanding the Concept:

This problem requires evaluating a definite integral involving an absolute value function. The key is to split the integral at the point where the expression inside the absolute value changes sign.


Step 2: Key Formula or Approach:

The definition of the absolute value function is:

We use the property of definite integrals: \(\int_{a}^{c} f(x)dx = \int_{a}^{b} f(x)dx + \int_{b}^{c} f(x)dx\).


Step 3: Detailed Explanation:

The expression inside the absolute value is \(x - 2\). This expression is zero when x = 2.


When \(x < 2\), \(x - 2\) is negative, so \(|x - 2| = -(x - 2) = 2 - x\).
When \(x \ge 2\), \(x - 2\) is non-negative, so \(|x - 2| = x - 2\).

Since the point x = 2 is within our integration interval [1, 4], we split the integral at x = 2:
\[ \int_{1}^{4} |x - 2| dx = \int_{1}^{2} |x - 2| dx + \int_{2}^{4} |x - 2| dx \]
Now substitute the appropriate expression for \(|x - 2|\) in each integral:
\[ = \int_{1}^{2} (2 - x) dx + \int_{2}^{4} (x - 2) dx \]
Evaluate the first integral:
\[ \int_{1}^{2} (2 - x) dx = \left[ 2x - \frac{x^2}{2} \right]_{1}^{2} = \left(2(2) - \frac{2^2}{2}\right) - \left(2(1) - \frac{1^2}{2}\right) \] \[ = (4 - 2) - \left(2 - \frac{1}{2}\right) = 2 - \frac{3}{2} = \frac{1}{2} \]
Evaluate the second integral:
\[ \int_{2}^{4} (x - 2) dx = \left[ \frac{x^2}{2} - 2x \right]_{2}^{4} = \left(\frac{4^2}{2} - 2(4)\right) - \left(\frac{2^2}{2} - 2(2)\right) \] \[ = \left(\frac{16}{2} - 8\right) - \left(\frac{4}{2} - 4\right) = (8 - 8) - (2 - 4) = 0 - (-2) = 2 \]
Add the results of the two integrals:
\[ Total Value = \frac{1}{2} + 2 = \frac{5}{2} \]

Step 4: Final Answer:

The value of the integral is \(\frac{5}{2}\).
Quick Tip: Geometrically, the integral of \(|x-2|\) from 1 to 4 represents the area of two triangles. The first has vertices at (1,1), (2,0), (2,1) with area \(\frac{1}{2} \times 1 \times 1 = \frac{1}{2}\). The second has vertices at (2,0), (4,2), (4,0) with area \(\frac{1}{2} \times 2 \times 2 = 2\). The total area is \(\frac{1}{2} + 2 = \frac{5}{2}\).

Step 1: Understanding the Concept:

This problem requires simplifying the integrand using logarithmic properties before performing the integration.


Step 2: Key Formula or Approach:

The key property to use is \(e^{a \ln x} = e^{\ln(x^a)} = x^a\).

After simplification, we use the power rule for integration: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\).


Step 3: Detailed Explanation:

First, simplify each term in the integrand using the property \(e^{a\ln x} = x^a\).


\(e^{5\log_e x} = x^5\)
\(e^{4\log_e x} = x^4\)
\(e^{3\log_e x} = x^3\)
\(e^{2\log_e x} = x^2\)

Substitute these back into the integral:
\[ I = \int \frac{x^5 - x^4}{x^3 - x^2} dx \]
Now, factor the numerator and the denominator:
\[ I = \int \frac{x^4(x - 1)}{x^2(x - 1)} dx \]
Assuming \(x \neq 1\) and \(x \neq 0\), we can cancel the common factor \((x-1)\):
\[ I = \int \frac{x^4}{x^2} dx \] \[ I = \int x^2 dx \]
Now, apply the power rule for integration:
\[ I = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C \]

Step 4: Final Answer:

The integral is equal to \(\frac{x^3}{3} + C\).
Quick Tip: Whenever you see expressions like \(e^{\ln f(x)}\), immediately simplify them to \(f(x)\). This is a common technique to make complex-looking integrals much simpler. Always look for opportunities to simplify by factoring.

Step 1: Understanding the Concept:

We need to find the area of a region enclosed by a curve and a line. This is a classic application of definite integration. The curve \(y^2 = 4x\) is a parabola opening to the right with its vertex at the origin.


Step 2: Key Formula or Approach:

The area of a region bounded by a curve \(y = f(x)\), the x-axis, and the lines \(x = a\) and \(x = b\) is given by \(\int_{a}^{b} f(x) dx\). Since the parabola is symmetric about the x-axis, we can find the area in the first quadrant and multiply it by 2.


Step 3: Detailed Explanation:

The given parabola is \(y^2 = 4x\). For the upper half of the parabola (in the first quadrant), we have \(y = \sqrt{4x} = 2\sqrt{x}\).

The region is bounded by \(x = 0\) (the y-axis, where the parabola starts) and \(x = 1\).

The area in the first quadrant is given by the integral:
\[ A_1 = \int_{0}^{1} y \, dx = \int_{0}^{1} 2\sqrt{x} \, dx \] \[ A_1 = 2 \int_{0}^{1} x^{1/2} \, dx \]
Using the power rule for integration:
\[ A_1 = 2 \left[ \frac{x^{1/2 + 1}}{1/2 + 1} \right]_{0}^{1} = 2 \left[ \frac{x^{3/2}}{3/2} \right]_{0}^{1} = 2 \left[ \frac{2}{3}x^{3/2} \right]_{0}^{1} \] \[ A_1 = \frac{4}{3} [x^{3/2}]_{0}^{1} = \frac{4}{3} (1^{3/2} - 0^{3/2}) = \frac{4}{3}(1 - 0) = \frac{4}{3} \]
This is the area of the region above the x-axis. Due to symmetry, the area of the region below the x-axis is also \(\frac{4}{3}\).

The total area is:
\[ A_{total} = 2 \times A_1 = 2 \times \frac{4}{3} = \frac{8}{3} \]

Step 4: Final Answer:

The total area of the region is \(\frac{8}{3}\) sq. units.
Quick Tip: Always sketch the curves to visualize the region. For curves symmetric about an axis (like \(y^2 = 4ax\)), calculating the area of one half and doubling it is often simpler and less prone to errors.

Step 1: Understanding the Concept:

A first-order differential equation is called linear if it can be expressed in the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\) or \(\frac{dx}{dy} + P(y)x = Q(y)\). In this form, the dependent variable (y in the first case, x in the second) and its derivative appear only to the first power and are not multiplied together.


Step 3: Detailed Explanation:

Let's analyze each equation:


(A) \(\frac{dy}{dx} + P(x)y = Q(x)\): This is the very definition of a linear first-order differential equation with y as the dependent variable. So, (A) is linear.

(B) \(\frac{dx}{dy} + P(y)x = Q(y)\): This is the standard form of a linear first-order differential equation with x as the dependent variable. So, (B) is linear.

(C) \((x - y)\frac{dy}{dx} = x + 2y\): We can rewrite this as \(\frac{dy}{dx} = \frac{x+2y}{x-y}\). This equation cannot be arranged into either of the standard linear forms. The terms involve products of y and \(\frac{dy}{dx}\), and it's a homogeneous equation, not linear.

(D) \((1 + x^2)\frac{dy}{dx} + 2xy = 2\): To check if this is linear, we try to put it in the standard form. Divide the entire equation by \((1 + x^2)\):

\[ \frac{dy}{dx} + \frac{2x}{1 + x^2}y = \frac{2}{1 + x^2} \]
This equation is exactly in the form \(\frac{dy}{dx} + P(x)y = Q(x)\), where \(P(x) = \frac{2x}{1 + x^2}\) and \(Q(x) = \frac{2}{1 + x^2}\). Therefore, (D) is a linear differential equation.



Step 4: Final Answer:

The equations (A), (B), and (D) are linear first-order differential equations. This corresponds to option (1).
Quick Tip: To test for linearity, always try to rearrange the equation into one of the two standard forms. If the dependent variable or its derivative appears with a power other than one, or in a non-linear function (like sin(y)), or are multiplied together, the equation is non-linear.

Step 1: Understanding the Concept:

This is a first-order differential equation that can be solved using the method of separation of variables.


Step 2: Key Formula or Approach:

The method involves rearranging the equation so that all terms involving y and dy are on one side, and all terms involving x and dx are on the other. Then, integrate both sides.


Step 3: Detailed Explanation:

The given differential equation is:
\[ \log_e\left(\frac{dy}{dx}\right) = 3x + 4y \]
To remove the logarithm, we take the exponential of both sides:
\[ \frac{dy}{dx} = e^{3x + 4y} \]
Using the property of exponents \(e^{a+b} = e^a \cdot e^b\), we can write:
\[ \frac{dy}{dx} = e^{3x} \cdot e^{4y} \]
Now, we separate the variables by moving all y-terms to the left side and all x-terms to the right side:
\[ \frac{dy}{e^{4y}} = e^{3x} dx \] \[ e^{-4y} dy = e^{3x} dx \]
Integrate both sides of the equation:
\[ \int e^{-4y} dy = \int e^{3x} dx \]
Performing the integration:
\[ \frac{e^{-4y}}{-4} = \frac{e^{3x}}{3} + C_1 \]
where \(C_1\) is the constant of integration. To match the form of the options, we can rearrange the terms. Move the x-term to the left side:
\[ -\frac{e^{-4y}}{4} - \frac{e^{3x}}{3} = C_1 \]
Multiply the entire equation by -12 to eliminate the fractions:
\[ (-12) \left(-\frac{e^{-4y}}{4}\right) - (-12) \left(\frac{e^{3x}}{3}\right) = -12 C_1 \] \[ 3e^{-4y} + 4e^{3x} = -12 C_1 \]
Let \(C = -12 C_1\). Since \(C_1\) is an arbitrary constant, C is also an arbitrary constant.
\[ 4e^{3x} + 3e^{-4y} = C \]
This can be written as:
\[ 4e^{3x} + 3e^{-4y} - C = 0 \]
Since C is an arbitrary constant, we can replace -C with a new constant, which we can also call C, giving the final form:
\[ 4e^{3x} + 3e^{-4y} + C = 0 \]

Step 4: Final Answer:

The solution to the differential equation is \(4e^{3x} + 3e^{-4y} + C = 0\).
Quick Tip: When solving differential equations, remember that the constant of integration C is arbitrary. An expression like `... = C` is equivalent to `... - C = 0` or `... + C = 0`, as the sign and magnitude of an arbitrary constant are also arbitrary. Match your final form to the given options.

Step 1: Understanding the Concept:

For any probability distribution, the sum of all probabilities for all possible values of the random variable must be equal to 1. That is, \(\sum P(X=x_i) = 1\).


Step 2: Key Formula or Approach:

1. Set up an equation by summing the probabilities and equating to 1.

2. Solve the equation for the unknown parameter 'a'.

3. Use the value of 'a' to calculate the required probability.


Step 3: Detailed Explanation:

The sum of the probabilities is:
\[ P(X=0) + P(X=1) + P(X=2) = 1 \] \[ (1 - 7a^2) + \left(\frac{1}{2}a + \frac{1}{4}\right) + (a^2) = 1 \]
Simplify the equation:
\[ 1 - 6a^2 + \frac{1}{2}a + \frac{1}{4} = 1 \]
Subtract 1 from both sides:
\[ -6a^2 + \frac{1}{2}a + \frac{1}{4} = 0 \]
To eliminate fractions, multiply the entire equation by 4:
\[ -24a^2 + 2a + 1 = 0 \]
Multiply by -1 to make the leading coefficient positive:
\[ 24a^2 - 2a - 1 = 0 \]
Solve this quadratic equation for 'a'. We can factor it:
\[ 24a^2 - 6a + 4a - 1 = 0 \] \[ 6a(4a - 1) + 1(4a - 1) = 0 \] \[ (6a + 1)(4a - 1) = 0 \]
This gives two possible values for a: \(a = -\frac{1}{6}\) or \(a = \frac{1}{4}\).

The problem states that \(a > 0\), so we must choose \(a = \frac{1}{4}\).

Now, we need to find \(P(0 < X \le 2)\). This is the sum of probabilities for X = 1 and X = 2.
\[ P(0 < X \le 2) = P(X=1) + P(X=2) \] \[ P(0 < X \le 2) = \left(\frac{1}{2}a + \frac{1}{4}\right) + (a^2) \]
Substitute \(a = \frac{1}{4}\) into this expression:
\[ P(0 < X \le 2) = \left(\frac{1}{2}\left(\frac{1}{4}\right) + \frac{1}{4}\right) + \left(\frac{1}{4}\right)^2 \] \[ = \left(\frac{1}{8} + \frac{1}{4}\right) + \frac{1}{16} \] \[ = \left(\frac{1}{8} + \frac{2}{8}\right) + \frac{1}{16} = \frac{3}{8} + \frac{1}{16} \] \[ = \frac{6}{16} + \frac{1}{16} = \frac{7}{16} \]

Step 4: Final Answer:

The value of P(0 < x \(\le\) 2) is \(\frac{7}{16}\).
Quick Tip: The two fundamental properties of a discrete probability distribution are: 1) \(0 \le P(X=x_i) \le 1\) for all \(x_i\), and 2) \(\sum P(X=x_i) = 1\). The second property is almost always the starting point for finding unknown parameters.

Step 1: Understanding the Concept:

In a Linear Programming Problem (LPP), if the optimal (maximum or minimum) value of the objective function occurs at two distinct corner points of the feasible region, then the optimal value also occurs at every point on the line segment joining these two points. This happens when the slope of the objective function is the same as the slope of the constraint line that forms the edge between those two points.


Step 2: Key Formula or Approach:

If the objective function Z has the same optimal value at two points, say \((x_1, y_1)\) and \((x_2, y_2)\), then:
\[ p x_1 + q y_1 = p x_2 + q y_2 \]

Step 3: Detailed Explanation:

The objective function is Z = px + qy.

The problem states that the optimum value occurs at both corner points (3, 4) and (0, 5).

This means the value of Z is the same at these two points.

Value of Z at (3, 4):
\[ Z_1 = p(3) + q(4) = 3p + 4q \]
Value of Z at (0, 5):
\[ Z_2 = p(0) + q(5) = 5q \]
Set the two values equal to each other:
\[ Z_1 = Z_2 \] \[ 3p + 4q = 5q \]
Now, solve for the relationship between p and q:
\[ 3p = 5q - 4q \] \[ 3p = q \]

Step 4: Final Answer:

The relationship between p and q is q = 3p.
Quick Tip: For an objective function Z = ax + by, the slope of the iso-profit/iso-cost line is -a/b. When multiple optimal solutions exist, this slope is equal to the slope of one of the boundary lines of the feasible region. In this case, the line connecting (3,4) and (0,5) has a slope of \((5-4)/(0-3) = -1/3\). The slope of Z = px + qy is -p/q. So, \(-p/q = -1/3\), which gives \(q=3p\).

Step 1: Understanding the Concept:

This is a minimization problem in Linear Programming. The optimal solution (minimum value) for a bounded feasible region occurs at one of the corner points. For an unbounded region, we must evaluate Z at the corner points and then verify if an even smaller value is possible within the region.


Step 2: Key Formula or Approach:

1. Identify the feasible region defined by the constraints.

2. Find the coordinates of the corner points of the feasible region.

3. Evaluate the objective function Z at each corner point.

4. The point that gives the minimum value is the optimal solution. If the minimum value occurs at more than one corner point, all points on the line segment connecting them are optimal solutions.


Step 3: Detailed Explanation:

The LPP is:

Minimize Z = x + 2y

Subject to:

1) \(2x + y \ge 3\)

2) \(x + 2y \ge 6\)

3) \(x \ge 0, y \ge 0\)


First, we find the corner points of the feasible region. The corner points are the intersections of the boundary lines.

Point A (Intersection with y-axis):

Let x = 0. The constraints become \(y \ge 3\) and \(2y \ge 6 \implies y \ge 3\). The intersection is at (0, 3).

Point B (Intersection with x-axis):

Let y = 0. The constraints become \(2x \ge 3 \implies x \ge 1.5\) and \(x \ge 6\). The intersection is at (6, 0).

Point C (Intersection of the lines 2x + y = 3 and x + 2y = 6):

From \(2x + y = 3\), we get \(y = 3 - 2x\). Substitute this into the second equation:
\(x + 2(3 - 2x) = 6\)
\(x + 6 - 4x = 6\)
\(-3x = 0 \implies x = 0\).

If \(x = 0\), then \(y = 3 - 2(0) = 3\). The intersection is (0, 3), which is Point A.


So, the corner points of the unbounded feasible region are (0, 3) and (6, 0).

Now, evaluate the objective function Z = x + 2y at these corner points.


At point (0, 3): \(Z = 0 + 2(3) = 6\).
At point (6, 0): \(Z = 6 + 2(0) = 6\).

The minimum value of Z at the corner points is 6. This value occurs at two different points, (0, 3) and (6, 0).

Since the feasible region is unbounded, we must check if Z can attain a value less than 6. We check the region \(x + 2y < 6\). This half-plane has no points in common with the feasible region, as one of the constraints is \(x + 2y \ge 6\). Therefore, the minimum value is indeed 6.


Step 4: Final Answer:

The minimum value of Z is 6, and it occurs at both corner points (0, 3) and (6, 0). Thus, the optimal feasible solution occurs at both (6, 0) and (0, 3).
Quick Tip: When the slope of the objective function is the same as the slope of a boundary constraint, and that boundary is part of the optimal solution edge, multiple optimal solutions exist. Here, Z = x + 2y has a slope of -1/2. The constraint x + 2y = 6 also has a slope of -1/2. This parallelism is the reason for multiple optimal solutions.

Step 1: Understanding the Concept:

To solve this problem, we need to check if the function f(x) = 10x satisfies the conditions for being one-one (injective) and onto (surjective).


One-one (Injective): A function f is one-one if for any two distinct elements x₁ and x₂ in the domain, their images f(x₁) and f(x₂) are also distinct. Mathematically, if f(x₁) = f(x₂), then x₁ = x₂.

Onto (Surjective): A function f from a set R to a set R is onto if for every element y in the codomain R, there exists at least one element x in the domain R such that f(x) = y. In other words, the range of the function is equal to its codomain.



Step 3: Detailed Explanation:

Checking for One-one:

Let's assume f(x₁) = f(x₂) for some x₁, x₂ \(\in\) R.
\[ 10x_1 = 10x_2 \]
Dividing both sides by 10, we get:
\[ x_1 = x_2 \]
Since f(x₁) = f(x₂) implies x₁ = x₂, the function is one-one.


Checking for Onto:

Let y be an arbitrary element in the codomain R. We need to check if there is an x in the domain R such that f(x) = y.
\[ f(x) = y \] \[ 10x = y \] \[ x = \frac{y}{10} \]
For any real number y, the value \(x = \frac{y}{10}\) is also a real number. Thus, for any y in the codomain, there exists a pre-image x in the domain. Therefore, the function is onto.


Step 4: Final Answer:

Since the function f(x) = 10x is both one-one and onto, option (1) is correct.
Quick Tip: A linear function of the form f(x) = ax + b, where a \(\neq\) 0, defined from R to R, is always both one-one and onto. Visualizing its graph, a straight line, shows that it passes both the horizontal line test (one-one) and covers the entire y-axis (onto).

Step 1: Understanding the Concept:

We need to construct a relation R on the set A = {1, 2, 3} that satisfies four conditions:
1. It contains (1, 2) and (1, 3).
2. It is reflexive.
3. It is symmetric.
4. It is not transitive.
We then need to count how many such unique relations exist.


Step 3: Detailed Explanation:

Let R be the relation we are looking for.

Condition 1 \& 2 (Reflexivity): For R to be reflexive on A, it must contain all pairs (a, a) for a \(\in\) A.

So, R must contain \{(1, 1), (2, 2), (3, 3)\.


