CBSE Class 10 Maths Sample Papers 2026: Practice Sets with Solutions

Question: If the HCF of $306$ and $657$ is $9$, what is their LCM?

(a) $22338$ (b) $23388$ (c) $23328$ (d) 22__(none)

Solution:
Using the formula:

**Answer: (a) $22338$**

Question: If $\alpha$ and $\beta$ are the zeroes of $x^2 - 5x + 6$, find $\alpha^2 + \beta^2$.

(a) $13$ (b) $25$ (c) $11$ (d) $37$

Solution:
$\alpha + \beta = 5$ and $\alpha \beta = 6$.

**Answer: (a) $13$**

Question: Find the distance between the points $A(3, -4)$ and $B(-3, 4)$.

Solution:

Question: Solve $2x^2 - 7x + 3 = 0$ using the quadratic formula.

Solution:
Here $a = 2$, $b = -7$, $c = 3$.

Since $D > 0$, two distinct real roots exist.

The roots are $x = 3$ and $x = \frac{1}{2}$.

Question: Find the mean of the following frequency distribution using the step deviation method.

Solution:
Let assumed mean $a = 25$, class width $h = 10$.

The mean is $27.8$.

Question: The $10$th term of the AP $2, 7, 12, 17, \ldots$ is:

(a) $42$ (b) $47$ (c) $52$ (d) $57$

Solution:
$a = 2$, $d = 5$.

**Answer: (b) $47$**

Question: If $\sin\theta = \frac{3}{5}$, then $\cos\theta$ is:

(a) $\frac{4}{5}$ (b) $\frac{3}{4}$ (c) $\frac{5}{3}$ (d) $\frac{5}{4}$

Solution:
Using $\sin^2\theta + \cos^2\theta = 1$:

**Answer: (a) $\frac{4}{5}$**

Question: Prove that $\frac{\sin\theta}{1 + \cos\theta} + \frac{1 + \cos\theta}{\sin\theta} = 2\csc\theta$.

Solution:
Taking LHS:

Taking LCM:

Expanding the numerator:

Using $\sin^2\theta + \cos^2\theta = 1$:

Hence proved.

Question: Find the sum of the first $20$ terms of the AP $3, 8, 13, 18, \ldots$

Solution:
$a = 3$, $d = 5$, $n = 20$.

The sum of the first $20$ terms is $1010$.

Question: The sum of a two-digit number and the number obtained by reversing its digits is $121$. If the digits differ by $3$, find the number.

Solution:
Let the tens digit be $x$ and the units digit be $y$.

The number is $10x + y$. The reversed number is $10y + x$.

Equation 1: $(10x + y) + (10y + x) = 121$

Equation 2: $x - y = 3$ (assuming $x > y$) $\ldots (ii)$

Adding $(i)$ and $(ii)$:

The number is $74$.

Verification: $74 + 47 = 121$ and $7 - 4 = 3$. Both conditions satisfied.

(If $y > x$, then $y - x = 3$, giving $x = 4, y = 7$, and the number is $47$.)

Question: From an external point $P$, two tangents $PA$ and $PB$ are drawn to a circle with centre $O$. If $\angle APB = 60°$, then $\angle AOB$ is:

(a) $90°$ (b) $120°$ (c) $150°$ (d) $60°$

Solution:
$\angle OAP = \angle OBP = 90°$ (radius perpendicular to tangent).
In quadrilateral $OAPB$:

**Answer: (b) $120°$**

Question: The area of a sector with radius $7$ cm and angle $60°$ is:

(a) $\frac{77}{3}$ cm$^2$ (b) $\frac{77}{6}$ cm$^2$ (c) $\frac{154}{3}$ cm$^2$ (d) $\frac{49}{3}$ cm$^2$

Solution:

**Answer: (a) $\frac{77}{3}$ cm$^2$**

Question: A solid is in the shape of a cone mounted on a hemisphere. The radius of each is $7$ cm and the height of the cone is $24$ cm. Find the total surface area of the solid.

Solution:
Radius $r = 7$ cm, height of cone $h = 24$ cm.

Slant height of cone:

CSA of cone $= \pi r l = \frac{22}{7} \times 7 \times 25 = 550$ cm$^2$

CSA of hemisphere $= 2\pi r^2 = 2 \times \frac{22}{7} \times 49 = 308$ cm$^2$

Total surface area $= 550 + 308 = 858$ cm$^2$

Note: The base of the cone sits on the hemisphere, so we do not include the base circle.

Case Study: A bag contains $5$ red balls, $3$ blue balls, and $2$ green balls. A ball is drawn at random.

(i) Find the probability of drawing a red ball. (1 mark)

Solution:
Total balls $= 5 + 3 + 2 = 10$.

(ii) Find the probability of drawing a ball that is NOT green. (1 mark)

Solution:

(iii) If $2$ more blue balls are added to the bag, what is the new probability of drawing a blue ball? (2 marks)

Solution:
New total $= 10 + 2 = 12$. Blue balls $= 3 + 2 = 5$.

Case Study: A garden is in the shape of a triangle with vertices at $A(0, 0)$, $B(8, 0)$, and $C(4, 6)$ on a coordinate grid (in metres).

(i) Find the length of side $AB$. (1 mark)

Solution:

(ii) Find the coordinates of the midpoint of $AC$. (1 mark)

Solution:

(iii) Find the area of the garden. (2 marks)

Solution: