CBSE 2026 Class 10 Mathematics Standard(041) Question Paper with Solutions PDF

Step 1: Understanding the Concept:

The Highest Common Factor (HCF) is the largest positive integer that divides each of the integers without leaving a remainder. We can find this using Prime Factorization or Euclid's Division Algorithm.


Step 2: Key Formula or Approach:

Euclid's Division Lemma: \( a = bq + r \), where \( 0 \leq r < b \). Repeat the process until \( r = 0 \); the last divisor is the HCF.


Step 3: Detailed Explanation:

1. Apply Euclid's Division Algorithm on 960 and 432: \[ 960 = 432 \times 2 + 96 \]
2. Now take 432 as the dividend and 96 as the divisor: \[ 432 = 96 \times 4 + 48 \]
3. Now take 96 as the dividend and 48 as the divisor: \[ 96 = 48 \times 2 + 0 \]
4. Since the remainder is now 0, the divisor at this stage is the HCF.


Step 4: Final Answer:

The HCF of 960 and 432 is 48. Quick Tip: To check your answer, divide both numbers by the HCF. \( 960/48 = 20 \) and \( 432/48 = 9 \). Since 20 and 9 have no common factors other than 1, 48 is indeed the HCF.

Step 1: Understanding the Concept:

A **prime number** is a natural number greater than 1 that has no positive divisors other than 1 and itself. A **composite number** has at least one divisor other than 1 and itself.


Step 2: Key Formula or Approach:

List the factors of the number 2.


Step 3: Detailed Explanation:

1. The factors of 2 are 1 and 2.
2. Since it has exactly two factors (1 and itself), it fits the definition of a prime number.
3. Note: 2 is the only **even** prime number. All other even numbers are divisible by 2, making them composite.


Step 4: Final Answer:

The natural number 2 is a prime number. Quick Tip: Remember that 1 is the only natural number that is neither prime nor composite. Every other natural number falls into one of those two categories.

Step 1: Understanding the Concept:

The "last digit" or "units digit" of a power follows a cyclical pattern. For the base 6, we can observe the powers to find the cycle.


Step 2: Key Formula or Approach:

Calculate the first few powers of 6 to identify the pattern.


Step 3: Detailed Explanation:

1. \( 6^1 = 6 \)
2. \( 6^2 = 36 \)
3. \( 6^3 = 216 \)
4. \( 6^4 = 1296 \)
Observation: Regardless of the power \( n \), the product of any number ending in 6 multiplied by 6 will always result in a number ending in 6 (\( 6 \times 6 = 36 \)).


Step 4: Final Answer:

For any natural number \( n \), \( 6^n \) ends with the digit 6. Quick Tip: The digits 0, 1, 5, and 6 always maintain their own value at the units place for any power \( n > 0 \). (e.g., \( 5^n \) always ends in 5).

Step 1: Understanding the Concept:

The "zeroes" of a function \( f(x) \) are the values of \( x \) for which \( f(x) = 0 \). Geometrically, these correspond to the points where the graph of the function intersects the \( x \)-axis.




Step 2: Key Formula or Approach:

Count the number of intersection points between the curve and the horizontal \( x \)-axis.


Step 3: Detailed Explanation:

1. Based on the description:
- Intersection 1: Left of the origin.
- Intersection 2: At/near the origin.
- Intersection 3: Right of the origin.
2. Total points of contact with the \( x \)-axis = 3.


Step 4: Final Answer:

The number of zeroes of \( f(x) \) is 3. Quick Tip: If a graph "touches" the x-axis (turns back) without crossing, it still counts as a zero (a repeated root). Always count every point where the graph meets the axis.

Step 1: Understanding the Concept:

A solution to a pair of linear equations corresponds to a point of intersection between their two lines.




Step 2: Key Formula or Approach:

Compare the ratios of coefficients for \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \).
For coincident lines: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \).


Step 3: Detailed Explanation:

1. Coincident lines lie exactly on top of each other.
2. This means every single point on the first line is also a point on the second line.
3. Since a line consists of an infinite number of points, there are infinitely many points that satisfy both equations simultaneously.


Step 4: Final Answer:

The pair of equations has an infinite number of solutions. Quick Tip: Summary of Linear Equation Solutions: * Intersecting lines = Unique solution. * Parallel lines = No solution. * Coincident lines = Infinite solutions.

Step 1: Understanding the Concept:

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (\(d\)).


Step 2: Key Formula or Approach:

The common difference is calculated as: \[ d = a_{n} - a_{n-1} \]
Commonly, \(d = a_2 - a_1\).