Condition 1 \& 3 (Symmetry): The relation must contain (1, 2) and (1, 3). For R to be symmetric, if (a, b) \(\in\) R, then (b, a) must also be in R.

Since (1, 2) \(\in\) R, we must have (2, 1) \(\in\) R.

Since (1, 3) \(\in\) R, we must have (3, 1) \(\in\) R.


Combining these, the smallest possible relation R that satisfies the first three properties is:
\[ R_{min} = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1)\} \]

Condition 4 (Not Transitive): Now we must check if R_min is transitive. A relation is transitive if (a, b) \(\in\) R and (b, c) \(\in\) R implies (a, c) \(\in\) R. Let's look for a counterexample.

Consider (2, 1) \(\in\) R_min and (1, 3) \(\in\) R_min. For transitivity, the pair (2, 3) must be in R_min. However, (2, 3) \(\notin\) R_min.

Similarly, consider (3, 1) \(\in\) R_min and (1, 2) \(\in\) R_min. For transitivity, (3, 2) must be in R_min. But (3, 2) \(\notin\) R_min.

Since we found a case where transitivity fails, R_min is not transitive.


So, R_min is one such relation. Are there any others?

The only pairs not in R_min are (2, 3) and (3, 2). If we add these to R_min, we must add both to maintain symmetry.

Let's define a new relation R' = R_min \(\cup\) \{(2, 3), (3, 2)\.

R' = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2)\. This is the universal relation A \(\times\) A.

Is R' transitive? Let's check the pair that failed before: (2, 1) \(\in\) R' and (1, 3) \(\in\) R'. Is (2, 3) \(\in\) R'? Yes, it is now. The universal relation on a set is always an equivalence relation, meaning it is reflexive, symmetric, and transitive. So R' is transitive.

This means we cannot add the pair \{(2, 3), (3, 2)\ because it makes the relation transitive.


Step 4: Final Answer:

The only relation that satisfies all the given conditions is R_min. Therefore, there is only 1 such relation.
Quick Tip: When asked to find the number of relations with certain properties, start by building the smallest possible relation that includes all the mandatory elements and satisfies the given properties (like reflexivity and symmetry). Then, check if this minimal relation meets the final condition (e.g., not transitive). Finally, see if adding any other allowed elements still keeps the properties valid.

Step 1: Understanding the Concept:

This problem involves finding the value of a trigonometric function of an inverse trigonometric function. A common method is to use a right-angled triangle to represent the inverse trigonometric function and then find the required trigonometric ratio.


Step 2: Key Formula or Approach:

Let \(\theta = \tan^{-1}x\). This implies \(\tan\theta = x\). We need to find \(\sin\theta\).

We can visualize this relationship using a right-angled triangle.


Step 3: Detailed Explanation:

Let \(\theta = \tan^{-1}x\). Then \(\tan\theta = x\).

We can write \(\tan\theta = \frac{x}{1}\).

In a right-angled triangle, \(\tan\theta = \frac{Opposite}{Adjacent}\).

So, we can let the side opposite to angle \(\theta\) be \(x\) and the adjacent side be 1.

Using the Pythagorean theorem, we can find the hypotenuse:
\[ Hypotenuse^2 = Opposite^2 + Adjacent^2 \] \[ Hypotenuse^2 = x^2 + 1^2 = 1 + x^2 \] \[ Hypotenuse = \sqrt{1 + x^2} \]
Now, we need to find \(\sin(\tan^{-1}x)\), which is \(\sin\theta\).

The formula for \(\sin\theta\) is \(\frac{Opposite}{Hypotenuse}\).
\[ \sin\theta = \frac{x}{\sqrt{1+x^2}} \]

Step 4: Final Answer:

Therefore, \(\sin(\tan^{-1}x) = \frac{x}{\sqrt{1+x^2}}\).
Quick Tip: Drawing a right-angled triangle is a very effective and quick method for simplifying expressions involving compositions of trigonometric and inverse trigonometric functions. Always label the sides based on the given inverse function and then use Pythagoras' theorem to find the third side.

Step 1: Understanding the Concept:

We need to calculate specific minors and cofactors of the given 3x3 matrix and check which of the given statements are true.

The minor M_ij of an element a_ij is the determinant of the submatrix formed by deleting the i-th row and j-th column.
The cofactor A_ij is related to the minor by the formula \( A_ij = (-1)^{i+j} M_ij\).


Step 3: Detailed Explanation:

The given matrix is .
Let's evaluate each statement.

(A) M₂₂= -1
M₂₂ is the minor of the element a₂₂ = 3. We delete the 2nd row and 2nd column.

Statement (A) is true.


(B) A₂₃ = 0
A₂₃  is the cofactor of the element a₂₃ = 2. First, find the minor M₂₃ .

Now, calculate the cofactor.
\[ A_{23} = (-1)^{2+3} M_{23} = (-1)^5 (0) = 0 \]
Statement (B) is true.

(C) A₃₂  = 3
A₃₂ is the cofactor of the element a₃₂ = 4. First, find the minor M₃₂.

Now, calculate the cofactor.
\[ A_{32} = (-1)^{3+2} M_{32} = (-1)^5 (1) = -1 \]
Statement (C) is false.

(D) M₃₂ = 1
From our calculation for statement (B), we found that M₃₂ = 0.

Statement (D) is false.


(E) M₃₂ = -3
The OCR here is likely a typo. Based on the options, let's assume it should have been A₃₂ = -1, or something related. From our calculation for statement (C), we found M₃₂ = 1.
Statement (E) is false.

Step 4: Final Answer:
The only true statements are (A) and (B). Therefore, the correct option is (1).
Quick Tip: Be very careful with the sign of the cofactor. The sign is determined by \( (-1)^{i+j}\), which follows a checkerboard pattern of signs: . For A₂₃, the position (2,3) has a '-' sign. For A₃₂, the position (3,2) also has a '-' sign.

Step 1: Understanding the Concept:

We are given a matrix equation involving a matrix A, its transpose \(A^{T}\), and the identity matrix I. We need to solve this equation to find the general value of the angle \(\theta\).


Step 2: Key Formula or Approach:

1. Find the transpose of matrix A, \(A^{T}\).

2. Calculate the sum \(A^{T} + A\).

3. Set this sum equal to the identity matrix I.

4. Solve the resulting trigonometric equation for \(\theta\).


Step 3: Detailed Explanation:

The given matrix is .
First, find the transpose of A, which is obtained by interchanging rows and columns.

Now, add \(A^{T}\) and A.

We are given that \(A^{T} + A = I\).

By equating the corresponding elements of the matrices, we get the equation:
\[ 2\cos\theta = 1 \] \[ \cos\theta = \frac{1}{2} \]
The principal value for \(\theta\) is \(\frac{\pi}{3}\).

The general solution for \(\cos\theta = \cos\alpha\) is \(\theta = 2n\pi \pm \alpha\), where n is an integer.

So, the general solution for \(\cos\theta = \frac{1}{2}\) is \(\theta = 2n\pi \pm \frac{\pi}{3}\), n \(\in\) Z.

Looking at the options provided, the option \(\theta = 2n\pi + \frac{\pi}{3}\) is one part of the general solution. It is the most appropriate choice among the given options.


Step 4: Final Answer:

The value of \(\theta\) is given by \(\theta = 2n\pi + \frac{\pi}{3}, n \in Z\).
Quick Tip: The matrix A is a standard rotation matrix. Knowing its properties can sometimes be helpful. In this case, direct calculation is straightforward. Remember the general solutions for trigonometric equations: for \(\cos x = \cos \alpha\), \(x = 2n\pi \pm \alpha\). Always check if the provided options are a subset of your general solution.

Step 1: Understanding the Concept:
A matrix M is symmetric if M^T= M.

A matrix M is skew-symmetric if M^T = -M.

We are given that A and B are skew-symmetric, so \( A^T = -A and B^T = -B\). We need to check the properties of combinations of these matrices. A useful property is that for a skew-symmetric matrix M, M^k is symmetric if k is even, and skew-symmetric if k is odd.

Step 3: Detailed Explanation:
Let's analyze each option.

1. A³ + B⁵ is skew-symmetric

Let P = A³ + B⁵. Let's find its transpose.
\[ P^T = (A^3 + B^5)^T = (A^3)^T + (B^5)^T = (A^T)^3 + (B^T)^5 \]
Since A and B are skew-symmetric, A^T = -A and B^T = -B.
\[ P^T = (-A)^3 + (-B)^5 = -A^3 - B^5 = -(A^3 + B^5) = -P \]
Since P^T = -P, the matrix is skew-symmetric. This statement is true.

2. A¹⁹ is skew-symmetric
Let Q = A¹⁹. The power 19 is odd.
\[ Q^T = (A^{19})^T = (A^T)^{19} = (-A)^{19} = -A^{19} = -Q \]
The matrix is skew-symmetric. This statement is true.

3. B¹⁴ is symmetric
Let R = B¹⁴. The power 14 is even.
\[ R^T = (B^{14})^T = (B^T)^{14} = (-B)^{14} = B^{14} = R \]
The matrix is symmetric. This statement is true.


4. A³ + B⁵ is symmetric

Let S = A³ + B⁵. Let's find its transpose.
\[ S^T = (A^4 + B^5)^T = (A^4)^T + (B^5)^T = (A^T)^4 + (B^T)^5 \] \[ S^T = (-A)^4 + (-B)^5 = A^4 - B^5 \]
For S to be symmetric, S^T must be equal to S.
\[ A^4 - B^5 = A^4 + B^5 \]
This implies \(-B^5 = B^5\), or \(2B^5 = 0\), which means \(B^5\) must be the zero matrix. This is not true for all skew-symmetric matrices B. Therefore, S is not generally symmetric. This statement is NOT true.


Step 4: Final Answer:

The statement that is not true is A³ + B⁵ is symmetric.
Quick Tip: For a skew-symmetric matrix M: M^{odd power} is skew-symmetric. M^{even power} is symmetric. This rule can help you quickly evaluate the options in such questions.

Step 1: Understanding the Concept:

We need to identify the incorrect statement among the given properties of invertible matrices. An invertible matrix is a square matrix that has a non-zero determinant.

Step 3: Detailed Explanation:
Let's analyze each statement.


1. adjA = |A|A^-1
The formula for the inverse of a matrix A is given by:
\[ A^{-1} = \frac{1}{|A|} adj(A) \]
Multiplying both sides by |A| (which is non-zero since A is invertible), we get:
\[ |A|A^{-1} = adj(A) \]
This statement is correct.

2. (A + B)^-1 = A^-1 + B^-1
This property states that the inverse of a sum is the sum of the inverses. This is generally not true for matrices. We can show this with a counterexample.
Let A = I and B = I (where I is the identity matrix). Both are invertible.
LHS = (A + B)^-1 = (I + I)^-1 = (2I)^-1 = \(\frac{1}{2}\)I^-1= \(\frac{1}{2}\)I = \(\begin{bmatrix} 1/2 & 0
0 & 1/2 \end{bmatrix}\).

RHS = A^-1 + B^-1 = I^-1 + I^-1 = I + I = 2I = \(\begin{bmatrix} 2 & 0
0 & 2 \end{bmatrix}\).

Since LHS \(\neq\) RHS, the statement is NOT correct.

3. |A^-1| = |A|^-1
We know that a matrix and its inverse satisfy the relation AA^-1 = I.
Taking the determinant of both sides:
\[ |AA^{-1}| = |I| \]
Using the property |XY| = |X||Y|, we get:
\[ |A||A^{-1}| = 1 \]
Since A is invertible, |A| \(\neq\) 0. We can divide by |A|:
\[ |A^{-1}| = \frac{1}{|A|} = |A|^{-1} \]
This statement is correct.

4. (AB)^-1 = B^-1A^-1
This is the well-known reversal law for the inverse of a product of matrices. It is a standard property.
This statement is correct.

Step 4: Final Answer:
The statement that is not correct is (A + B)^-1 = A^-1 + B^-1.
Quick Tip: Remember that matrix algebra often differs from scalar algebra. Properties like \((a+b)^{-1} = a^{-1} + b^{-1}\) or \(ab = ba\) do not generally hold for matrices. Be especially skeptical of properties involving addition and inversion/multiplication.

Step 1: Understanding the Concept:

This question involves properties of matrix multiplication and determinants. We need to determine the size of the product matrix AB and then apply the property for the determinant of a scalar multiple of a matrix.


Step 2: Key Formula or Approach:

1. Determine the order (dimensions) of the product matrix AB.
2. Use the property of determinants: For a square matrix M of order n and a scalar k, |kM| = kⁿ|M|.


Step 3: Detailed Explanation:

First, let's find the order of the product matrix AB.

The order of matrix A is 2 x 3.

The order of matrix B is 3 x 2.

For the product AB to be defined, the number of columns in A must equal the number of rows in B. Here, it is 3, so the product is defined.

The order of the resulting matrix AB is (number of rows of A) x (number of columns of B), which is 2 x 2.

So, AB is a square matrix of order n = 2.


Now we need to find the determinant of 5AB. We use the property |kM| = kⁿ|M|.

Here, M = AB, k = 5, and the order n = 2.
\[ |5AB| = 5^2 |AB| = 25|AB| \]

Note that options (1) and (2) are incorrect because the determinant is only defined for square matrices. Since A (2x3) and B (3x2) are not square, |A| and |B| are not defined.


Step 4: Final Answer:

The value of |5AB| is 5² |AB|.
Quick Tip: Remember two crucial rules: 1. The order of a product of matrices (m x n) * (n x p) is (m x p). 2. The determinant of a scalar multiple |kM| is kⁿ|M|, where 'n' is the order of the square matrix M, not necessarily the scalar's exponent in the original expression.

Step 1: Understanding the Concept:

This question tests the conditions for the consistency and nature of solutions for a system of linear equations AX = B, where A is a square matrix. The determinant of A, |A|, plays a crucial role.


Step 3: Detailed Explanation:

Let's analyze the conditions for the system of equations AX = B.


Case 1: |A| \(\neq\) 0 (A is non-singular)

If the determinant of the coefficient matrix is non-zero, the matrix A is invertible. The system has a unique solution given by X = A^-1B.

- Statement (B) says the system has a unique solution if |A| \(\neq\) 0. This is true.

- Statement (A) says the system has a unique solution if |A| = 0. This is false.


Case 2: |A| = 0 (A is singular)

If the determinant is zero, the system may have no solution or infinitely many solutions. To determine which, we calculate (adj A)B. The solution is given by X = A^-1B = \(\frac{(adj A)B}{|A|}\).

If (adj A)B \(\neq\) 0 (the zero vector), then the system is inconsistent and has no solution.
If (adj A)B = 0 (the zero vector), then the system is consistent and has infinitely many solutions.

- Statement (C) says the system has no solution if |A| = 0 and (adj A)B \(\neq\) 0. This is true.

- Statement (D) says the system has infinitely many solutions if |A| = 0 and (adj A)B = 0. This is true.


Step 4: Final Answer:

The correct statements are (B), (C), and (D). Therefore, the correct option is (2).
Quick Tip: A summary of conditions for solving AX=B: If |A| \(\neq\) 0 \(\rightarrow\) Unique solution (consistent). If |A| = 0: Calculate (adj A)B. If (adj A)B \(\neq\) 0 \(\rightarrow\) No solution (inconsistent). If (adj A)B = 0 \(\rightarrow\) Infinitely many solutions (consistent).

Step 1: Understanding the Concept:

For a function to be continuous at a point x = a, the limit of the function as x approaches a must exist and be equal to the value of the function at that point. That is, \(\lim_{x \to a} f(x) = f(a)\).


Step 2: Key Formula or Approach:

We will apply the condition for continuity at x = \(\frac{\pi}{2}\).
\[ \lim_{x \to \frac{\pi}{2}} f(x) = f\left(\frac{\pi}{2}\right) \]
The limit on the left side is an indeterminate form (0/0), so we can evaluate it using L'Hôpital's Rule.


Step 3: Detailed Explanation:

Given the function is continuous at x = \(\frac{\pi}{2}\), we have:
\[ \lim_{x \to \frac{\pi}{2}} \frac{k\cos x}{\pi - 2x} = f\left(\frac{\pi}{2}\right) \]
We are given that \(f(\frac{\pi}{2}) = 3\).

Now we evaluate the limit. As x \(\to \frac{\pi}{2}\), cos(x) \(\to\) 0 and (\(\pi\) - 2x) \(\to\) 0. This is a \(\frac{0}{0}\) indeterminate form.

Applying L'Hôpital's Rule, we differentiate the numerator and the denominator with respect to x:
\[ Derivative of numerator: \frac{d}{dx}(k\cos x) = -k\sin x \] \[ Derivative of denominator: \frac{d}{dx}(\pi - 2x) = -2 \]
So, the limit becomes:
\[ \lim_{x \to \frac{\pi}{2}} \frac{-k\sin x}{-2} = \lim_{x \to \frac{\pi}{2}} \frac{k\sin x}{2} \]
Now, substitute x = \(\frac{\pi}{2}\) into the simplified expression:
\[ \frac{k\sin(\frac{\pi}{2})}{2} = \frac{k(1)}{2} = \frac{k}{2} \]
For continuity, this limit must equal \(f(\frac{\pi}{2})\).
\[ \frac{k}{2} = 3 \] \[ k = 6 \]

Step 4: Final Answer:

The value of k is 6.
Quick Tip: L'Hôpital's Rule is a powerful tool for evaluating limits of indeterminate forms \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). Alternatively, one could use a substitution like \(h = x - \frac{\pi}{2}\) to solve the limit using standard trigonometric limits.

Step 1: Understanding the Concept:

A function of the form f(x) = |g(x)| is generally not differentiable at the points where g(x) = 0, because these points often correspond to sharp corners or "cusps" in the graph. We need to find these points for each function in List-I.


Step 3: Detailed Explanation:

Let's analyze each function in List-I.


(A) f(x) = |x|

The function is not differentiable where the expression inside the absolute value is zero.
\(x = 0\).

So, f(x) = |x| is not differentiable at x = 0 only. This matches with (II).


(B) f(x) = |x + 2|

Set the expression inside the absolute value to zero.
\(x + 2 = 0 \implies x = -2\).

So, f(x) = |x + 2| is not differentiable at x = -2 only. This matches with (I).


(C) f(x) = \(|x^2 - 4|\)

Set the expression inside the absolute value to zero.
\(x^2 - 4 = 0 \implies (x - 2)(x + 2) = 0\).

The solutions are x = 2 and x = -2.

So, f(x) = \(|x^2 - 4|\) is not differentiable at x = 2 and x = -2. This matches with (IV).


(D) f(x) = |x - 2|

Set the expression inside the absolute value to zero.
\(x - 2 = 0 \implies x = 2\).

So, f(x) = |x - 2| is not differentiable at x = 2 only. This matches with (III).


Step 4: Final Answer:

The correct matching is:

(A) \(\rightarrow\) (II)

(B) \(\rightarrow\) (I)

(C) \(\rightarrow\) (IV)

(D) \(\rightarrow\) (III)

This corresponds to option (2).
Quick Tip: To quickly find points of non-differentiability for an absolute value function f(x) = |g(x)|, solve the equation g(x) = 0. The roots of this equation are the candidates for points where the function is not differentiable, especially if g'(x) is not zero at those roots.

Step 1: Understanding the Concept:

We need to find the derivative of a composite function, which requires the use of the chain rule multiple times. After finding the derivative, we will evaluate it at the specified point.

Step 2: Key Formula or Approach:
The chain rule states that if \(y = f(g(h(x)))\), then \(\frac{dy}{dx} = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)\).
We have y = sin(u), u = cos(v), and v = x².

Step 3: Detailed Explanation:
The function is y = sin(cos(x²)).

Let's apply the chain rule to find \(\frac{dy}{dx}\).
\[ \frac{dy}{dx} = \frac{d}{dx} \sin(\cos(x^2)) \] \[ = \cos(\cos(x^2)) \cdot \frac{d}{dx}(\cos(x^2)) \] \[ = \cos(\cos(x^2)) \cdot (-\sin(x^2)) \cdot \frac{d}{dx}(x^2) \] \[ = \cos(\cos(x^2)) \cdot (-\sin(x^2)) \cdot (2x) \] \[ \frac{dy}{dx} = -2x \sin(x^2) \cos(\cos(x^2)) \]
Now, we evaluate this derivative at \(x = \frac{\sqrt{\pi}}{2}\).

First, let's find the values of the terms involving x:
\[ x^2 = \left(\frac{\sqrt{\pi}}{2}\right)^2 = \frac{\pi}{4} \]
So we have:
\[ \sin(x^2) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \] \[ \cos(x^2) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Substitute these values back into the expression for the derivative:
\[ \frac{dy}{dx} \bigg|_{x=\frac{\sqrt{\pi}}{2}} = -2\left(\frac{\sqrt{\pi}}{2}\right) \cdot \sin\left(\frac{\pi}{4}\right) \cdot \cos\left(\cos\left(\frac{\pi}{4}\right)\right) \] \[ = -\sqrt{\pi} \cdot \left(\frac{1}{\sqrt{2}}\right) \cdot \cos\left(\frac{1}{\sqrt{2}}\right) \] \[ = -\frac{\sqrt{\pi}}{\sqrt{2}} \cos\left(\frac{1}{\sqrt{2}}\right) \]
This can also be written as \(-\sqrt{\frac{\pi}{2}} \cos(\frac{1}{\sqrt{2}})\).


Step 4: Final Answer:

The value of the derivative at \(x = \frac{\sqrt{\pi}}{2}\) is \(-\frac{\sqrt{\pi}}{\sqrt{2}} \cos(\frac{1}{\sqrt{2}})\). This matches option (A).
Quick Tip: When differentiating nested functions, apply the chain rule from the outside in. Differentiate the outermost function, keeping the inside intact, then multiply by the derivative of the next function inside, and so on, until you reach the innermost variable. Be careful with signs, especially when differentiating cosine.

Step 1: Understanding the Concept:

We need to find the minimum or maximum value for each function in List-I by analyzing their properties. This involves understanding the range of basic functions like squares, absolute values, and trigonometric functions.


Step 3: Detailed Explanation:


(A) f(x) = \((2x - 1)^2 + 3\):
The term \((2x - 1)^2\) is a square, so its minimum value is 0. This occurs when \(2x - 1 = 0\).

Therefore, the minimum value of the entire function is \(0 + 3 = 3\). This matches with (III).


(B) f(x) = \(-|x + 1| + 4\): (Assuming a typo in the OCR, which likely missed the negative sign before the absolute value).

The term \(|x + 1|\) is always greater than or equal to 0. Its minimum value is 0.

Therefore, the term \(-|x + 1|\) has a maximum value of 0.

The maximum value of the entire function is \(0 + 4 = 4\). This matches with (I).


(C) f(x) = sin(2x) + 6:
The range of the sine function, sin(2x), is [-1, 1].

The minimum value of sin(2x) is -1.

Therefore, the minimum value of the entire function is \(-1 + 6 = 5\). This matches with (IV).


(D) f(x) = \(-(x - 1)^2 + 10\):
The term \((x - 1)^2\) has a minimum value of 0.

Therefore, the term \(-(x - 1)^2\) has a maximum value of 0.

The maximum value of the entire function is \(0 + 10 = 10\). This matches with (II).



Step 4: Final Answer:

The correct matching is (A) - (III), (B) - (I), (C) - (IV), (D) - (II). This corresponds to option (3).
Quick Tip: To find the max/min of simple quadratic and absolute value functions, find the value that makes the squared or absolute part zero. For trigonometric functions, use their known range (e.g., [-1, 1] for sine and cosine).

Step 1: Understanding the Concept:

To determine if a function is increasing or decreasing on an interval, we examine the sign of its first derivative on that interval. If f'(x) > 0, the function is increasing. If f'(x) < 0, it is decreasing.


Step 2: Key Formula or Approach:

1. Find the derivative of f(x), which is f'(x).

2. Analyze the sign of f'(x) in the given interval [0, \(\frac{\pi}{2}\)).


Step 3: Detailed Explanation:

The given function is f(x) = tanx - x.

First, find the derivative f'(x):
\[ f'(x) = \frac{d}{dx}(\tan x - x) = \sec^2 x - 1 \]
We know the trigonometric identity \(1 + \tan^2 x = \sec^2 x\).

Substituting this into the derivative expression:
\[ f'(x) = (1 + \tan^2 x) - 1 = \tan^2 x \]
Now, we need to analyze the sign of f'(x) = \(\tan^2 x\) on the interval [0, \(\frac{\pi}{2}\)).

For any value of x in this interval, tanx is defined. Since \(\tan^2 x\) is a square, its value is always greater than or equal to zero.
\[ f'(x) = \tan^2 x \ge 0 \]
The derivative is zero only at x = 0 (since tan(0) = 0). For all other values of x in (0, \(\frac{\pi}{2}\)), tanx is positive, so \(\tan^2 x\) is strictly positive.

Since f'(x) \(\ge\) 0 on the interval [0, \(\frac{\pi}{2}\)) and is not identically zero over any subinterval, the function f(x) is increasing on this interval.


Step 4: Final Answer:

The function f(x) = tanx - x is an increasing function on [0, \(\frac{\pi}{2}\)).
Quick Tip: When checking for increasing/decreasing behavior, finding the first derivative is the standard method. Remembering basic trigonometric identities is crucial for simplifying the derivative and analyzing its sign.

Step 1: Understanding the Concept:

We are asked to find the derivative of the area (A) with respect to the circumference (C), which is represented as \(\frac{dA}{dC}\). This is a related rates problem where both A and C are functions of the radius (r).


Step 2: Key Formula or Approach:

Let A be the area and C be the circumference of a circle with radius r.
\[ A = \pi r^2 \] \[ C = 2\pi r \]
We can use the chain rule to find \(\frac{dA}{dC}\):
\[ \frac{dA}{dC} = \frac{dA/dr}{dC/dr} \]

Step 3: Detailed Explanation:

First, we find the derivatives of A and C with respect to r.
\[ \frac{dA}{dr} = \frac{d}{dr}(\pi r^2) = 2\pi r \] \[ \frac{dC}{dr} = \frac{d}{dr}(2\pi r) = 2\pi \]
Now, we can find \(\frac{dA}{dC}\) using the chain rule:
\[ \frac{dA}{dC} = \frac{2\pi r}{2\pi} = r \]
The problem asks for this rate of change when the radius is 4 cm.

Substituting r = 4 cm:
\[ \frac{dA}{dC} = 4 cm \]
The units of this rate are area units divided by length units, which is cm²/cm. So the result should just be cm. However, the options provided all have units cm²/cm, which is a bit redundant but guides us to the numerical answer.

The final numerical value is 4.

The answer is 4 cm²/cm.

Step 4: Final Answer:

The rate of change of the area with respect to its circumference is 4 cm²/cm.
Quick Tip: An alternative method is to express A directly as a function of C. From C = 2\( \pi \)r, we get r = C/(2\(\pi\)). Substitute this into the area formula: A = \(\pi\)(C/(2\(\pi\)))\textsuperscript{2} = \(\pi\)(C\textsuperscript{2}/(4\(\pi\)\textsuperscript{2})) = C\textsuperscript{2}/(4\(\pi\)). Now differentiate A with respect to C: dA/dC = 2C/(4\(\pi\)) = C/(2\(\pi\)). Substitute C = 2\(\pi\)r = 2\(\pi\)(4) = 8\(\pi\). So dA/dC = 8\(\pi\)/(2\(\pi\)) = 4.

Step 1: Understanding the Concept:

This definite integral can be solved efficiently using a property of definite integrals, often known as the "King's property".


Step 2: Key Formula or Approach:

The property states that \(\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx\).

Here, \(a = \frac{\pi}{6}\) and \(b = \frac{\pi}{3}\). So, \(a+b = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi + 2\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}\).


Step 3: Detailed Explanation:

Let the given integral be I.
\[ I = \int_{\pi/6}^{\pi/3} \frac{\tan x}{\tan x + \cot x} dx \quad \cdots (1) \]
Using the property \(\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx\), we replace x with \(a+b-x = \frac{\pi}{2}-x\).
\[ I = \int_{\pi/6}^{\pi/3} \frac{\tan(\frac{\pi}{2}-x)}{\tan(\frac{\pi}{2}-x) + \cot(\frac{\pi}{2}-x)} dx \]
We know that \(\tan(\frac{\pi}{2}-x) = \cot x\) and \(\cot(\frac{\pi}{2}-x) = \tan x\).
\[ I = \int_{\pi/6}^{\pi/3} \frac{\cot x}{\cot x + \tan x} dx \quad \cdots (2) \]
Now, add equation (1) and equation (2):
\[ I + I = \int_{\pi/6}^{\pi/3} \frac{\tan x}{\tan x + \cot x} dx + \int_{\pi/6}^{\pi/3} \frac{\cot x}{\cot x + \tan x} dx \] \[ 2I = \int_{\pi/6}^{\pi/3} \frac{\tan x + \cot x}{\tan x + \cot x} dx \] \[ 2I = \int_{\pi/6}^{\pi/3} 1 \cdot dx \] \[ 2I = [x]_{\pi/6}^{\pi/3} \] \[ 2I = \frac{\pi}{3} - \frac{\pi}{6} = \frac{2\pi - \pi}{6} = \frac{\pi}{6} \] \[ I = \frac{\pi}{12} \]

Step 4: Final Answer:

The value of the integral is \(\frac{\pi}{12}\).
Quick Tip: For definite integrals of the form \(\int_{a}^{b} \frac{f(x)}{f(x)+f(a+b-x)} dx\), the result is always \(\frac{b-a}{2}\). Here, \(f(x) = \tan x\), so \(f(a+b-x) = f(\frac{\pi}{2}-x) = \tan(\frac{\pi}{2}-x) = \cot x\). The integral is in this form, so the answer is \((\frac{\pi}{3} - \frac{\pi}{6})/2 = (\frac{\pi}{6})/2 = \frac{\pi}{12}\).

Step 1: Understanding the Concept:

We need to evaluate four different definite integrals and match them with their correct values from List-II. This involves using various integration techniques such as substitution, properties of odd/even functions, and partial fractions.


Step 3: Detailed Explanation:


(A) \(\int_{0}^{1} \frac{2x}{1+x^2} dx\):
Use substitution. Let \(u = 1 + x^2\). Then \(du = 2x dx\).

When \(x=0\), \(u=1\). When \(x=1\), \(u=2\).

The integral becomes \(\int_{1}^{2} \frac{1}{u} du = [\ln|u|]_{1}^{2} = \ln(2) - \ln(1) = \ln(2)\). This matches (III).


(B) \(\int_{-1}^{1} \sin^3x \cos^4x dx\):
Let \(f(x) = \sin^3x \cos^4x\). Check if it's an odd or even function.

\(f(-x) = \sin^3(-x) \cos^4(-x) = (-\sin x)^3 (\cos x)^4 = -\sin^3x \cos^4x = -f(x)\).

Since \(f(-x) = -f(x)\), the function is odd. The integral of an odd function over a symmetric interval \([-a, a]\) is 0. This matches (IV).


(C) \(\int_{0}^{\pi} \sin x dx\):
The antiderivative of \(\sin x\) is \(-\cos x\).

\(\int_{0}^{\pi} \sin x dx = [-\cos x]_{0}^{\pi} = (-\cos(\pi)) - (-\cos(0)) = (-(-1)) - (-1) = 1 + 1 = 2\). This matches (I).


(D) \(\int_{2}^{3} \frac{2}{x^2 - 1} dx\):
Use partial fraction decomposition: \(\frac{2}{x^2 - 1} = \frac{2}{(x-1)(x+1)} = \frac{1}{x-1} - \frac{1}{x+1}\).

The integral is \(\int_{2}^{3} (\frac{1}{x-1} - \frac{1}{x+1}) dx = [\ln|x-1| - \ln|x+1|]_{2}^{3} = [\ln|\frac{x-1}{x+1}|]_{2}^{3}\).

\(= \ln|\frac{3-1}{3+1}| - \ln|\frac{2-1}{2+1}| = \ln|\frac{2}{4}| - \ln|\frac{1}{3}| = \ln(\frac{1}{2}) - \ln(\frac{1}{3}) = \ln(\frac{1/2}{1/3}) = \ln(\frac{3}{2})\). This matches (II).



Step 4: Final Answer:

The correct matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II). This corresponds to option (4).
Quick Tip: Before starting a definite integral, always check for simplifying properties. Is the interval symmetric (like [-a, a])? If so, check if the function is odd (integral is 0) or even. Is it of the form that allows using King's property? These checks can save a lot of calculation time.

Step 1: Understanding the Concept:

This integral is in a special form that matches the property \(\int e^x (f(x) + f'(x)) dx = e^x f(x) + C\). We need to manipulate the integrand to fit this form.


Step 2: Key Formula or Approach:

Identify a function f(x) and its derivative f'(x) within the parentheses.


Step 3: Detailed Explanation:

First, let's rewrite the expression inside the parentheses by splitting the fraction:
\[ \frac{x-1}{3x^2} = \frac{x}{3x^2} - \frac{1}{3x^2} = \frac{1}{3x} - \frac{1}{3x^2} \]
So the integral becomes:
\[ I = \int e^x \left(\frac{1}{3x} - \frac{1}{3x^2}\right) dx \]
Now let's check if this is in the form \(\int e^x (f(x) + f'(x)) dx\).

Let's try setting \(f(x) = \frac{1}{3x} = \frac{1}{3}x^{-1}\).

Now find the derivative of f(x):
\[ f'(x) = \frac{d}{dx}\left(\frac{1}{3}x^{-1}\right) = \frac{1}{3}(-1)x^{-2} = -\frac{1}{3x^2} \]
This matches the second term in the parentheses perfectly.

So, our integral is indeed in the form \(\int e^x (f(x) + f'(x)) dx\) with \(f(x) = \frac{1}{3x}\).

The result of this integration is \(e^x f(x) + C\).
\[ I = e^x \left(\frac{1}{3x}\right) + C = \frac{e^x}{3x} + C \]

Step 4: Final Answer:

The value of the integral is \(\frac{1}{3x}e^x + C\).
Quick Tip: Whenever you see an integral involving a product of \(e^x\) and another function, always try to check if the other function can be expressed as a sum of a function and its derivative, i.e., \(f(x) + f'(x)\). This pattern is very common in competitive exams.

Note: The provided question with y = x⁵yields an answer of 1/3, which is not among the options. It is highly likely that the question intended to ask for y = x³, as handwritten '3' and '5' can look similar, and the correct answer for y = x³ is one of the options. We will proceed with this assumption.


Step 1: Understanding the Concept:

We need to find the area bounded by a curve and the x-axis. Since area is always a non-negative quantity, we must integrate the absolute value of the function, \(|f(x)|\). The function \(y = x^3\) is negative for \(x < 0\) and positive for \(x > 0\).


Step 2: Key Formula or Approach:

The area A is given by the definite integral of \(|x^3|\) from -1 to 1.
\[ A = \int_{-1}^{1} |x^3| dx \]
We must split the integral at x = 0 where the function's sign changes.


Step 3: Detailed Explanation:

Let's split the integral into two parts:
\[ A = \int_{-1}^{0} |x^3| dx + \int_{0}^{1} |x^3| dx \]
For \(x \in [-1, 0)\), \(x^3\) is negative, so \(|x^3| = -x^3\).

For \(x \in [0, 1]\), \(x^3\) is positive, so \(|x^3| = x^3\).
\[ A = \int_{-1}^{0} (-x^3) dx + \int_{0}^{1} (x^3) dx \]
Now, we integrate:
\[ A = \left[ -\frac{x^4}{4} \right]_{-1}^{0} + \left[ \frac{x^4}{4} \right]_{0}^{1} \]
Evaluate the first part:
\[ \left( -\frac{0^4}{4} \right) - \left( -\frac{(-1)^4}{4} \right) = 0 - \left( -\frac{1}{4} \right) = \frac{1}{4} \]
Evaluate the second part:
\[ \left( \frac{1^4}{4} \right) - \left( \frac{0^4}{4} \right) = \frac{1}{4} - 0 = \frac{1}{4} \]
The total area is the sum of the two parts:
\[ A = \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]

Step 4: Final Answer:

Assuming the intended curve was y = x³, the area is \(\frac{1}{2}\) sq. units.
Quick Tip: For functions that are symmetric about the origin (odd functions like \(x^3\), \(x^5\), sin(x)), the area from -a to a is twice the area from 0 to a. So, \(A = 2 \int_{0}^{a} f(x) dx\). This simplifies the calculation. For this problem, \(A = 2 \int_{0}^{1} x^3 dx = 2[\frac{x^4}{4}]_{0}^{1} = 2(\frac{1}{4}) = \frac{1}{2}\).

Step 1: Understanding the Concept:

The given curve \(y = 2\sqrt{1-x^2}\) represents a portion of an ellipse. We can find the area either by setting up a definite integral or by using the geometric formula for the area of an ellipse.


Step 2: Key Formula or Approach:

Method 1: Geometric Interpretation

Rearrange the equation of the curve:
\[ y = 2\sqrt{1-x^2} \implies \frac{y}{2} = \sqrt{1-x^2} \]
Square both sides (assuming y \(\ge\) 0):
\[ \left(\frac{y}{2}\right)^2 = 1 - x^2 \implies x^2 + \frac{y^2}{4} = 1 \]
This is the standard equation of an ellipse centered at the origin with semi-minor axis a = 1 and semi-major axis b = 2. The area of a full ellipse is \(\pi ab\). The given constraints are \(y \ge 0\) (from the positive square root) and \(x \in [0,1]\), which means we are finding the area of the ellipse in the first quadrant.

Method 2: Integration

The area is given by the integral \(A = \int_{0}^{1} 2\sqrt{1-x^2} dx\).


Step 3: Detailed Explanation:

Using Method 1 (Geometry):

The total area of the ellipse \(x^2 + \frac{y^2}{4} = 1\) is \(\pi ab = \pi(1)(2) = 2\pi\).

The required area is the portion in the first quadrant (where x is from 0 to 1 and y is positive). This is exactly one-quarter of the total area of the ellipse.
\[ A = \frac{1}{4} (Total Area) = \frac{1}{4} (2\pi) = \frac{\pi}{2} \]

Using Method 2 (Integration):
\[ A = \int_{0}^{1} 2\sqrt{1-x^2} dx = 2 \int_{0}^{1} \sqrt{1-x^2} dx \]
We use the standard integral formula \(\int \sqrt{a^2-x^2} dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}(\frac{x}{a})\). Here, a = 1.
\[ A = 2 \left[ \frac{x}{2}\sqrt{1-x^2} + \frac{1}{2}\sin^{-1}(x) \right]_{0}^{1} \] \[ A = 2 \left[ \left( \frac{1}{2}\sqrt{1-1^2} + \frac{1}{2}\sin^{-1}(1) \right) - \left( \frac{0}{2}\sqrt{1-0^2} + \frac{1}{2}\sin^{-1}(0) \right) \right] \] \[ A = 2 \left[ \left( 0 + \frac{1}{2} \cdot \frac{\pi}{2} \right) - (0 + 0) \right] \] \[ A = 2 \left[ \frac{\pi}{4} \right] = \frac{\pi}{2} \]

Step 4: Final Answer:

The area of the region is \(\frac{\pi}{2}\) sq. units.
Quick Tip: Recognizing that the equation represents a standard geometric shape (like a circle or ellipse) can be much faster than performing the integration. Always try to identify the curve first before jumping into calculus.

Step 1: Understanding the Concept:

The given differential equation is a linear first-order differential equation. To solve it, we first need to convert it to the standard form \(\frac{dy}{dx} + P(x)y = Q(x)\) and then find the integrating factor (I.F.).


Step 2: Key Formula or Approach:

The standard form of a linear differential equation is \(\frac{dy}{dx} + P(x)y = Q(x)\).