Step 3: Detailed Explanation:

1. Let the first term \(a_1 = \sqrt{2}\).
2. Let the second term \(a_2 = 2\sqrt{2}\).
3. Calculate the difference: \[ d = 2\sqrt{2} - \sqrt{2} \] \[ d = \sqrt{2}(2 - 1) = \sqrt{2} \]
4. Verification: \(3\sqrt{2} - 2\sqrt{2} = \sqrt{2}\). The difference is consistent.


Step 4: Final Answer:

The common difference is \(\sqrt{2}\). Quick Tip: When working with surds (roots), treat the radical like a variable. For example, \(2\sqrt{2} - \sqrt{2}\) is just like \(2x - x\).

Step 1: Understanding the Concept:

When two triangles are similar (\(\Delta ABC \sim \Delta DEF\)), the ratios of their corresponding sides are equal.




Step 2: Key Formula or Approach:
\[ \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} \]


Step 3: Detailed Explanation:

1. Given \(2 AB = DE\). This can be written as the ratio: \[ \frac{AB}{DE} = \frac{1}{2} \]
2. Since the triangles are similar, use the property of corresponding sides: \[ \frac{AB}{DE} = \frac{BC}{EF} \]
3. Substitute the known values (\(\frac{1}{2}\) and \(BC = 8\)): \[ \frac{1}{2} = \frac{8}{EF} \]
4. Cross-multiply to solve for \(EF\): \[ EF = 8 \times 2 = 16 cm \]


Step 4: Final Answer:

The length of \(EF\) is 16 cm. Quick Tip: Always ensure you are matching the correct corresponding sides. In \(\Delta ABC \sim \Delta DEF\), \(BC\) corresponds to \(EF\), and \(AB\) corresponds to \(DE\).

Step 1: Understanding the Concept:

The midpoint is the point exactly halfway between two endpoints. A point lies on the **x-axis** if its y-coordinate is 0, and on the **y-axis** if its x-coordinate is 0.


Step 2: Key Formula or Approach:

Midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)


Step 3: Detailed Explanation:

1. Coordinates are \((x_1, y_1) = (5, -4)\) and \((x_2, y_2) = (6, 4)\).
2. Calculate the x-coordinate of the midpoint: \[ x = \frac{5 + 6}{2} = \frac{11}{2} = 5.5 \]
3. Calculate the y-coordinate of the midpoint: \[ y = \frac{-4 + 4}{2} = \frac{0}{2} = 0 \]
4. The midpoint is \((5.5, 0)\). Since the y-coordinate is 0, the point lies on the x-axis.


Step 4: Final Answer:

The midpoint lies on the x-axis. Quick Tip: If two points have y-coordinates that are negatives of each other (like -4 and 4), their midpoint will always have a y-coordinate of 0 and thus lie on the x-axis.

Step 1: Understanding the Concept:

Trigonometric ratios are related via the Pythagorean identity. In a right-angled triangle, if \(\sin \theta = \frac{Opposite}{Hypotenuse}\), we can find the adjacent side using Pythagoras' theorem.




Step 2: Key Formula or Approach:

Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \implies \cos \theta = \sqrt{1 - \sin^2 \theta} \)


Step 3: Detailed Explanation:

1. Substitute \(\sin \theta = \frac{a}{b}\) into the identity: \[ \cos \theta = \sqrt{1 - \left(\frac{a}{b}\right)^2} \]
2. Simplify the expression: \[ \cos \theta = \sqrt{1 - \frac{a^2}{b^2}} = \sqrt{\frac{b^2 - a^2}{b^2}} \]
3. Extract the denominator from the square root: \[ \cos \theta = \frac{\sqrt{b^2 - a^2}}{b} \]


Step 4: Final Answer:
\(\cos \theta = \frac{\sqrt{b^2 - a^2}}{b}\). Quick Tip: Using the "SOH CAH TOA" mnemonic: \(\sin\) is Opposite/Hypotenuse (\(a/b\)). By Pythagoras, the Adjacent side is \(\sqrt{b^2 - a^2}\). \(\cos\) is Adjacent/Hypotenuse.

Step 1: Understanding the Concept:

The expression \(\sin^2 \theta + \cos^2 \theta\) is a fundamental trigonometric identity. It is constant for any valid angle \(\theta\).


Step 2: Key Formula or Approach:

Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \) for any value of \(\theta\).


Step 3: Detailed Explanation:

1. The question provides \(\cos A = 1/2\). While we could find \(A = 60^\circ\) and then find \(\sin A\), it is unnecessary.
2. The identity \(\sin^2 A + \cos^2 A\) is independent of the value of \(A\).
3. Therefore, regardless of whether \(\cos A\) is \(1/2\), \(1/3\), or any other value, the sum of squares is always 1.


Step 4: Final Answer:

The value is 1. Quick Tip: This is a "trick" question. Don't waste time calculating the specific values of sine and cosine if you recognize a fundamental identity!