The integrating factor is given by the formula: I.F. = \(e^{\int P(x)dx}\).


Step 3: Detailed Explanation:

The given differential equation is:
\[ (x \log_e x) \frac{dy}{dx} + y = 2\log_e x \]
To convert this to the standard form, divide the entire equation by \((x \log_e x)\):
\[ \frac{dy}{dx} + \frac{1}{x \log_e x} y = \frac{2\log_e x}{x \log_e x} \] \[ \frac{dy}{dx} + \left(\frac{1}{x \log_e x}\right)y = \frac{2}{x} \]
Comparing this with the standard form, we have:
\[ P(x) = \frac{1}{x \log_e x} \]
Now, we calculate the integrating factor:
\[ I.F. = e^{\int P(x)dx} = e^{\int \frac{1}{x \log_e x} dx} \]
To evaluate the integral \(\int \frac{1}{x \log_e x} dx\), we use the substitution method.

Let \(t = \log_e x\). Then, \(dt = \frac{1}{x} dx\).

The integral becomes:
\[ \int \frac{1}{t} dt = \ln|t| = \ln(\log_e x) \]
Now, substitute this back into the formula for the integrating factor:
\[ I.F. = e^{\ln(\log_e x)} \]
Using the property \(e^{\ln u} = u\), we get:
\[ I.F. = \log_e x \]

Step 4: Final Answer:

The integrating factor of the given differential equation is \(\log_e x\).
Quick Tip: When finding the integrating factor, always ensure the differential equation is in the standard form with the coefficient of \(\frac{dy}{dx}\) being 1. The integral in the exponent often requires a substitution, especially when logarithmic or trigonometric functions are involved.

Step 1: Understanding the Concept:

We need to analyze the given differential equation based on its type (linear, homogeneous), find its general solution, and then determine a particular solution given an initial condition.


Step 3: Detailed Explanation:

The given equation is \(x \frac{dy}{dx} = y(\log_e y - \log_e x + 1)\).

Using properties of logarithms, we can rewrite it as:
\[ x \frac{dy}{dx} = y\left(\log_e\left(\frac{y}{x}\right) + 1\right) \] \[ \frac{dy}{dx} = \frac{y}{x}\left(\log_e\left(\frac{y}{x}\right) + 1\right) \]

(A) Is it linear? A linear DE is of the form \(\frac{dy}{dx} + P(x)y = Q(x)\). Due to the term \(\log_e y\), this equation is not linear. So, (A) is false.


(B) Is it homogeneous? A DE is homogeneous if it can be written in the form \(\frac{dy}{dx} = F(\frac{y}{x})\). Our equation is in this form. So, (B) is true.


(C) & (D) General Solution: Since it is homogeneous, we use the substitution \(y = vx\). This implies \(\frac{dy}{dx} = v + x \frac{dv}{dx}\).

Substituting into the equation:
\[ v + x \frac{dv}{dx} = v(\log_e v + 1) = v\log_e v + v \] \[ x \frac{dv}{dx} = v\log_e v \]
Separating the variables:
\[ \frac{dv}{v\log_e v} = \frac{dx}{x} \]
Integrating both sides: \(\int \frac{dv}{v\log_e v} = \int \frac{dx}{x}\).

For the left integral, let \(u = \log_e v\), so \(du = \frac{1}{v} dv\).
\[ \int \frac{du}{u} = \ln(u) = \ln(\log_e v) \]
For the right integral, \(\int \frac{dx}{x} = \ln(x) + \ln(C) = \ln(Cx)\).

Equating the results:
\[ \ln(\log_e v) = \ln(Cx) \implies \log_e v = Cx \]
Substitute back \(v = \frac{y}{x}\):
\[ \log_e\left(\frac{y}{x}\right) = Cx \]
This matches statement (C). So, (C) is true and (D) is false.


(E) Particular Solution: Given the initial condition \(y(1) = 1\).

Using the general solution from (C): \(\log_e(\frac{y}{x}) = Cx\).

Substitute \(x=1\) and \(y=1\):
\[ \log_e\left(\frac{1}{1}\right) = C(1) \implies \log_e(1) = C \implies C=0 \]
The particular solution is \(\log_e(\frac{y}{x}) = 0\).

Taking the exponential of both sides:
\[ \frac{y}{x} = e^0 = 1 \implies y=x \]
So, (E) is true.


Step 4: Final Answer:

The true statements are (B), (C), and (E). This corresponds to option (4).
Quick Tip: To quickly identify a homogeneous differential equation, check if all terms have the same degree, or if the equation can be written entirely in terms of \(y/x\). The substitution \(y=vx\) is the standard method for solving them.

Step 1: Understanding the Concept:

This question tests the fundamental properties of the standard basis vectors (\(\hat{i}, \hat{j}, \hat{k}\)) in a 3D Cartesian coordinate system, specifically their dot products and cross products.


Step 3: Detailed Explanation:

Let's evaluate each statement based on the definitions of dot and cross products.


(A) \(\hat{i} \times \hat{i} = \vec{0}\)

The cross product of any vector with itself is the zero vector (\(\vec{0}\)). This is because the angle between the vectors is 0, and \(\sin(0) = 0\). The magnitude is \(|\hat{i}||\hat{i}|\sin(0) = 1 \cdot 1 \cdot 0 = 0\). So, statement (A) is true.


(B) \(\hat{i} \times \hat{k} = \hat{j}\)

The cross products of the basis vectors follow a cyclic rule for a right-handed coordinate system: \(\hat{i} \times \hat{j} = \hat{k}\), \(\hat{j} \times \hat{k} = \hat{i}\), and \(\hat{k} \times \hat{i} = \hat{j}\). Reversing the order of a cross product introduces a negative sign.

Therefore, \(\hat{i} \times \hat{k} = -(\hat{k} \times \hat{i}) = -\hat{j}\). So, statement (B) is false.


(C) \(\hat{i} \cdot \hat{i} = 1\)

The dot product of a vector with itself is the square of its magnitude. Since \(\hat{i}\) is a unit vector, its magnitude is 1. Thus, \(\hat{i} \cdot \hat{i} = |\hat{i}|^2 = 1^2 = 1\). So, statement (C) is true.


(D) \(\hat{i} \cdot \hat{j} = 0\)

The vectors \(\hat{i}\) and \(\hat{j}\) are orthogonal (perpendicular), meaning the angle between them is 90 degrees. The dot product of orthogonal vectors is zero, as \(\cos(90^\circ) = 0\). So, statement (D) is true.


Step 4: Final Answer:

The true statements are (A), (C), and (D). This corresponds to option (2).
Quick Tip: Remember the geometric interpretations: the dot product measures projection and is related to the cosine of the angle, while the cross product measures the area of the parallelogram formed by the vectors and is related to the sine of the angle, yielding a vector perpendicular to both.

Step 1: Understanding the Concept:

If three points A, B, and C are collinear, it means they lie on the same straight line. This implies that the vector from A to B (\(\vec{AB}\)) must be parallel to the vector from B to C (\(\vec{BC}\)). Two vectors are parallel if one is a scalar multiple of the other.


Step 2: Key Formula or Approach:

1. Find the vectors \(\vec{AB}\) and \(\vec{BC}\) by subtracting the position vectors of the initial points from the terminal points.
\(\vec{AB} = \vec{OB} - \vec{OA}\)
\(\vec{BC} = \vec{OC} - \vec{OB}\)
2. For two vectors \(\vec{p} = p_x\hat{i} + p_y\hat{j}\) and \(\vec{q} = q_x\hat{i} + q_y\hat{j}\) to be parallel, the ratio of their corresponding components must be equal: \(\frac{p_x}{q_x} = \frac{p_y}{q_y}\).


Step 3: Detailed Explanation:

Let the position vectors be \(\vec{OA} = 20\hat{i} + \lambda\hat{j}\), \(\vec{OB} = 5\hat{i} - \hat{j}\), and \(\vec{OC} = 10\hat{i} - 13\hat{j}\).

First, calculate the vector \(\vec{AB}\):
\[ \vec{AB} = \vec{OB} - \vec{OA} = (5\hat{i} - \hat{j}) - (20\hat{i} + \lambda\hat{j}) = (5-20)\hat{i} + (-1-\lambda)\hat{j} = -15\hat{i} - (1+\lambda)\hat{j} \]
Next, calculate the vector \(\vec{BC}\):
\[ \vec{BC} = \vec{OC} - \vec{OB} = (10\hat{i} - 13\hat{j}) - (5\hat{i} - \hat{j}) = (10-5)\hat{i} + (-13 - (-1))\hat{j} = 5\hat{i} - 12\hat{j} \]
Since A, B, and C are collinear, \(\vec{AB}\) is parallel to \(\vec{BC}\). We can equate the ratios of their components:
\[ \frac{x-component of \vec{AB}}{x-component of \vec{BC}} = \frac{y-component of \vec{AB}}{y-component of \vec{BC}} \] \[ \frac{-15}{5} = \frac{-(1+\lambda)}{-12} \] \[ -3 = \frac{1+\lambda}{12} \]
Multiply both sides by 12:
\[ -3 \times 12 = 1 + \lambda \] \[ -36 = 1 + \lambda \] \[ \lambda = -36 - 1 = -37 \]

Step 4: Final Answer:

The value of \(\lambda\) is -37.
Quick Tip: For collinearity problems in 2D, setting the ratios of components equal is often the fastest method. Another way is to set the determinant of a matrix formed by the coordinates (with an extra column of 1s) to zero, which represents the area of the triangle formed by the points being zero.

Step 1: Understanding the Concept:

We are given a relationship between three vectors and their magnitudes. To find the angle between two of the vectors, \(\vec{a}\) and \(\vec{b}\), we can use the dot product formula, which relates the dot product to the magnitudes and the cosine of the angle.


Step 2: Key Formula or Approach:

The angle \(\theta\) between two vectors \(\vec{a}\) and \(\vec{b}\) is given by \(\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}\). We can find the value of \(\vec{a} \cdot \vec{b}\) by manipulating the given vector equation.


Step 3: Detailed Explanation:

We are given \(\vec{a} + \vec{b} + \vec{c} = \vec{0}\).

To find the angle between \(\vec{a}\) and \(\vec{b}\), we should isolate the third vector, \(\vec{c}\).
\[ \vec{a} + \vec{b} = -\vec{c} \]
Now, take the dot product of each side with itself. This is equivalent to squaring the magnitude of each side.
\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (-\vec{c}) \cdot (-\vec{c}) \] \[ |\vec{a} + \vec{b}|^2 = |-\vec{c}|^2 = |\vec{c}|^2 \]
Expand the left side:
\[ \vec{a} \cdot \vec{a} + 2(\vec{a} \cdot \vec{b}) + \vec{b} \cdot \vec{b} = |\vec{c}|^2 \] \[ |\vec{a}|^2 + 2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2 = |\vec{c}|^2 \]
Now, substitute the given magnitudes: \(|\vec{a}| = 3, |\vec{b}| = 5, |\vec{c}| = 7\).
\[ 3^2 + 2(\vec{a} \cdot \vec{b}) + 5^2 = 7^2 \] \[ 9 + 2(\vec{a} \cdot \vec{b}) + 25 = 49 \] \[ 34 + 2(\vec{a} \cdot \vec{b}) = 49 \] \[ 2(\vec{a} \cdot \vec{b}) = 49 - 34 = 15 \] \[ \vec{a} \cdot \vec{b} = \frac{15}{2} \]
Now we can find the angle \(\theta\) between \(\vec{a}\) and \(\vec{b}\).
\[ \cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} = \frac{15/2}{3 \times 5} = \frac{15/2}{15} = \frac{1}{2} \]
The angle \(\theta\) in the range \([0, \pi]\) for which \(\cos\theta = \frac{1}{2}\) is \(\theta = \frac{\pi}{3}\).


Step 4: Final Answer:

The angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\).
Quick Tip: This is a classic problem. When given a vector sum equal to zero, to find the angle between any two vectors (say A and B), isolate the third vector (C) on one side of the equation and then square the magnitudes of both sides. This introduces the dot product A.B, which contains the angle information.

Step 1: Understanding the Concept:

A vector that is perpendicular to two other vectors, \(\vec{a}\) and \(\vec{b}\), must be parallel to their cross product, \(\vec{a} \times \vec{b}\). We can express \(\vec{d}\) as a scalar multiple of this cross product and then use the given dot product condition to find the scalar. Finally, we calculate the magnitude of \(\vec{d}\).


Step 2: Key Formula or Approach:

1. Calculate the cross product \(\vec{n} = \vec{a} \times \vec{b}\).
2. Let \(\vec{d} = \lambda \vec{n}\) for some scalar \(\lambda\).
3. Use the condition \(\vec{c} \cdot \vec{d} = 16\) to solve for \(\lambda\).
4. Calculate the magnitude \(|\vec{d}|\).


Step 3: Detailed Explanation:

Given vectors are \(\vec{a} = \hat{i} + 4\hat{j} + 0\hat{k}\) and \(\vec{b} = 0\hat{i} + 4\hat{j} + \hat{k}\).

First, find the cross product \(\vec{a} \times \vec{b}\).

Since \(\vec{d}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\), it must be parallel to their cross product.

Let \(\vec{d} = \lambda(4\hat{i} - \hat{j} + 4\hat{k})\) for some scalar \(\lambda\).

We are given that \(\vec{c} \cdot \vec{d} = 16\), where \(\vec{c} = \hat{i} - 2\hat{k}\).
\[ (\hat{i} + 0\hat{j} - 2\hat{k}) \cdot (\lambda(4\hat{i} - \hat{j} + 4\hat{k})) = 16 \] \[ \lambda [ (1)(4) + (0)(-1) + (-2)(4) ] = 16 \] \[ \lambda [ 4 + 0 - 8 ] = 16 \] \[ \lambda [ -4 ] = 16 \implies \lambda = -4 \]
Now we have the vector \(\vec{d}\):
\[ \vec{d} = -4(4\hat{i} - \hat{j} + 4\hat{k}) = -16\hat{i} + 4\hat{j} - 16\hat{k} \]
Finally, find the magnitude of \(\vec{d}\):
\[ |\vec{d}| = \sqrt{(-16)^2 + 4^2 + (-16)^2} = \sqrt{256 + 16 + 256} = \sqrt{528} \]
To simplify \(\sqrt{528}\): \(528 = 16 \times 33\).
\[ |\vec{d}| = \sqrt{16 \times 33} = 4\sqrt{33} \]
Alternatively, \(|\vec{d}| = |\lambda| |\vec{a} \times \vec{b}| = |-4| |4\hat{i} - \hat{j} + 4\hat{k}| = 4\sqrt{4^2+(-1)^2+4^2} = 4\sqrt{16+1+16} = 4\sqrt{33}\).


Step 4: Final Answer:

The magnitude of \(\vec{d}\) is \(4\sqrt{33}\).
Quick Tip: The cross product \(\vec{a} \times \vec{b}\) gives a vector perpendicular to the plane containing \(\vec{a}\) and \(\vec{b}\). Any vector perpendicular to this plane must be a scalar multiple of the cross product. This is a fundamental concept for solving such problems.

Step 1: Understanding the Concept:

The angles \(\alpha, \beta, \gamma\) are the direction angles of a line. The cosines of these angles, \(\cos\alpha, \cos\beta, \cos\gamma\), are called the direction cosines of the line. There is a fundamental identity relating these direction cosines.


Step 2: Key Formula or Approach:

The fundamental identity for direction cosines (\(l, m, n\)) is:
\[ l^2 + m^2 + n^2 = 1 \]
where \(l = \cos\alpha\), \(m = \cos\beta\), and \(n = \cos\gamma\).

So, \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\).

We also use the Pythagorean identity from trigonometry: \(\sin^2\theta + \cos^2\theta = 1\).


Step 3: Detailed Explanation:

We need to find the value of the expression \(\sin^2\alpha + \sin^2\beta + \sin^2\gamma\).

Using the identity \(\sin^2\theta = 1 - \cos^2\theta\), we can rewrite the expression:
\[ \sin^2\alpha + \sin^2\beta + \sin^2\gamma = (1 - \cos^2\alpha) + (1 - \cos^2\beta) + (1 - \cos^2\gamma) \]
Rearranging the terms:
\[ = (1 + 1 + 1) - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) \] \[ = 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma) \]
Now, substitute the fundamental identity for direction cosines, \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\):
\[ = 3 - 1 = 2 \]

Step 4: Final Answer:

The value of \(\sin^2\alpha + \sin^2\beta + \sin^2\gamma\) is 2.
Quick Tip: This is a standard identity in 3D geometry. It's useful to memorize both forms: \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\) and \(\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 2\).

Step 1: Understanding the Concept:

The vector equation of a line is given in the form \(\vec{r} = \vec{a} + \lambda\vec{b}\), where \(\vec{a}\) is the position vector of a point on the line and \(\vec{b}\) is a vector parallel to the line, whose components give the direction ratios.


Step 3: Detailed Explanation:

The given equation is \(\vec{r} = (\hat{i} - 2\hat{j} + 4\hat{k}) + \lambda(-\hat{i} + 2\hat{j} - 4\hat{k})\).

By comparing with \(\vec{r} = \vec{a} + \lambda\vec{b}\), we have:
\(\vec{a} = \hat{i} - 2\hat{j} + 4\hat{k}\)
\(\vec{b} = -\hat{i} + 2\hat{j} - 4\hat{k}\)


(A) A point on the given line:

The position vector \(\vec{a}\) corresponds to a point on the line. The coordinates of this point are (1, -2, 4). This matches with (III).


(B) Direction ratios of the line:

The components of the parallel vector \(\vec{b}\) are the direction ratios of the line. These are (-1, 2, -4). This matches with (IV).


(C) Direction cosines of the line:

Direction cosines are the components of the unit vector in the direction of \(\vec{b}\). First, find the magnitude of \(\vec{b}\).
\(|\vec{b}| = \sqrt{(-1)^2 + 2^2 + (-4)^2} = \sqrt{1 + 4 + 16} = \sqrt{21}\).

The unit vector is \(\hat{b} = \frac{\vec{b}}{|\vec{b}|} = \frac{-\hat{i} + 2\hat{j} - 4\hat{k}}{\sqrt{21}}\).

The direction cosines are \((-\frac{1}{\sqrt{21}}, \frac{2}{\sqrt{21}}, -\frac{4}{\sqrt{21}})\). This matches with (I).


(D) Direction ratios of a line perpendicular to given line:

Let the direction ratios of a perpendicular line be \((l, m, n)\). The dot product of its direction vector with \(\vec{b}\) must be zero.
\(\vec{b} \cdot (l\hat{i} + m\hat{j} + n\hat{k}) = 0\)
\((-1)(l) + (2)(m) + (-4)(n) = 0 \implies -l + 2m - 4n = 0\).

We need to check which direction ratios from List-II satisfy this condition. Let's test option (II) (4, -2, -2).
\(-(4) + 2(-2) - 4(-2) = -4 - 4 + 8 = 0\).

The condition is satisfied. So, (4, -2, -2) are direction ratios of a perpendicular line. This matches with (II).


Step 4: Final Answer:

The correct matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II). This corresponds to option (3).
Quick Tip: For a line \(\vec{r} = \vec{a} + \lambda\vec{b}\): The components of \(\vec{a}\) give a point on the line. The components of \(\vec{b}\) give the direction ratios. The components of the unit vector \(\hat{b}\) give the direction cosines. Any vector whose dot product with \(\vec{b}\) is zero is perpendicular to the line.

Step 1: Understanding the Concept:

We are asked to find the shortest distance between two lines given in Cartesian form. First, we need to determine if the lines are parallel, intersecting, or skew.


Step 2: Key Formula or Approach:

The Cartesian form of a line is \(\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}\). The vector form is \(\vec{r} = (x_1\hat{i} + y_1\hat{j} + z_1\hat{k}) + \lambda(a\hat{i} + b\hat{j} + c\hat{k})\).
The direction ratios of the first line are \((a_1, b_1, c_1)\) and for the second line are \((a_2, b_2, c_2)\).
Two lines are parallel if their direction ratios are proportional, i.e., \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\).
If lines are parallel, the shortest distance is the perpendicular distance from a point on one line to the other line. If the lines are not parallel, they are either intersecting or skew.


Step 3: Detailed Explanation:

Let the first line be L1 and the second line be L2.

L1: \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}\). The direction ratios are \(\vec{d_1} = (2, 3, 4)\).
L2: \(\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}\). The direction ratios are \(\vec{d_2} = (4, 6, 8)\).


First, check if the lines are parallel by comparing their direction ratios.
\[ \frac{4}{2} = 2 \] \[ \frac{6}{3} = 2 \] \[ \frac{8}{4} = 2 \]
Since the ratios are equal, the direction vectors are proportional (\(\vec{d_2} = 2\vec{d_1}\)). Therefore, the lines are parallel.


Now, to find the distance between them, we need to check if they are the same line (coincident) or distinct parallel lines. We can do this by taking a point from one line and checking if it lies on the other.

A point on L1 is \(P_1 = (1, 2, 3)\). Let's check if this point satisfies the equation of L2.

Substitute the coordinates of \(P_1\) into the equation for L2:
\[ \frac{1-2}{4} = \frac{-1}{4} \] \[ \frac{2-4}{6} = \frac{-2}{6} = \frac{-1}{3} \] \[ \frac{3-5}{8} = \frac{-2}{8} = \frac{-1}{4} \]
Since \(\frac{-1}{4} \neq \frac{-1}{3}\), the point (1, 2, 3) does not lie on the second line. Therefore, the lines are distinct and parallel.


The shortest distance between two parallel lines \(\vec{r} = \vec{a_1} + \lambda\vec{d}\) and \(\vec{r} = \vec{a_2} + \mu\vec{d}\) is given by the formula:
\[ D = \frac{|(\vec{a_2} - \vec{a_1}) \times \vec{d}|}{|\vec{d}|} \]
From the equations:
\(\vec{a_1} = \hat{i} + 2\hat{j} + 3\hat{k}\)
\(\vec{a_2} = 2\hat{i} + 4\hat{j} + 5\hat{k}\)
\(\vec{d} = 2\hat{i} + 3\hat{j} + 4\hat{k}\) (we can use the simpler direction vector)
\(\vec{a_2} - \vec{a_1} = (2-1)\hat{i} + (4-2)\hat{j} + (5-3)\hat{k} = \hat{i} + 2\hat{j} + 2\hat{k}\)


Now calculate the cross product:

Magnitude of the cross product: \(|2\hat{i} - \hat{k}| = \sqrt{2^2 + (-1)^2} = \sqrt{5}\).

Magnitude of the direction vector: \(|\vec{d}| = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{4+9+16} = \sqrt{29}\).

Distance: \(D = \frac{\sqrt{5}}{\sqrt{29}} = \sqrt{\frac{5}{29}}\).
Quick Tip: When finding the distance between lines, always check if they are parallel first by comparing their direction ratios. If they are, use the formula for parallel lines. If not, use the formula for skew lines. If the skew line distance formula gives zero, the lines are intersecting. In an exam, if your calculated answer is not in the options, re-check your calculations, then re-read the question for typos. If it still doesn't match, the question may be flawed.

Step 1: Understanding the Concept:

We need to determine the system of linear inequalities that defines the given shaded feasible region. We do this by finding the equation of each boundary line and then determining the direction of the inequality (e.g., \(\le\) or \(\ge\)) by testing a point within the shaded region, like the origin (0,0) if it's not on the line.


Step 3: Detailed Explanation:

First, let's identify the boundary lines from the graph.

(A) Line passing through (15, 0) and (0, 15):
The equation is \(\frac{x}{15} + \frac{y}{15} = 1\), which simplifies to \(x + y = 15\). The shaded region is above this line (away from the origin). Testing the point (15, 20) which is in the region: \(15 + 20 = 35 \ge 15\). So the inequality is \(x + y \ge 15\).

(B) Line passing through (30, 0) and (0, 30):
The equation is \(\frac{x}{30} + \frac{y}{30} = 1\), which simplifies to \(x + y = 30\). The shaded region is below this line (towards the origin). Testing the origin (0,0): \(0+0=0 \le 30\). So the inequality is \(x + y \le 30\).

(C) Vertical line passing through x = 15:
The equation is \(x = 15\). The shaded region is to the left of this line. So the inequality is \(x \le 15\).

(D) Horizontal line passing through y = 20:
The equation is \(y = 20\). The shaded region is below this line. So the inequality is \(y \le 20\).

(E) Non-negativity constraints:
The shaded region is in the first quadrant, which means \(x \ge 0\) and \(y \ge 0\).


Combining all these inequalities, we get the set of constraints: \[ x + y \le 30, \quad x + y \ge 15, \quad x \le 15, \quad y \le 20, \quad x \ge 0, \quad y \ge 0 \]
This set matches the constraints given in option (1).


Step 4: Final Answer:

The correct set of constraints is \(x + y \le 30, x + y \ge 15, x \le 15, y \le 20, x, y \ge 0\).
Quick Tip: To quickly find the equation of a line from its intercepts, use the intercept form: \(\frac{x}{a} + \frac{y}{b} = 1\), where 'a' is the x-intercept and 'b' is the y-intercept. To determine the inequality, pick a test point (the origin is easiest) and see if it satisfies the condition for the shaded region.

Given the objective function and the constraints:


1. \( 2x - y \geq -5 \)

2. \( 3x + y \geq 3 \)

3. \( 2x - 3y \leq 12 \)

4. \( x \geq 0, y \geq 0 \)


We need to graph these inequalities to find the feasible region and then determine the corner points of this region.


- From the first constraint: \( 2x - y = -5 \), we can rearrange it to express in slope-intercept form:

\( y = 2x + 5 \).


- For the second constraint: \( 3x + y = 3 \), rearranging gives:
\( y = -3x + 3 \).


- For the third constraint: \( 2x - 3y = 12 \), we rearrange to express as:
\( y = \frac{2x - 12}{3} \).


After graphing these three lines along with the \( x \geq 0 \) and \( y \geq 0 \) conditions, we identify the corner points of the feasible region. These points are the vertices where the constraints intersect. By solving the system of equations of the lines and checking which points lie in the feasible region, we determine that the corner points are:
\[ (0, 3), (0, 5), (1, 0), (6, 0) \] Quick Tip: Graph the constraints to find the feasible region. Then, solve for the corner points where the constraints intersect. These points will help in determining the optimal solution for the objective function.

Step 1: Understanding the Concept:

We are given a condition about the probabilities of two events, B and the intersection of A and B (A and B is the same as A \(\cap\) B). We need to determine the value of a conditional probability.


Step 2: Key Formula or Approach:

The definition of conditional probability is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (assuming P(B) > 0) \] \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \quad (assuming P(A) > 0) \]

Step 3: Detailed Explanation:

We are given the condition P(B) = P(A \(\cap\) B).

Let's use this condition to find P(A|B).

From the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Substitute the given condition P(A \(\cap\) B) = P(B) into the formula:
\[ P(A|B) = \frac{P(B)}{P(B)} \]
Assuming P(B) > 0 (if P(B)=0, the conditional probability P(A|B) is undefined), we get:
\[ P(A|B) = 1 \]
This means that given event B has occurred, event A is certain to occur. This makes sense, as the condition P(B) = P(A \(\cap\) B) implies that the event B is a subset of the event A. Whenever B happens, A must also happen.


Let's check P(B|A): \[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{P(B)}{P(A)} \]
We cannot determine the value of this without knowing P(A). So, option (1) and (3) are not necessarily correct.


Step 4: Final Answer:

The correct statement is P(A|B) = 1.
Quick Tip: The condition \(P(B) = P(A \cap B)\) implies that event B is a subset of event A (\(B \subseteq A\)). If you visualize this with a Venn diagram, the entire circle for B is inside the circle for A. Therefore, if B occurs, A must also occur, making \(P(A|B) = 1\).

Step 1: Understanding the Concept:

This question is about the theorem of total probability and Bayes' theorem. The events E₁, E₂, E₃ form a partition of the sample space, as they are mutually exclusive (they cannot happen at the same time) and exhaustive (one of them must happen).


Step 3: Detailed Explanation:

Let's analyze each statement.


(A) \(P(A) = P(E_1)P(E_1|A) + P(E_2)P(E_2|A) + P(E_3)P(E_3|A)\)

This is not the standard formula for the law of total probability. The formula relates P(A) to conditional probabilities of A given E_i, not the other way around. From the definition of conditional probability, \(P(E_i)P(E_i|A) = P(E_i \cap A)/P(A) * P(E_i)\), which doesn't simplify nicely to P(A). So, (A) is false.


(B) \(P(A) = P(A|E_1)P(E_1) + P(A|E_2)P(E_2) + P(A|E_3)P(E_3)\)

This is the correct statement of the Law of Total Probability. It expresses the total probability of an event A in terms of the probabilities of A conditional on each event in a partition of the sample space. Since \(P(A|E_i)P(E_i) = P(A \cap E_i)\), this formula is equivalent to \(P(A) = P(A \cap E_1) + P(A \cap E_2) + P(A \cap E_3)\), which is true because E_i are mutually exclusive and exhaustive. So, (B) is true.


(C) \(P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{j=1}^{3} P(A|E_j)P(E_j)}, i=1,2,3\)

This is the correct statement of Bayes' Theorem. The numerator, by the multiplication rule, is \(P(A \cap E_i)\). The denominator, by the law of total probability (from statement B), is P(A). So, the formula becomes \(P(E_i|A) = \frac{P(A \cap E_i)}{P(A)}\), which is the definition of conditional probability. So, (C) is true.


(D) \(P(A|E_i) = \frac{P(E_i|A)P(E_i)}{\sum_{j=1}^{3} P(E_i|A)P(E_j)}, i=1,2,3\)

This is an incorrect formulation. It seems to be a misstatement of Bayes' theorem, with the roles of A and E_i confused. So, (D) is false.


Step 4: Final Answer:

The true statements are (B) and (C). This corresponds to option (4).
Quick Tip: Memorize the statements of the Law of Total Probability and Bayes' Theorem. \(\textbf{Total Probability}\): Used to find the probability of an event A by considering all possible scenarios (the partition \(E_i\)). \(P(A) = \sum P(A|E_i)P(E_i)\). \(\textbf{Bayes' Theorem}\): Used to "reverse" the conditioning. If you know \(P(A|E_i)\), you can find \(P(E_i|A)\). \(P(E_i|A) = \frac{P(A|E_i)P(E_i)}{P(A)}\).

Step 1: Understanding the Concept:

This question requires the application of basic probability formulas, including the multiplication rule for conditional probability, the formula for conditional probability, the addition rule, and the complement rule.


Step 3: Detailed Explanation:

Given: P(A) = 0.8, P(B) = 0.5, P(B|A) = 0.4


(A) P(A \(\cap\) B)

Using the multiplication rule: \(P(A \cap B) = P(B|A) \times P(A)\).
\(P(A \cap B) = 0.4 \times 0.8 = 0.32\). This matches (II).


(B) P(A|B)

Using the formula for conditional probability: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).
\(P(A|B) = \frac{0.32}{0.5} = \frac{32}{50} = \frac{16}{25} = 0.64\). This matches (III).


(C) P(A \(\cup\) B)

Using the addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
\(P(A \cup B) = 0.8 + 0.5 - 0.32 = 1.3 - 0.32 = 0.98\). This matches (IV).


(D) P(A')

Using the complement rule: \(P(A') = 1 - P(A)\).
\(P(A') = 1 - 0.8 = 0.2\). This matches (I).


Step 4: Final Answer:

The correct matching is:

(A) \(\rightarrow\) (II)

(B) \(\rightarrow\) (III)

(C) \(\rightarrow\) (IV)

(D) \(\rightarrow\) (I)

This corresponds to option (2).
Quick Tip: The first step in many conditional probability problems is to find \(P(A \cap B)\). It can usually be found from one of the two multiplication rules: \(P(A \cap B) = P(A|B)P(B) = P(B|A)P(A)\). Once you have the intersection probability, most other quantities can be calculated.

Step 1: Understanding the Concept:

This is a problem of conditional probability. We are given some information (the outcome of the black die) which reduces our sample space, and we need to find the probability of another event within this new, smaller sample space.


Step 2: Key Formula or Approach:

Let A be the event "the sum is greater than 9".

Let B be the event "the black die resulted in a 5".

We need to find P(A|B), the probability of A given B.

The formula is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\).
Alternatively, and more simply, we can directly consider the reduced sample space.


Step 3: Detailed Explanation:

Let's use the reduced sample space method.
The event B, "the black die resulted in a 5", has occurred. This means the possible outcomes are now only those where the first die is a 5.
The reduced sample space S' is:

S' = (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)

The total number of outcomes in this reduced sample space is n(S') = 6.


Now, within this sample space, we want to find the outcomes that satisfy the event A, "the sum is greater than 9". Let's check the sum for each outcome in S':


(5, 1): sum = 6
(5, 2): sum = 7
(5, 3): sum = 8
(5, 4): sum = 9 (not greater than 9)
(5, 5): sum = 10 (greater than 9)
(5, 6): sum = 11 (greater than 9)

The favorable outcomes are (5, 5), (5, 6).

The number of favorable outcomes is 2.


The probability is the ratio of favorable outcomes to the total outcomes in the reduced sample space.
\[ P(A|B) = \frac{Number of favorable outcomes}{Total number of outcomes in reduced sample space} = \frac{2}{6} = \frac{1}{3} \]

Step 4: Final Answer:

The required probability is \(\frac{1}{3}\).
Quick Tip: For "given that" problems involving dice, cards, or coins, it's often easiest to work with the reduced sample space. Simply list all the outcomes that are possible given the condition, and then count how many of those satisfy the event you're interested in.

Step 1: Understanding the Concept:

A matrix is singular if its determinant is equal to zero. We are given three singular matrices P, Q, and R. We will use this property to find the values of the variables a, b, and c.


Step 2: Key Formula or Approach:

For a 2x2 matrix , the determinant is given by \(|M| = ps - qr\).

If M is singular, then \(|M| = 0\), which implies \(ps - qr = 0\).


Step 3: Detailed Explanation:

For matrix P:

Given is singular, so \(|P| = 0\).
\[ (2)(3) - (3a)(4) = 0 \] \[ 6 - 12a = 0 \] \[ 12a = 6 \] \[ a = \frac{6}{12} = \frac{1}{2} \]

For matrix Q:

Given is singular, so \(|Q| = 0\).
\[ (b)(6) - (5)(2a) = 0 \] \[ 6b - 10a = 0 \]
Substitute the value of \(a = 1/2\):
\[ 6b - 10\left(\frac{1}{2}\right) = 0 \] \[ 6b - 5 = 0 \] \[ 6b = 5 \] \[ b = \frac{5}{6} \]

For matrix R:

Given is singular, so \(|R| = 0\).
\[ (a^2 + b^2 - c)(c) - (1-c)(c+1) = 0 \] \[ c(a^2 + b^2) - c^2 - (1 - c^2) = 0 \] \[ c(a^2 + b^2) - c^2 - 1 + c^2 = 0 \] \[ c(a^2 + b^2) - 1 = 0 \] \[ c(a^2 + b^2) = 1 \]
Substitute the values of a and b:
\[ a^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \] \[ b^2 = \left(\frac{5}{6}\right)^2 = \frac{25}{36} \] \[ c\left(\frac{1}{4} + \frac{25}{36}\right) = 1 \] \[ c\left(\frac{9}{36} + \frac{25}{36}\right) = 1 \] \[ c\left(\frac{34}{36}\right) = 1 \] \[ c\left(\frac{17}{18}\right) = 1 \] \[ c = \frac{18}{17} \]

Step 4: Final Answer:

We need to find the value of \((2a + 6b + 17c)\).
\[ 2a = 2\left(\frac{1}{2}\right) = 1 \] \[ 6b = 6\left(\frac{5}{6}\right) = 5 \] \[ 17c = 17\left(\frac{18}{17}\right) = 18 \] \[ 2a + 6b + 17c = 1 + 5 + 18 = 24 \]
The value is 24.
Quick Tip: The most important piece of information for this question is the term "singular matrix". Always remember that the determinant of a singular matrix is zero. This is the key to setting up the equations needed to solve for the variables.

Step 1: Understanding the Concept:

The question asks for the determinant of the adjugate (or adjoint) of a matrix. There is a direct relationship between the determinant of a matrix, its order, and the determinant of its adjugate.


Step 2: Key Formula or Approach:

For any non-singular square matrix A of order n, the determinant of its adjugate is given by the formula:
\[ |adj(A)| = |A|^{n-1} \]

Step 3: Detailed Explanation:

We are given the following information:

The matrix A is non-singular.

The order of the matrix A is \(n = 3\).

The determinant of the matrix A is \(|A| = 15\).

Using the formula from Step 2:
\[ |adj(A)| = |A|^{n-1} \]
Substitute the given values into the formula:
\[ |adj(A)| = (15)^{3-1} \] \[ |adj(A)| = (15)^2 \]

Step 4: Final Answer:

Calculate the final value:
\[ (15)^2 = 225 \]
Therefore, \(|adj A|\) is equal to 225.
Quick Tip: This is a direct formula-based question. Memorize the properties of determinants and adjugate matrices, such as \(|adj(A)| = |A|^{n-1}\) and \(A(adj A) = (adj A)A = |A|I\). These are frequently tested.

Step 1: Understanding the Concept:

The problem involves solving a matrix equation \(AX = B\). This can be solved by performing the matrix multiplication on the left-hand side and then equating the corresponding elements of the resulting matrix with the matrix on the right-hand side. This will yield a system of linear equations.


Step 2: Key Formula or Approach:

The multiplication of a 2x2 matrix with a 2x1 matrix is as follows:

Step 3: Detailed Explanation:

We are given the equation \(AX = B\). Substituting the given matrices:

Perform the matrix multiplication on the left side:

By equating the corresponding elements, we get a system of two linear equations:

1) \(3\alpha - 14 = 7\)

2) \(4\alpha + 4 = 32\)

Let's solve the first equation for \(\alpha\):
\[ 3\alpha = 7 + 14 \] \[ 3\alpha = 21 \] \[ \alpha = \frac{21}{3} = 7 \]
We can verify this result using the second equation:
\[ 4(7) + 4 = 28 + 4 = 32 \]
The value \(\alpha = 7\) satisfies both equations.


Step 4: Final Answer:

The value of \(\alpha\) is 7.
Quick Tip: When solving matrix equations like \(AX = B\), you can either set up and solve the system of linear equations (as shown here) or find the inverse of A and compute \(X = A^{-1}B\). For simple systems, direct multiplication and comparison is often faster.

Step 1: Understanding the Concept:

This question tests the fundamental properties of determinants. We need to evaluate each statement to identify the one that is not a valid property.


Step 3: Detailed Explanation:

Let's analyze each statement:

(A) This is a standard property of determinants. If two rows or two columns of a matrix are identical, the rows/columns are linearly dependent, and the determinant is zero. This statement is correct.


(B) This is also a correct property. If any row or column of a determinant contains all zeros, its value is zero. This can be easily seen by expanding the determinant along that row or column. Each term in the expansion would have a zero factor. This statement is correct.


(C) Interchanging the rows and columns of a matrix is equivalent to taking its transpose. A fundamental property of determinants is that the determinant of a matrix is equal to the determinant of its transpose, i.e., \(|A| = |A^T|\). This statement is correct.


(D) This statement is incorrect. A key property of determinants is that if any two rows or any two columns are interchanged, the sign of the determinant is changed (it is multiplied by -1). The statement claims the sign "remains unchanged," which is false.


Step 4: Final Answer:

The incorrect statement is (D) because interchanging two rows or columns reverses the sign of the determinant.
Quick Tip: The properties of determinants are crucial for both computational problems and theoretical questions. It is highly recommended to memorize the 6-7 key properties, especially those related to row/column operations.

Step 1: Understanding the Concept:

The task is to find the inverse of each matrix in List-I and match it with the correct inverse from List-II.


Step 2: Key Formula or Approach:

For a 2x2 matrix \(M = \begin{bmatrix} a & b
c & d \end{bmatrix}\), the inverse is given by the formula:
\[ M^{-1} = \frac{1}{det(M)} adj(M) = \frac{1}{ad-bc} \begin{bmatrix} d & -b
-c & a \end{bmatrix} \]

Step 3: Detailed Explanation:

We will calculate the inverse for each matrix in List-I.


(A) For matrix \(A = \begin{bmatrix} 1 & 7
4 & -2 \end{bmatrix}\):


Determinant: \(|A| = (1)(-2) - (7)(4) = -2 - 28 = -30\).


Inverse: \(A^{-1} = \frac{1}{-30} \begin{bmatrix} -2 & -7
-4 & 1 \end{bmatrix} = \begin{bmatrix} \frac{-2}{-30} & \frac{-7}{-30}
\frac{-4}{-30} & \frac{1}{-30} \end{bmatrix} = \begin{bmatrix} 1/15 & 7/30
2/15 & -1/30 \end{bmatrix}\).

This matches with (III).


(B) For matrix \(B = \begin{bmatrix} 6 & -3
2 & 4 \end{bmatrix}\):


Determinant: \(|B| = (6)(4) - (-3)(2) = 24 + 6 = 30\).


Inverse: \(B^{-1} = \frac{1}{30} \begin{bmatrix} 4 & 3
-2 & 6 \end{bmatrix} = \begin{bmatrix} \frac{4}{30} & \frac{3}{30}
\frac{-2}{30} & \frac{6}{30} \end{bmatrix} = \begin{bmatrix} 2/15 & 1/10
-1/15 & 1/5 \end{bmatrix}\).


This matches with (I).


(C) For matrix \(C = \begin{bmatrix} 5 & 2
-5 & 4 \end{bmatrix}\):

Determinant: \(|C| = (5)(4) - (2)(-5) = 20 + 10 = 30\).


Inverse: \(C^{-1} = \frac{1}{30} \begin{bmatrix} 4 & -2
5 & 5 \end{bmatrix} = \begin{bmatrix} \frac{4}{30} & \frac{-2}{30}
\frac{5}{30} & \frac{5}{30} \end{bmatrix} = \begin{bmatrix} 2/15 & -1/15
1/6 & 1/6 \end{bmatrix}\).


This matches with (IV).


(D) For matrix \(D = \begin{bmatrix} 7 & 4
3 & 6 \end{bmatrix}\):

Determinant: \(|D| = (7)(6) - (4)(3) = 42 - 12 = 30\).


Inverse: \(D^{-1} = \frac{1}{30} \begin{bmatrix} 6 & -4
-3 & 7 \end{bmatrix} = \begin{bmatrix} \frac{6}{30} & \frac{-4}{30}
\frac{-3}{30} & \frac{7}{30} \end{bmatrix} = \begin{bmatrix} 1/5 & -2/15
-1/10 & 7/30 \end{bmatrix}\).


This matches with (II).


Step 4: Final Answer:

The correct matching is:

(A) \(\rightarrow\) (III)

(B) \(\rightarrow\) (I)

(C) \(\rightarrow\) (IV)

(D) \(\rightarrow\) (II)

This corresponds to option (C).
Quick Tip: For finding the inverse of a 2x2 matrix, remember the simple rule: swap the elements on the main diagonal, change the sign of the off-diagonal elements, and divide the resulting matrix by the determinant. This is much faster than using the full cofactor method.

Step 1: Understanding the Concept:

The question tests a fundamental property of modular arithmetic. The expression `A mod B` gives the remainder when A is divided by B. The property in question is `(A + kB) mod B = A mod B`, where k is any integer. This is because `kB` is a multiple of `B`, and adding a multiple of the divisor does not change the remainder.


Step 2: Key Formula or Approach:

We need to evaluate both sides of the equation `X mod Y = (X + aY) mod Y` with the given values and see for which values of 'a' the equality holds.

The key property of modular arithmetic is: \[ (A + B) \mod N = ((A \mod N) + (B \mod N)) \mod N \]

Step 3: Detailed Explanation:

Given values are X = 11 and Y = 3.

First, let's calculate the left-hand side (LHS) of the equation: \[ LHS = X \mod Y = 11 \mod 3 \]
When 11 is divided by 3, the quotient is 3 and the remainder is 2. (Since \(11 = 3 \times 3 + 2\)).

So, LHS = 2.


Now, let's analyze the right-hand side (RHS) of the equation: \[ RHS = (X + aY) \mod Y = (11 + a \times 3) \mod 3 \]
Here, 'a' is an integer. The term `3a` is always a multiple of 3, regardless of whether 'a' is even, odd, positive, negative, or zero.

According to the property of modular arithmetic: \[ (11 + 3a) \mod 3 = ((11 \mod 3) + (3a \mod 3)) \mod 3 \]
We know that `11 mod 3 = 2`.

And since `3a` is a multiple of 3, the remainder when `3a` is divided by 3 is always 0. \[ 3a \mod 3 = 0 \]
Substituting these values back: \[ RHS = (2 + 0) \mod 3 = 2 \mod 3 = 2 \]

Step 4: Final Answer:

We see that LHS = 2 and RHS = 2. The equality holds true for any integer value of 'a'. Therefore, the statement is true for all integral values of 'a'.
Quick Tip: For any integers X, Y, and 'a', the expression `(X + aY) mod Y` will always be equal to `X mod Y`. This is because adding a multiple of the modulus (Y) does not change the remainder.

Step 1: Understanding the Concept:

This problem requires finding the value of \(3^{128} \mod 7\). This can be solved efficiently using modular exponentiation, often with the help of Fermat's Little Theorem.


Step 2: Key Formula or Approach:

Fermat's Little Theorem states that if 'p' is a prime number, then for any integer 'a' not divisible by 'p', we have: \[ a^{p-1} \equiv 1 \pmod{p} \]
Here, p = 7 (a prime number) and a = 3 (not divisible by 7).

So, we can apply the theorem: \[ 3^{7-1} \equiv 1 \pmod{7} \] \[ 3^6 \equiv 1 \pmod{7} \]

Step 3: Detailed Explanation:

Our goal is to find the remainder of \(3^{128}\) when divided by 7. We can use the result from Fermat's Little Theorem, \(3^6 \equiv 1 \pmod{7}\).

We need to express the exponent 128 in terms of 6. We do this by dividing 128 by 6.
\[ 128 \div 6 = 21 with a remainder of 2 \]
So, we can write the exponent 128 as: \[ 128 = 6 \times 21 + 2 \]
Now, we can rewrite the original expression: \[ 3^{128} = 3^{(6 \times 21 + 2)} = 3^{(6 \times 21)} \times 3^2 = (3^6)^{21} \times 3^2 \]
Now we apply the modulus 7 to this expression: \[ 3^{128} \pmod{7} \equiv ((3^6)^{21} \times 3^2) \pmod{7} \]
Since we know \(3^6 \equiv 1 \pmod{7}\), we can substitute this in: \[ \equiv (1^{21} \times 3^2) \pmod{7} \] \[ \equiv (1 \times 9) \pmod{7} \] \[ \equiv 9 \pmod{7} \]
Now, we find the remainder when 9 is divided by 7. \[ 9 = 7 \times 1 + 2 \]
The remainder is 2.


Step 4: Final Answer:

The least non-negative remainder when \(3^{128}\) is divided by 7 is 2.
Quick Tip: When dealing with large exponents in modular arithmetic, always look for a pattern or apply theorems like Fermat's Little Theorem or Euler's Totient Theorem. This avoids calculating the large power directly. The goal is to find a smaller power that gives a remainder of 1 or -1, as this simplifies the calculation significantly.

Step 1: Understanding the Concept:

This is a problem of mixtures where a certain quantity of the original substance is removed and replaced with another substance repeatedly. The key is to calculate the fraction of the original substance that remains after each operation.


Step 2: Key Formula or Approach:

The formula for calculating the final quantity of the original substance after 'n' successive replacements is: \[ Final Quantity = Initial Quantity \times \left(1 - \frac{Quantity Replaced}{Total Volume}\right)^n \]
Where 'n' is the number of times the operation is performed.


Step 3: Detailed Explanation:

Let's identify the given values:

Initial Quantity of milk = 60 litres.

Total Volume of the mixture = 60 litres (this remains constant).

Quantity Replaced in each step = 6 litres.

The process was performed once, and then "repeated further two more times". So, the total number of operations is \(n = 1 + 2 = 3\).


Now, we apply the formula: \[ Amount of milk left = 60 \times \left(1 - \frac{6}{60}\right)^3 \]
First, simplify the fraction inside the parenthesis: \[ \frac{6}{60} = \frac{1}{10} = 0.1 \]
So the expression becomes: \[ Amount of milk left = 60 \times (1 - 0.1)^3 \] \[ = 60 \times (0.9)^3 \]
Calculate \( (0.9)^3 \): \[ (0.9)^3 = 0.9 \times 0.9 \times 0.9 = 0.81 \times 0.9 = 0.729 \]
Finally, multiply this by the initial quantity: \[ Amount of milk left = 60 \times 0.729 \] \[ = 43.74 \]

Step 4: Final Answer:

After three operations, the amount of milk remaining in the tub is 43.74 litres.
Quick Tip: In mixture replacement problems, focus on the fraction of the original liquid that remains after one operation. In this case, \(1 - 6/60 = 9/10\) of the milk remains. For three operations, you simply calculate \( Initial \times (\frac{9}{10}) \times (\frac{9}{10}) \times (\frac{9}{10}) \).

Step 1: Understanding the Concept:

This is a classic 'Boats and Streams' problem. The speed of the boat is affected by the speed of the river's current. When rowing downstream, the speeds add up. When rowing upstream, the river's speed is subtracted from the boat's speed.


Step 2: Key Formula or Approach:

Let \(u\) be the speed of the boat in still water.

Let \(v\) be the speed of the stream.

Speed downstream (\(S_d\)) = \(u + v\).

Speed upstream (\(S_u\)) = \(u - v\).

Time = Distance / Speed.

The problem states that the time to go upstream is 4 times the time to go downstream for the same distance. \[ T_{upstream} = 4 \times T_{downstream} \]

Step 3: Detailed Explanation:

Given values:

Speed of the boat in still water, \(u = 5\) km/hr.

Let the speed of the stream be \(v\) km/hr.

Let the distance be 'd' km.


Using the time, distance, and speed relationship: \[ T_{downstream} = \frac{d}{S_d} = \frac{d}{5 + v} \] \[ T_{upstream} = \frac{d}{S_u} = \frac{d}{5 - v} \]
Now, we use the given condition \( T_{upstream} = 4 \times T_{downstream} \): \[ \frac{d}{5 - v} = 4 \times \left(\frac{d}{5 + v}\right) \]
The distance 'd' is the same on both sides, so it can be cancelled out: \[ \frac{1}{5 - v} = \frac{4}{5 + v} \]
Now, we cross-multiply to solve for \(v\): \[ 1 \times (5 + v) = 4 \times (5 - v) \] \[ 5 + v = 20 - 4v \]
Bring all the terms with \(v\) to one side and constants to the other: \[ v + 4v = 20 - 5 \] \[ 5v = 15 \] \[ v = \frac{15}{5} \] \[ v = 3 \]

Step 4: Final Answer:

The speed of the flow of the stream is 3 km/hr.
Quick Tip: For problems where the ratio of upstream and downstream times is given for the same distance, you can use the direct formula: \( \frac{u}{v} = \frac{T_u + T_d}{T_u - T_d} \). Here, \( \frac{T_u}{T_d} = 4 \), so \( \frac{5}{v} = \frac{4+1}{4-1} = \frac{5}{3} \). This gives \(v = 3\) km/hr.

Step 1: Understanding the Concept:

"Ajay defeats Vijay by X meters" means that when Ajay (the winner) finishes the race, Vijay is X meters behind the finish line. To find this distance, we need to calculate how far Vijay has traveled in the time it took Ajay to complete the race.


Step 2: Key Formula or Approach:

1. Calculate the speed of the slower runner (Vijay).
\[ Speed = \frac{Distance}{Time} \]
2. Calculate the distance covered by Vijay in the time taken by the faster runner (Ajay).
\[ Distance = Speed \times Time \]
3. Find the difference between the total race distance and the distance covered by Vijay.
\[ Defeat Margin = Total Distance - Distance covered by loser \]

Step 3: Detailed Explanation:

Given information:

Race distance = 600 m.

Time taken by Ajay = 38 seconds.

Time taken by Vijay = 48 seconds.


First, let's calculate Vijay's speed: \[ Speed_{Vijay} = \frac{Total Distance}{Time_{Vijay}} = \frac{600 m}{48 s} \] \[ Speed_{Vijay} = 12.5 m/s \]
Now, we need to find out where Vijay is when Ajay crosses the finish line. This happens at t = 38 seconds.
Let's calculate the distance Vijay covers in 38 seconds: \[ Distance_{Vijay in 38s} = Speed_{Vijay} \times Time_{Ajay} \] \[ Distance_{Vijay in 38s} = 12.5 m/s \times 38 s \] \[ Distance_{Vijay in 38s} = 475 m \]
This means when Ajay is at the 600 m mark (finish line), Vijay is at the 475 m mark.

The distance by which Ajay defeats Vijay is the difference: \[ Defeat Margin = 600 m - 475 m = 125 m \]

Step 4: Final Answer:

Ajay will defeat Vijay by 125 meters.
Quick Tip: In "defeat by distance" race problems, the key is to freeze the moment the winner crosses the finish line and calculate the position of the other runner at that exact time. The difference in their positions is the answer.

Step 1: Understanding the Concept:

This question requires evaluating four separate mathematical statements (inequalities and equations) to determine their validity.


Step 2: Detailed Explanation:

Let's analyze each statement one by one.


Statement (A): \(\sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2}\)

Since all terms are positive, we can square both sides to compare them. The inequality direction will be preserved.

LHS squared: \((\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + (\sqrt{3})^2 + 2(\sqrt{5})(\sqrt{3}) = 5 + 3 + 2\sqrt{15} = 8 + 2\sqrt{15}\).

RHS squared: \((\sqrt{6} + \sqrt{2})^2 = (\sqrt{6})^2 + (\sqrt{2})^2 + 2(\sqrt{6})(\sqrt{2}) = 6 + 2 + 2\sqrt{12} = 8 + 2\sqrt{12}\).

The inequality becomes: \(8 + 2\sqrt{15} > 8 + 2\sqrt{12}\).

Subtracting 8 from both sides gives: \(2\sqrt{15} > 2\sqrt{12}\).

Dividing by 2 gives: \(\sqrt{15} > \sqrt{12}\).

Since \(15 > 12\), this is true. Therefore, statement (A) is true.


Statement (B): If \(a > b\) and \(c < 0\), then \(\frac{a}{c} < \frac{b}{c}\)

This is a fundamental rule of inequalities. When you multiply or divide both sides of an inequality by a negative number (since \(c < 0\)), you must reverse the inequality sign.

Starting with \(a > b\), and dividing by a negative number \(c\), the `>` sign flips to `<`.

So, \(\frac{a}{c} < \frac{b}{c}\). Therefore, statement (B) is true.


Statement (C): \(\frac{1}{x^2} > \frac{1}{x} > 1\), if \(0 < x < 1\)

Let's take a sample value for x in the given range, for example, \(x = 0.5\).

Then \(x^2 = (0.5)^2 = 0.25\).

The inequality becomes: \(\frac{1}{0.25} > \frac{1}{0.5} > 1\).

Calculating the values: \(4 > 2 > 1\). This is correct.

Logically, for any number \(x\) between 0 and 1, \(x^2\) is smaller than \(x\) (e.g., \(0.5^2 < 0.5\)).

Since \(0 < x^2 < x\), taking the reciprocal of these positive numbers will reverse the inequalities: \(\frac{1}{x^2} > \frac{1}{x}\).

Also, since \(x < 1\), its reciprocal \(\frac{1}{x}\) will be greater than 1.

Combining these results, we get \(\frac{1}{x^2} > \frac{1}{x} > 1\). Therefore, statement (C) is true.


Statement (D): If a and b are positive integers and \(\frac{a-b}{6.25} = \frac{4}{2.5}\) then \(b > a\)

Let's solve the given equation for \(a-b\).
\[ \frac{4}{2.5} = \frac{40}{25} = \frac{8}{5} = 1.6 \]
So the equation is: \(\frac{a-b}{6.25} = 1.6\).
\[ a - b = 1.6 \times 6.25 \] \[ a - b = 1.6 \times \frac{25}{4} = \frac{8}{5} \times \frac{25}{4} = \frac{8 \times 25}{5 \times 4} = \frac{200}{20} = 10 \]
So, \(a - b = 10\). This implies \(a = b + 10\).

Since a and b are positive integers, this means 'a' is 10 more than 'b', so \(a > b\).

The statement claims \(b > a\), which is the opposite. Therefore, statement (D) is false.


Step 3: Final Answer:

Statements (A), (B), and (C) are true, while (D) is false. The correct choice is the one that lists (A), (B), and (C) only.
Quick Tip: When testing inequalities with variables, plugging in a simple test value that fits the condition (like \(x=0.5\) for \(0

Step 1: Understanding the Concept:

The problem requires finding a differential equation that is satisfied by the given function. This involves finding the first and second derivatives of y with respect to x using implicit differentiation and then substituting them into the given options to find the correct relationship.


Step 2: Detailed Explanation:

Given the equation: \[ e^y = \log x \]
First Derivative:
Differentiate both sides with respect to x: \[ \frac{d}{dx}(e^y) = \frac{d}{dx}(\log x) \]
Using the chain rule on the left side: \[ e^y \frac{dy}{dx} = \frac{1}{x} \]
Rearranging this equation, we get: \[ x e^y \frac{dy}{dx} = 1 \quad \cdots (1) \]

Second Derivative:
Now, differentiate equation (1) with respect to x. We use the product rule for three functions (u.v.w)' = u'vw + uv'w + uvw' on the left side, where u=x, v=\(e^y\), w=dy/dx. \[ \frac{d}{dx}\left(x e^y \frac{dy}{dx}\right) = \frac{d}{dx}(1) \] \[ \left(\frac{d}{dx}x\right) \left(e^y \frac{dy}{dx}\right) + x \left(\frac{d}{dx}e^y\right) \left(\frac{dy}{dx}\right) + x e^y \left(\frac{d}{dx}\frac{dy}{dx}\right) = 0 \] \[ (1) \left(e^y \frac{dy}{dx}\right) + x \left(e^y \frac{dy}{dx}\right) \left(\frac{dy}{dx}\right) + x e^y \left(\frac{d^2y}{dx^2}\right) = 0 \] \[ e^y \frac{dy}{dx} + x e^y \left(\frac{dy}{dx}\right)^2 + x e^y \frac{d^2y}{dx^2} = 0 \]
Since \(e^y\) is always positive, we can divide the entire equation by \(e^y\): \[ \frac{dy}{dx} + x \left(\frac{dy}{dx}\right)^2 + x \frac{d^2y}{dx^2} = 0 \]

Step 3: Final Answer:

Rearranging the terms to match the options, we get: \[ x \frac{d^2y}{dx^2} + x \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0 \]
This matches option (D).
Quick Tip: When performing implicit differentiation a second time, it's often easier to differentiate a simplified form of the first derivative. Here, differentiating \(x e^y \frac{dy}{dx} = 1\) is simpler than differentiating \(\frac{dy}{dx} = \frac{1}{x e^y}\) which would require the quotient rule and lead to more complex substitutions.

Step 1: Understanding the Concept:

Marginal cost is the rate of change of the total cost with respect to the number of units produced. In calculus, this is represented by the first derivative of the total cost function, C(x). The marginal cost at a specific production level (x=10) is found by evaluating the derivative at that point.


Step 2: Key Formula or Approach:

Marginal Cost (MC) = \(\frac{dC}{dx}\).


Step 3: Detailed Explanation:

The given total cost function is: \[ C(x) = 0.007x^3 + 26x^2 + 15x + 400 \]
First, we find the derivative of C(x) with respect to x to get the marginal cost function, MC(x). \[ MC(x) = \frac{dC}{dx} = \frac{d}{dx}(0.007x^3 + 26x^2 + 15x + 400) \]
Using the power rule for differentiation (\(\frac{d}{dx}(ax^n) = anx^{n-1}\)): \[ MC(x) = 0.007(3x^2) + 26(2x) + 15(1) + 0 \] \[ MC(x) = 0.021x^2 + 52x + 15 \]
Now, we need to find the marginal cost when 10 items are produced, so we substitute x = 10 into the MC(x) function. \[ MC(10) = 0.021(10)^2 + 52(10) + 15 \] \[ MC(10) = 0.021(100) + 520 + 15 \] \[ MC(10) = 2.1 + 520 + 15 \] \[ MC(10) = 537.1 \]


Step 4: Final Answer:

The marginal cost when 10 items are produced is Rs. 537.1.
Quick Tip: In economics and business calculus, "marginal" almost always means "derivative of". Marginal cost is the derivative of the cost function, marginal revenue is the derivative of the revenue function, and so on. This is a key concept to remember for such application-based problems.

Step 1: Understanding the Concept:

The slope of the tangent to a curve at a point is given by the value of its derivative (\(dy/dx\)) at that point. The normal line is perpendicular to the tangent line at that same point. The slopes of two perpendicular lines are negative reciprocals of each other.


Step 2: Key Formula or Approach:

1. Find the derivative of the curve's equation: \(m_{tangent} = \frac{dy}{dx}\).

2. Evaluate the derivative at the given x-value to find the slope of the tangent at that point.

3. The slope of the normal is the negative reciprocal of the tangent's slope: \(m_{normal} = -\frac{1}{m_{tangent}}\).


Step 3: Detailed Explanation:

The equation of the curve is given by: \[ y = 2x^2 \]
First, we find the derivative of y with respect to x: \[ \frac{dy}{dx} = \frac{d}{dx}(2x^2) = 2(2x) = 4x \]
This expression, \(4x\), gives the slope of the tangent at any point x on the curve.

We need to find the slope at \(x = 1\). So, we substitute \(x = 1\) into the derivative: \[ m_{tangent} = \frac{dy}{dx}\bigg|_{x=1} = 4(1) = 4 \]
The slope of the tangent line at \(x=1\) is 4.

Now, we find the slope of the normal line, which is the negative reciprocal of the tangent's slope. \[ m_{normal} = -\frac{1}{m_{tangent}} = -\frac{1}{4} \]

Step 4: Final Answer:

The slope of the normal to the curve at x = 1 is -1/4.
Quick Tip: Remember the relationship between the slopes of perpendicular lines. If a line has slope 'm', any line perpendicular to it will have a slope of '-1/m'. Be careful not to confuse the slope of the tangent with the slope of the normal.

Step 1: Understanding the Concept:

This problem involves integration by recognizing a specific form. The integral form \(\int e^x (g(x) + g'(x)) dx = e^x g(x) + C\) is a standard and very useful result derived from the product rule of differentiation. The goal is to manipulate the given integrand into this form.


Step 2: Key Formula or Approach:

The key formula to use is: \[ \int e^x (g(x) + g'(x)) dx = e^x g(x) + C \]
We need to rewrite the given integral to match this pattern.


Step 3: Detailed Explanation:

The given integral is: \[ I = \int \frac{(1 + x \log x)}{xe^{-x}} dx \]
First, let's simplify the integrand. The term \(e^{-x}\) in the denominator is equivalent to \(e^x\) in the numerator. \[ I = \int e^x \frac{(1 + x \log x)}{x} dx \]
Now, let's split the fraction inside the integral: \[ I = \int e^x \left(\frac{1}{x} + \frac{x \log x}{x}\right) dx \] \[ I = \int e^x \left(\frac{1}{x} + \log x\right) dx \]
Now we check if this expression fits the form \(\int e^x (g(x) + g'(x)) dx\).

Let's try setting \(g(x) = \log x\).

If \(g(x) = \log x\), then its derivative is \(g'(x) = \frac{1}{x}\).

Substituting these into our integral, we get: \[ I = \int e^x (g'(x) + g(x)) dx \]
This perfectly matches the standard form.

Therefore, the result of the integration is: \[ I = e^x g(x) + C = e^x \log x + C \]
The problem states that the integral is equal to \(e^x f(x) + C\).

By comparing our result \(e^x \log x + C\) with the given form, we can see that: \[ f(x) = \log x \]

Step 4: Final Answer:

The function f(x) is \(\log x\).
Quick Tip: Whenever you see an integral involving \(e^x\) multiplied by a function, always try to check if the function can be expressed as the sum of another function and its derivative, i.e., \(g(x) + g'(x)\). This shortcut can save a lot of time compared to using integration by parts.

Step 1: Understanding the Concept:

This problem requires solving a first-order differential equation using the method of separation of variables. After finding the general solution, we use the given point (1, 1) to find the particular solution. Finally, we compare the coefficients of our solution with the given form to determine the values of \(\alpha, \beta, \gamma, \delta\), and C, and compute the required expression.


Step 2: Key Formula or Approach:

The given differential equation is: \[ x(e^{2y} - 1)dy + (x^2 - 1)e^y dx = 0 \]
We will separate the variables (terms with y and dy on one side, terms with x and dx on the other) and then integrate both sides.


Step 3: Detailed Explanation:

Start by rearranging the equation: \[ x(e^{2y} - 1)dy = -(x^2 - 1)e^y dx \]
Now, group the y-terms on the left and x-terms on the right. Divide by \(xe^y\): \[ \frac{e^{2y} - 1}{e^y} dy = -\frac{x^2 - 1}{x} dx \]
Simplify both sides: \[ \left(\frac{e^{2y}}{e^y} - \frac{1}{e^y}\right) dy = -\left(\frac{x^2}{x} - \frac{1}{x}\right) dx \] \[ (e^y - e^{-y}) dy = -(x - \frac{1}{x}) dx \]
Integrate both sides: \[ \int (e^y - e^{-y}) dy = -\int (x - \frac{1}{x}) dx \] \[ e^y - (-e^{-y}) = -\left(\frac{x^2}{2} - \log|x|\right) + K \]
where K is the constant of integration. \[ e^y + e^{-y} = -\frac{x^2}{2} + \log|x| + K \]
To match the given form, let's move all terms to one side: \[ e^y + e^{-y} + \frac{1}{2}x^2 - \log|x| - K = 0 \]
Now, compare this with the given form \(e^{\alpha y} + e^{\beta y} + \gamma x^2 + \delta \log|x| + C = 0\).
By comparing the coefficients, we get: \(\alpha = 1\)
\(\beta = -1\)
\(\gamma = \frac{1}{2}\)
\(\delta = -1\)
\(C = -K\)

The solution passes through the point (1, 1). Substitute x=1, y=1 into our solution to find K: \[ e^1 + e^{-1} + \frac{1}{2}(1)^2 - \log|1| = K \] \[ e + \frac{1}{e} + \frac{1}{2} - 0 = K \]
So, \(K = e + \frac{1}{e} + \frac{1}{2}\).
The constant C in the given form is \(C = -K\), so: \[ C = -\left(e + \frac{1}{e} + \frac{1}{2}\right) \]
We need to find the value of \((\alpha + \beta + \gamma + \delta - C)\): \[ = \left(1 + (-1) + \frac{1}{2} + (-1)\right) - \left(-\left(e + \frac{1}{e} + \frac{1}{2}\right)\right) \] \[ = \left(0 + \frac{1}{2} - 1\right) + \left(e + \frac{1}{e} + \frac{1}{2}\right) \] \[ = -\frac{1}{2} + e + \frac{1}{e} + \frac{1}{2} \] \[ = e + \frac{1}{e} \]

Step 4: Final Answer:

The value of the expression \((\alpha + \beta + \gamma + \delta - C)\) is \(e + \frac{1}{e}\).
Quick Tip: For variable separable differential equations, the main goal is to isolate all 'y' terms with 'dy' and all 'x' terms with 'dx' on opposite sides of the equation. Be careful with algebraic manipulations, especially with negative signs and logarithms.

Step 1: Understanding the Concept:

The problem asks for the variance of a discrete random variable given its probability distribution. First, we must find the value of the unknown 'k' using the property that the sum of all probabilities in a distribution is 1. Then, we calculate the mean (Expected Value, E[X]) and the expectation of the square of the variable (E[X²]). Finally, we use the variance formula.

Step 2: Key Formula or Approach:

1. Sum of probabilities: \(\sum P(X_i) = 1\).
2. Mean (Expected Value): \(E[X] = \mu = \sum X_i P(X_i)\).
3. Expectation of X²: \(E[X^2] = \sum X_i^2 P(X_i)\).
4. Variance: \(Var(X) = \sigma^2 = E[X^2] - (E[X])^2\).

Step 3: Detailed Explanation:
Find k:
The sum of all probabilities must be 1. \[ 0.2 + k + 2k + 2k = 1 \] \[ 0.2 + 5k = 1 \] \[ 5k = 1 - 0.2 = 0.8 \] \[ k = \frac{0.8}{5} = 0.16 \]
Now we can complete the probability distribution table: 

Calculate Mean (E[X]): \[ E[X] = (0 \times 0.2) + (1 \times 0.16) + (2 \times 0.32) + (3 \times 0.32) \] \[ E[X] = 0 + 0.16 + 0.64 + 0.96 = 1.76 \]

Calculate E[X²]: \[ E[X^2] = (0^2 \times 0.2) + (1^2 \times 0.16) + (2^2 \times 0.32) + (3^2 \times 0.32) \] \[ E[X^2] = (0 \times 0.2) + (1 \times 0.16) + (4 \times 0.32) + (9 \times 0.32) \] \[ E[X^2] = 0 + 0.16 + 1.28 + 2.88 = 4.32 \]

Calculate Variance (Var(X)): \[ Var(X) = E[X^2] - (E[X])^2 \] \[ Var(X) = 4.32 - (1.76)^2 \] \[ Var(X) = 4.32 - 3.0976 = 1.2224 \]

Convert to Fraction:
Now, let's check the options.
Option (A): \(\frac{764}{625}\).

Let's calculate this value: \(764 \div 625 = 1.2224\).

This matches our calculated variance.


Step 4: Final Answer:

The variance of the random variable X is 1.2224, which is equal to \(\frac{764}{625}\).
Quick Tip: To avoid errors in calculation, it's helpful to construct a table with columns for X, P(X), X*P(X), and X²*P(X). Summing the last two columns will directly give you E[X] and E[X²], making the final variance calculation straightforward.

Step 1: Understanding the Concept:

The problem asks for the minimum number of coin tosses, 'n', required to make the probability of a certain event (getting at least one head) exceed a specific value (90% or 0.9). The key is to use the complement rule of probability, as calculating the probability of "at least one" is often easier by calculating 1 minus the probability of "none".


Step 2: Key Formula or Approach:

Let 'n' be the number of times the coin is tossed.

The probability of getting a head (H) in a single toss of a fair coin is \(P(H) = 0.5\).

The probability of getting a tail (T) is \(P(T) = 0.5\).

The event "at least one head" is the complement of the event "no heads" (i.e., all tails). \[ P(at least one head) = 1 - P(no heads) \]
We are given the condition: \[ P(at least one head) > 0.90 \]

Step 3: Detailed Explanation:

First, let's find the probability of getting "no heads" in 'n' tosses. This means getting tails on all 'n' tosses.
Since the tosses are independent events, the probability of getting n tails in a row is: \[ P(no heads) = P(T and T and \dots n times) = (P(T))^n = (0.5)^n = \left(\frac{1}{2}\right)^n \]
Now, substitute this into our inequality: \[ 1 - \left(\frac{1}{2}\right)^n > 0.90 \]
Rearrange the inequality to solve for n: \[ 1 - 0.90 > \left(\frac{1}{2}\right)^n \] \[ 0.10 > \left(\frac{1}{2}\right)^n \] \[ \frac{1}{10} > \frac{1}{2^n} \]
To get rid of the fractions, we can take the reciprocal of both sides. When we do this, we must reverse the inequality sign: \[ 10 < 2^n \]
Now we need to find the smallest integer 'n' that satisfies this condition. We can test values of n:
If n = 1, \(2^1 = 2\), which is not greater than 10.

If n = 2, \(2^2 = 4\), which is not greater than 10.

If n = 3, \(2^3 = 8\), which is not greater than 10.

If n = 4, \(2^4 = 16\), which is greater than 10.


Step 4: Final Answer:

The minimum number of times the coin must be tossed is 4.
Quick Tip: Problems involving "at least one" are strong candidates for using the complement rule: P(A) = 1 - P(not A). Calculating the probability of the event *not* happening is often much simpler. In this case, 'not getting at least one head' means 'getting zero heads', which is a single, easy-to-calculate outcome.

Step 1: Understanding the Concept:

This question tests the fundamental properties of the Cumulative Distribution Function (CDF) for a standard normal variable Z, which has a mean of 0 and a standard deviation of 1.


Step 2: Detailed Explanation:

Let's evaluate each statement:

(A) \(F(Z) = \int_{-\infty}^{Z} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz, -\infty < Z < \infty\):

This is the mathematical definition of the CDF of a standard normal distribution. It represents the area under the standard normal curve from \(-\infty\) up to a specific value Z. This statement is correct. (Note: The OCR for the exponent was corrected to the standard form \(-z^2/2\)).


(B) \(F(-Z) = 1 - F(Z)\):

The standard normal distribution is symmetric about its mean, which is 0. The CDF \(F(-Z)\) represents the probability \(P(z \le -Z)\). Due to symmetry, this is equal to the probability \(P(z \ge Z)\). We know that \(P(z \ge Z) = 1 - P(z < Z) = 1 - F(Z)\). This statement is correct.


(C) \(F(0) = 0\):
\(F(0)\) is the area under the standard normal curve to the left of Z=0. Since the curve is symmetric about 0, the area to the left of 0 is exactly half of the total area. The total area is 1, so \(F(0) = 0.5\). This statement is incorrect.


(D) \(F(\infty) = 1\):

The CDF evaluated at positive infinity, \(F(\infty)\), represents the cumulative probability over the entire range of the variable, i.e., \(P(Z \le \infty)\). The total area under any probability density function must be equal to 1. This statement is correct.


Step 3: Final Answer:

Statements (A), (B), and (D) are correct properties of the standard normal CDF. Therefore, the correct option includes only these three statements.
Quick Tip: For the standard normal distribution, remember these key CDF values: \(F(-\infty) = 0\), \(F(0) = 0.5\), and \(F(\infty) = 1\). The symmetry property \(F(-z) = 1 - F(z)\) is also crucial and frequently tested.

Step 1: Understanding the Concept:

The "mean" of the numbers obtained from a random event like throwing a die refers to the expected value of the outcome. The expected value is a weighted average of all possible outcomes, where each outcome is weighted by its probability.


Step 2: Key Formula or Approach:

The mean or expected value E[X] of a discrete random variable X is calculated as: \[ E[X] = \sum_{i} x_i P(X=x_i) \]
where \(x_i\) are the possible outcomes and \(P(X=x_i)\) are their respective probabilities.


Step 3: Detailed Explanation:

First, we need to determine the probability of each outcome. The die has a total of 6 faces.

- The number 1 is on 3 faces, so \(P(X=1) = \frac{3}{6} = \frac{1}{2}\).

- The number 2 is on 2 faces, so \(P(X=2) = \frac{2}{6} = \frac{1}{3}\).

- The number 5 is on 1 face, so \(P(X=5) = \frac{1}{6}\).

(Check: The probabilities sum to 1: \(\frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1\)).


Now, we apply the formula for the expected value: \[ E[X] = (1 \times P(X=1)) + (2 \times P(X=2)) + (5 \times P(X=5)) \] \[ E[X] = \left(1 \times \frac{3}{6}\right) + \left(2 \times \frac{2}{6}\right) + \left(5 \times \frac{1}{6}\right) \] \[ E[X] = \frac{3}{6} + \frac{4}{6} + \frac{5}{6} \] \[ E[X] = \frac{3+4+5}{6} = \frac{12}{6} = 2 \]

Step 4: Final Answer:

The mean of the numbers obtained on throwing the die is 2.
Quick Tip: To calculate the mean of a discrete distribution, think of it as a weighted average. Multiply each possible value by its probability and sum the results. This is a fundamental concept in probability and statistics.

Step 1: Understanding the Concept:

A moving average is a technique used to smooth out short-term fluctuations in data and highlight longer-term trends or cycles. A "three-year moving average" is calculated by taking the average of three consecutive data points.


Step 2: Key Formula or Approach:

For a data series \(y_1, y_2, y_3, y_4, \dots\), the three-period moving averages are calculated as:
- First moving average = \(\frac{y_1 + y_2 + y_3}{3}\)
- Second moving average = \(\frac{y_2 + y_3 + y_4}{3}\)
- Third moving average = \(\frac{y_3 + y_4 + y_5}{3}\), and so on.


Step 3: Detailed Explanation:

The given data values are 15, 18, 21, 27, 39.

First three-year moving average:
This will be the average of the first three values (15, 18, 21). \[ Average_1 = \frac{15 + 18 + 21}{3} = \frac{54}{3} = 18 \]
Second three-year moving average:
This will be the average of the next three values (18, 21, 27). \[ Average_2 = \frac{18 + 21 + 27}{3} = \frac{66}{3} = 22 \]
Third three-year moving average:
This will be the average of the last three values (21, 27, 39). \[ Average_3 = \frac{21 + 27 + 39}{3} = \frac{87}{3} = 29 \]

Step 4: Final Answer:

The sequence of the three-year moving averages is 18, 22, 29.
Quick Tip: When calculating moving averages, the "window" of data slides one period at a time. For a 3-period moving average, you average periods 1-3, then 2-4, then 3-5, and so on. Note that you lose some data points at the beginning and end of the series (in this case, for 5 data points, you only get 3 moving averages).

Step 1: Understanding the Concept:

This question assesses the understanding of the purpose and applications of time series analysis. Time series analysis involves analyzing a sequence of data points collected over time to identify patterns and make forecasts.


Step 2: Detailed Explanation:

Let's analyze each statement:

(A) Time series analysis does not help to understand the behavior of a variable in the past.
This statement is incorrect. The very first step and a primary goal of time series analysis is to study past data to identify patterns like trends, seasonality, and cycles. Understanding the past is crucial for forecasting the future.


(B) Time series predict the future behavior of variable.
This statement is correct. One of the main applications of time series analysis is forecasting. By identifying the underlying patterns in past data, models can be built to extrapolate these patterns and predict future values.


(C) Time series helps to plan future operations.
This statement is correct. Businesses heavily rely on time series forecasts (e.g., for sales, demand) to plan for the future. This includes planning inventory levels, production schedules, staffing, and budgeting.


(D) The main aim of the time series analysis is to derive conclusions after arranging the time series in a systematic manner.
This statement is correct. Time series data is chronological. By arranging it systematically and analyzing it, we can derive conclusions about the underlying forces that affect the variable and its behavior over time.


Step 3: Final Answer:

Statements (B), (C), and (D) are correct descriptions of the purpose and utility of time series analysis. Statement (A) is incorrect. Therefore, the correct option is (C).
Quick Tip: Remember the core objectives of time series analysis: (1) Understand the past by identifying components like trend and seasonality. (2) Use this understanding to forecast the future. (3) Use the forecasts to make better decisions and plans.

Step 1: Understanding the Concept:

A time series is typically decomposed into several components that represent the different sources of variation in the data over time. The question asks to identify which option is not one of these standard components.


Step 2: Detailed Explanation:

The classical components of a time series are:

1. Trend (T): The long-term, general direction of the data (e.g., upward, downward, or stable). It represents the underlying growth or decline in the series over a long period.

2. Cyclical Component (C): Long-term wave-like patterns or fluctuations around the trend. These cycles are longer than a year and are often associated with business or economic cycles.

3. Seasonal Component (S): Short-term, regular patterns of variation that repeat within a year (e.g., quarterly, monthly, or weekly). Examples include higher retail sales in December or increased ice cream sales in summer.

4. Irregular or Random Component (I): The unpredictable, erratic variations in the data that are not accounted for by the other components. These are caused by random, unforeseen events.


Now let's evaluate the given options:

- (A) Trend component: This is a standard component.

- (B) Cyclical Component: This is a standard component.

- (C) Seasonal Component: This is a standard component.

- (D) Average Component: This is not a standard component of time series decomposition. The average (or mean) is a statistical measure of the entire series' central tendency, but it is not considered one of the dynamic components that describe the variation over time.


Step 3: Final Answer:

"Average Component" is not a standard component of a time series.
Quick Tip: Remember the four main components of a time series: Trend, Cyclical, Seasonal, and Irregular (T, C, S, I). Any option not on this list is likely the answer to a "which is not a component" question.

Step 1: Understanding the Concept:

The problem asks to find the parameters 'a' and 'b' for a linear trend line fitted using the method of least squares. The equation is given in a coded form, which simplifies calculations by shifting the origin of the time variable 'x'.


Step 2: Key Formula or Approach:

The trend line is \(y = a + bX\), where \(X = x - 2022\).
The normal equations for finding 'a' and 'b' are: \[ \sum y = na + b \sum X \] \[ \sum Xy = a \sum X + b \sum X^2 \]
Since the time variable is coded such that \(\sum X = 0\), the formulas simplify to: \[ a = \frac{\sum y}{n} \] \[ b = \frac{\sum Xy}{\sum X^2} \]

Step 3: Detailed Explanation:


Let's create a calculation table with the coded time variable \(X = x - 2022\).


The number of data points is n = 5.


Now, we can calculate 'a' and 'b' using the simplified formulas: \[ a = \frac{\sum y}{n} = \frac{16}{5} = 3.2 \] \[ b = \frac{\sum Xy}{\sum X^2} = \frac{2}{10} = 0.2 \]
The question asks for the value of the ratio \(\frac{a}{b}\). \[ \frac{a}{b} = \frac{3.2}{0.2} = \frac{32}{2} = 16 \]

Step 4: Final Answer:

The value of \(\frac{a}{b}\) is 16.
Quick Tip: When fitting a linear trend, always check if you can simplify the time variable. If the number of years is odd, choosing the middle year as the origin (as done here with `X = x - 2022`) makes \(\sum X = 0\). This greatly simplifies the calculations for the least squares parameters 'a' and 'b'.

Step 1: Understanding the Concept:

This question tests the definitions of fundamental terms used in inferential statistics, which involves drawing conclusions about a population based on data from a sample.


Step 2: Detailed Explanation:

Let's match each term in List-I with its correct definition in List-II.

(A) An observed set of population selected for analysis: This is the definition of a Sample. A sample is a subset of a larger population that is collected and analyzed.
Therefore, (A) matches with (IV).


(B) A specific characteristic of the population: A numerical value that describes a characteristic of an entire population (like the population mean, \(\mu\), or population standard deviation, \(\sigma\)) is called a Parameter.
Therefore, (B) matches with (I).


(C) A specific characteristic of the sample: A numerical value that describes a characteristic of a sample (like the sample mean, \(\bar{x}\), or sample standard deviation, s) is called a Statistic. Statistics are used to estimate population parameters.
Therefore, (C) matches with (III).


(D) A statement made about a population parameter for testing: In statistics, a claim or statement about a population parameter that is subject to verification is called a Hypothesis (specifically, a statistical hypothesis).
Therefore, (D) matches with (II).


Step 3: Final Answer:

The correct matching is: A-IV, B-I, C-III, D-II. This corresponds to option (D).
Quick Tip: A simple way to remember the difference between a parameter and a statistic is the "P-P" and "S-S" rule: \(\textbf{Parameters}\) describe \(\textbf{P}\)opulations, and \(\textbf{S}\)tatistics describe \(\textbf{S}\)amples.

Step 1: Understanding the Concept:

The t-distribution is used in hypothesis testing and for constructing confidence intervals when the population variance is unknown and the sample size is small. The question asks for the key assumptions required for its valid application.


Step 2: Detailed Explanation:

Let's analyze each statement:

(A) The variance of population is known.

This statement is incorrect. The t-distribution is specifically used when the population variance (\(\sigma^2\)) is unknown and must be estimated from the sample variance (\(s^2\)). If the population variance were known, the Z-distribution would be used.


(B) The samples are drawn from a normally distributed population.

This statement is correct. For the t-distribution to be applicable, especially with small sample sizes, the underlying population from which the sample is drawn should be normal or approximately normal.


(C) Sample standard deviation is an unbiased estimate of the population variance.

This statement is incorrect. The sample variance (\(s^2\)) is an unbiased estimator of the population variance (\(\sigma^2\)). The sample standard deviation (s) is a biased estimator of the population standard deviation (\(\sigma\)).


(D) It depends on a parameter known as degree of freedom.

This statement is correct. The shape of the t-distribution is determined by its degrees of freedom (df), which for a single sample is typically calculated as n-1 (where n is the sample size). As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.


Step 3: Final Answer:

The correct assumptions are that the samples are drawn from a normally distributed population (B) and that the distribution depends on degrees of freedom (D).
Quick Tip: Remember the key difference between using a Z-test and a t-test: a Z-test is used when the population standard deviation (\(\sigma\)) is known, while a t-test is used when \(\sigma\) is unknown and estimated by the sample standard deviation (s).

Step 1: Understanding the Concept:

A confidence interval provides a range of plausible values for a population parameter. The width of this interval depends on the confidence level, the sample variability, and the sample size. We can use the formula for the confidence interval to work backward and solve for the sample size.


Step 2: Key Formula or Approach:

The formula for a confidence interval for the mean is: \[ CI = \bar{x} \pm Margin of Error (ME) \]
Where the Margin of Error is given by: \[ ME = Z_{\alpha/2} \times \frac{s}{\sqrt{n}} \]
We can find the sample mean (\(\bar{x}\)) and the margin of error from the given interval.


Step 3: Detailed Explanation:

The given 95% confidence interval is [132, 160].

1. Calculate the sample mean (\(\bar{x}\)): The sample mean is the midpoint of the confidence interval. \[ \bar{x} = \frac{Upper Limit + Lower Limit}{2} = \frac{160 + 132}{2} = \frac{292}{2} = 146 \]
2. Calculate the Margin of Error (ME): The margin of error is half the width of the interval. \[ ME = \frac{Upper Limit - Lower Limit}{2} = \frac{160 - 132}{2} = \frac{28}{2} = 14 \]
3. Use the ME formula to find n:
We have:
- ME = 14
- Sample standard deviation, s = 50
- For a 95% confidence level, the critical Z-value is \(Z_{0.025}\) = 1.96.
Now, substitute these values into the formula: \[ 14 = 1.96 \times \frac{50}{\sqrt{n}} \]
Rearrange the equation to solve for \(\sqrt{n}\): \[ \sqrt{n} = \frac{1.96 \times 50}{14} \] \[ \sqrt{n} = \frac{98}{14} = 7 \]
Finally, square both sides to find n: \[ n = 7^2 = 49 \]

Step 4: Final Answer:

The size of the sample in the study is 49.
Quick Tip: The margin of error is the key link between the confidence interval, standard deviation, and sample size. If you are given the interval, you can always calculate the ME and then use its formula to solve for any unknown component.

Step 1: Understanding the Concept:

This question requires knowledge of the definitions of different financial instruments and concepts related to annuities. An annuity is a series of equal payments made at regular intervals.


Step 2: Detailed Explanation:

Let's define the given terms:

- Sinking Fund: A fund created by a company to set aside money over time to retire its debts or bonds. It involves making periodic payments for a fixed term to accumulate a specific future sum. It has a definite end date.

- Perpetuity: A type of annuity where the stream of cash flows continues indefinitely, or "forever". It does not have an end date. This perfectly matches the description in the question.

- Coupon payment: This is the periodic interest payment made by the issuer of a bond to the bondholder. While it's a periodic payment, it is a component of a bond, not the name of the annuity itself, and it typically ends when the bond matures.

- Bond: A debt instrument where an investor loans money to an entity (corporate or governmental) which borrows the funds for a defined period of time at a variable or fixed interest rate. It has a fixed maturity date.


Step 3: Final Answer:

An annuity with payments that continue forever is called a perpetuity.
Quick Tip: The word "perpetuity" comes from "perpetual," which means everlasting or never-ending. This linguistic link can help you remember its definition in finance.

Step 1: Understanding the Concept:

A sinking fund is a financial tool used primarily by corporations and governments to accumulate money for a specific future purpose, typically to repay a long-term debt or to replace a depreciating asset.


Step 2: Detailed Explanation:

Let's analyze each statement:

(A) It is a fixed term account.

This is correct. A sinking fund is established to meet a financial obligation at a specific future date, so it operates over a fixed term.


(B) It is a set-up for a particular upcoming expense.

This is correct. The purpose of a sinking fund is to earmark funds for a specific, predetermined future liability or capital expenditure, such as redeeming a bond issue or buying new machinery.


(C) A fixed amount at regular intervals is deposited in the Sinking Fund.

This is correct. The fund grows through a series of regular, periodic contributions (like an annuity), allowing the required amount to be accumulated systematically.


(D) It can be used in any emergency.

This is incorrect. A sinking fund is a restricted fund, meaning the money is designated for a specific purpose and cannot be used for general purposes or unrelated emergencies. This is different from a contingency fund or emergency fund.


Step 3: Final Answer:

Statements (A), (B), and (C) accurately describe a sinking fund. Statement (D) is incorrect. Therefore, the correct option is (B).
Quick Tip: Think of a sinking fund as a "targeted savings account" for a big, specific future expense. It's not a general-purpose rainy day fund. Its structure (regular payments over a fixed term) is the key to its function.

Step 1: Understanding the Concept:

This problem requires the calculation of an Equated Monthly Installment (EMI) for a loan. The reducing balance method means that interest is calculated each month on the outstanding principal.


Step 2: Key Formula or Approach:

The formula to calculate EMI is: \[ EMI = P \times r \times \frac{(1+r)^n}{(1+r)^n - 1} \]
where:
- \(P\) is the principal loan amount.
- \(r\) is the monthly interest rate.
- \(n\) is the number of monthly installments.


Step 3: Detailed Explanation:

1. Calculate the Principal Loan Amount (P): \[ P = Total House Cost - Down Payment \] \[ P = 39,65,000 - 5,00,000 = 34,65,000 \]
2. Calculate the Monthly Interest Rate (r):
The annual rate is 6%, compounded monthly. \[ r = \frac{6%}{12} = 0.5% = 0.005 \]
3. Calculate the Number of Installments (n):
The loan term is 25 years. \[ n = 25 years \times 12 months/year = 300 months \]
4. Calculate the EMI:
We are given that \((1+r)^n = (1.005)^{300} = 4.465\).
Now, substitute the values into the EMI formula: \[ EMI = 34,65,000 \times 0.005 \times \frac{(1.005)^{300}}{(1.005)^{300} - 1} \] \[ EMI = 17,325 \times \frac{4.465}{4.465 - 1} \] \[ EMI = 17,325 \times \frac{4.465}{3.465} \] \[ EMI \approx 17,325 \times 1.2886002886 \] \[ EMI \approx 22324.59 \]

Step 4: Final Answer:

Rounding to the nearest rupee, the EMI is Rupees 22,325.
Quick Tip: When an exam question provides a calculated value like \((1.005)^{300} = 4.465\), it's a strong hint that you are on the right track and should use this value directly in your formula. This saves you from performing complex exponentiation.

Step 1: Understanding the Concept:

This question tests the basic terminology of accounting, specifically related to the valuation of assets over time.


Step 2: Detailed Explanation:

Let's define the key terms:

- Original Value (or Cost): The amount paid to acquire an asset.

- Accumulated Depreciation: The total amount of depreciation expense that has been charged against an asset since it was put into service.

- Book Value: The value of an asset as it appears on the balance sheet. It is calculated as the original cost of the asset minus its accumulated depreciation. This exactly matches the definition in the question.

- Salvage Value (or Scrap Value): The estimated residual value of an asset at the end of its useful life. It's the amount the asset is expected to be sold for after it's fully depreciated.

- Lost Value: This is not a standard accounting term for asset valuation. The decline in value is captured by depreciation.


Step 3: Final Answer:

The original value of an asset minus the accumulated depreciation is the Book Value.
Quick Tip: Remember the fundamental accounting equation for an asset's worth on the books: **Book Value = Original Cost - Accumulated Depreciation**. The book value decreases over time as the asset is depreciated.

Step 1: Understanding the Concept:

This problem involves the straight-line (linear) method of depreciation. This method assumes that an asset loses an equal amount of value each year over its useful life. The scrap value is the remaining value after all depreciation has been accounted for.


Step 2: Key Formula or Approach:

The relationships under the linear method are:
1. Total Depreciation = Annual Depreciation \(\times\) Useful Life
2. Scrap Value = Original Cost - Total Depreciation


Step 3: Detailed Explanation:

Given values are:
- Original Cost = Rupees 36,000
- Useful Life = 10 years
- Annual Depreciation = Rupees 3,000


First, calculate the total depreciation over the asset's useful life: \[ Total Depreciation = 3,000 per year \times 10 years = 30,000 \]
Next, calculate the scrap value by subtracting the total depreciation from the original cost: \[ Scrap Value = 36,000 - 30,000 = 6,000 \]

Step 4: Final Answer:

The scrap value of the sofa set is Rupees 6,000.
Quick Tip: The formula for annual depreciation in the straight-line method is: (Cost - Scrap Value) / Useful Life. You can rearrange this formula to find any of the components if the others are known, as was required in this problem.

Step 1: Understanding the Concept:

The Compound Annual Growth Rate (CAGR) is the mean annual growth rate of an investment over a specified period longer than one year. It represents the constant rate at which the investment would have grown if it had compounded at the same rate each year.


Step 2: Key Formula or Approach:

The formula for CAGR is: \[ CAGR = \left( \frac{Ending Value}{Beginning Value} \right)^{1/n} - 1 \]
where:
- Ending Value is the value of the investment at the end of the period.
- Beginning Value is the value of the investment at the start of the period.
- \(n\) is the number of years.


Step 3: Detailed Explanation:

From the problem statement:
- Beginning Value = Rupees 10,000 (the initial investment)
- Ending Value = Rupees 14,000 (the value at the end of the last year, 2023)
- Number of years (\(n\)) = 6 years (the investment was held for 6 years)


Now, substitute these values into the CAGR formula: \[ CAGR = \left( \frac{14,000}{10,000} \right)^{1/6} - 1 \] \[ CAGR = (1.4)^{1/6} - 1 \]
The problem provides the value for \((1.4)^{1/6} \approx 1.058\). \[ CAGR \approx 1.058 - 1 = 0.058 \]
To express this as a percentage, multiply by 100: \[ CAGR = 0.058 \times 100% = 5.8% \]

Step 4: Final Answer:

The compound annual growth rate (CAGR) of the investment is 5.8%.
Quick Tip: CAGR is a smoothing metric. Notice that the investment value fluctuates year to year (it even decreased in 2021). CAGR ignores this volatility and provides a single, representative growth rate over the entire period. It only considers the starting and ending values.

Step 1: Understanding the Concept:

This question asks about the fundamental assumptions and requirements that define a Linear Programming Problem (LPP). An LPP is a mathematical technique for optimizing (maximizing or minimizing) a linear objective function, subject to a set of linear constraints.


Step 2: Detailed Explanation:


Let's review the options based on the definition of an LPP:

1. All the elements of an LPP should be quantifiable. This is a basic requirement. The objective function and constraints must be expressed in terms of numerical values (coefficients, constants). This is the "linearity" and "programmable" aspect. So, this is a requirement.


2. All decision variables should assume non-negative values. This is the non-negativity constraint (\(x_i \ge 0\)), a standard requirement in most LPPs, ensuring that the variables represent real-world quantities like production units, which cannot be negative. So, this is a requirement.


3. There are a finite number of decision variables and a finite number of constraints. This is the finiteness requirement. The problem must be bounded in scope with a specific number of variables to solve for and a specific number of conditions to meet. So, this is a requirement.


4. It deals with optimizing number of objectives more than one. This is not a requirement. A standard LPP is defined by having a single objective function to optimize. Problems that involve optimizing multiple objectives simultaneously are known as multi-objective optimization problems, which is a different field from standard linear programming.


Step 3: Final Answer:

The statement that is NOT a basic requirement of an LPP is that it deals with optimizing more than one objective.
Quick Tip: Remember the core components of an LPP: (1) a single linear objective function, (2) a set of linear constraints, and (3) non-negativity constraints for the decision variables. Any statement contradicting these core components is incorrect.

Step 1: Understanding the Concept:

This problem requires solving a two-variable LPP by finding the feasible region, identifying its corner points, and then evaluating the objective function at these points to find the maximum value and determine the nature of the optimal solution.


Step 2: Detailed Explanation:

1. Find the Corner Points of the Feasible Region:

The feasible region is defined by the constraints \(3x + 5y \le 15\), \(5x + 2y \le 10\), \(x \ge 0\), and \(y \ge 0\).

- Point 1 (Origin): (0, 0).

- Point 2 (y-intercept of 3x+5y=15): Set x=0, \(5y=15 \Rightarrow y=3\). Point is (0, 3).

- Point 3 (x-intercept of 5x+2y=10): Set y=0, \(5x=10 \Rightarrow x=2\). Point is (2, 0).

- Point 4 (Intersection of 3x+5y=15 and 5x+2y=10):

Multiply the first equation by 2: \(6x + 10y = 30\).

Multiply the second equation by 5: \(25x + 10y = 50\).

Subtract the new first from the new second: \(19x = 20 \Rightarrow x = 20/19\).

Substitute x back into \(5x+2y=10\): \(5(20/19) + 2y = 10 \Rightarrow 100/19 + 2y = 10 \Rightarrow 2y = 10 - 100/19 = 90/19 \Rightarrow y = 45/19\).

Point is (20/19, 45/19).


2. Analyze the Statements:

- (D) The feasible region is unbounded. This is false. The region is bounded by the axes and the two lines in the first quadrant.

- (B) The feasible region is bounded with corner points (0, 0), (2, 0), (20/19, 45/19) and (0, 3). This is true, as calculated above.


3. Evaluate the Objective Function Z = 5x + 2y at Corner Points:

- Z at (0, 0) = \(5(0) + 2(0) = 0\).

- Z at (0, 3) = \(5(0) + 2(3) = 6\).

- Z at (2, 0) = \(5(2) + 2(0) = 10\).

- Z at (20/19, 45/19) = \(5(20/19) + 2(45/19) = (100+90)/19 = 190/19 = 10\).


4. Analyze Optimality:

The maximum value of Z is 10. This maximum value occurs at two adjacent corner points, (2, 0) and (20/19, 45/19). When the optimal value is achieved at more than one corner point, it is also achieved at every point on the line segment connecting them.


- (A) The LPP has a unique optimal solution at (2, 0) only. This is false. While (2,0) is an optimal solution, it is not unique.

- (C) The optimal value is unique, but there are an infinite number of optimal solutions. This is true. The unique maximum value is 10, and there are infinite solutions on the line segment \(5x+2y=10\) between x=20/19 and x=2.


Step 3: Final Answer:

The correct statements are (B) and (C). Therefore, option (D) is the correct choice.
Quick Tip: A key indicator of multiple optimal solutions in an LPP is when the slope of the objective function line is the same as the slope of one of the boundary lines of the feasible region. Here, the slope of Z is -5/2, which is the same as the slope of the constraint line \(5x+2y=10\).