CBSE Class 10 Maths Standard 2026 Question Paper 30/2/1 with Solutions

The LCM of 960 and 240 is :

1. (A)960
2. (B)240
3. (C)60
4. (D)15

The natural number 1 is :

1. (A)a prime number.
2. (B)a composite number.
3. (C)prime as well as composite.
4. (D)neither prime nor composite.

For any natural number n, $5^n$ ends with the digit :

1. (A)0
2. (B)5
3. (C)3
4. (D)2

The graph of $y=f(x)$ is given. The number of distinct zeroes of $y=f(x)$ is :

1. (A)0
2. (B)1
3. (C)2
4. (D)3

If $\alpha$ and $\beta$ are two zeroes of a polynomial $f(x)=p{x}^2-2x+3p$ and $\alpha +\beta =\alpha \beta$, then value of p is :

1. (A)$-\frac{2}{3}$
2. (B)$\frac{2}{3}$
3. (C)$\frac{1}{3}$
4. (D)$-\frac{1}{3}$

If the pair of linear equations : $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ is consistent and dependent, then

1. (A)$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
2. (B)$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}=\frac{c_1}{c_2}$
3. (C)$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
4. (D)$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Which of the following sequence is not an A.P. ?

1. (A)$2,\frac{5}{2},3,\frac{7}{2},...$
2. (B)$-1.2,-3.2,-5.2,-7.2,...$
3. (C)$\sqrt{2},\sqrt{8},\sqrt{18},...$
4. (D)$1^2,3^2,5^2,7^2,...$

In triangles ABC and PQR, $\angle A=\angle Q$ and $\angle B=\angle R$, then AB : AC is equal to :

1. (A)PQ : PR
2. (B)PQ : QR
3. (C)QR : QP
4. (D)PR : QR

The distance of the point A(4a, 3a) from x-axis is :

1. (A)3a
2. (B)$-3a$
3. (C)4a
4. (D)$-4a$

If $\cos A=\frac{4}{5}$, then the value of $\tan A$ is :

1. (A)$\frac{3}{5}$
2. (B)$\frac{3}{4}$
3. (C)$\frac{4}{3}$
4. (D)$\frac{5}{3}$

If $2\sin A=1$, then the value of $\tan A+\cot A$ is :

1. (A)$\sqrt{3}$
2. (B)$\frac{4}{\sqrt{3}}$
3. (C)$\frac{\sqrt{3}}{2}$
4. (D)1

From a point on the ground, which is 60 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be $45^{\circ }$. The height (in metres) of the tower is :

1. (A)$10\sqrt{3}$
2. (B)$30\sqrt{3}$
3. (C)60
4. (D)30

In the given figure, PA and PB are tangents to a circle centred at O. If $\angle OAB=15^{\circ }$, then $\angle APB$ equals :

1. (A)$30^{\circ }$
2. (B)$15^{\circ }$
3. (C)$45^{\circ }$
4. (D)$10^{\circ }$

In the given figure, PA and PB are tangents to a circle centred at O. If $\angle AOB=130^{\circ }$, then $\angle APB$ is equal to :

1. (A)$130^{\circ }$
2. (B)$50^{\circ }$
3. (C)$120^{\circ }$
4. (D)$90^{\circ }$

Area of a segment of a circle of radius 'r' and central angle $60^{\circ }$ is :

1. (A)$\frac{\pi r^2}{2}-\frac{1}{2}{r}^2$
2. (B)$\frac{2\pi r}{4}-\frac{\sqrt{3}}{4}{r}^2$
3. (C)$\frac{\pi r^2}{6}-\frac{\sqrt{3}}{4}{r}^2$
4. (D)$\frac{2\pi r}{4}-r^2\sin 60^{\circ }$

A hemispherical bowl is made of steel of thickness 1 cm. The outer radius of the bowl is 6 cm. The volume of steel used (in $c{m}^3$) is :

1. (A)$182\pi$
2. (B)$\frac{182}{3}\pi$
3. (C)$\frac{682}{3}\pi$
4. (D)$\frac{364}{3}\pi$

The mean and median of a frequency distribution are 43 and 43.4 respectively. The mode of the distribution is :

1. (A)43.4
2. (B)42.4
3. (C)44.2
4. (D)49.3

The probability for a randomly selected number out of 1, 2, 3, 4, ..., 25 to be a composite number is :

1. (A)$\frac{15}{25}$
2. (B)$\frac{10}{25}$
3. (C)$\frac{11}{25}$
4. (D)$\frac{9}{25}$

Assertion (A) : The surface area of the cuboid formed by joining two cubes of sides 4 cm each, end-to-end, is 160 $c{m}^2$.
Reason (R): The surface area of a cuboid of dimensions $l\times b\times h$ is $(lb+bh+hl)$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

Assertion (A) : The mean of first 'n' natural numbers is $\frac{n-1}{2}$.
Reason (R): The sum of first 'n' natural numbers is $\frac{n(n+1)}{2}$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. (B)Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

If $\alpha$, $\beta$ are the zeroes of the quadratic polynomial $p{x}^2+qx+r$, then find the value of ${\alpha }^3\beta +{\beta }^3\alpha$.

In the given figure, $△AHK\sim △ABC$. If AK = 10 cm, BC = 3.5 cm and HK = 7 cm, find the length of AC.

In the given figure, XY $\parallel$ QR, $\frac{PQ}{XQ}=\frac{7}{3}$ and PR = 6.3 cm. Find the length of YR.

If the points A(4, 5), B(m, 6), C(4, 3) and D(1, n) taken in this order are the vertices of a parallelogram ABCD, then find the values of m and n.

If $\tan \theta +\frac{1}{\tan \theta }=2$, find the value of ${\tan }^2\theta +\frac{1}{{\tan }^2\theta }$.

Prove that : $\sqrt{\frac{1-\sin \theta }{1+\sin \theta }}=\sec \theta -\tan \theta$

In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.

Prove that $\sqrt{5}$ is an irrational number.

Find the coordinates of the points of trisection of the line segment joining the points A(-1, 4) and B(-3, -2).

Prove that : $\frac{{\sec }^3\theta }{{\sec }^2\theta -1}+\frac{cose{c}^3\theta }{cose{c}^2\theta -1}=\sec \theta .cosec\theta (\sec \theta +cosec\theta )$

If $\frac{\sec \alpha }{cosec\beta }=p$ and $\frac{\tan \alpha }{cosec\beta }=q$, then prove that $(p^2-q^2){\sec }^2\alpha =p^2$.

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $\angle PTQ=2\angle OPQ$.

Find the area of the sector of a circle of radius 42 cm and of central angle $30^{\circ }$. Also, find the area of the corresponding major sector. [Use $\pi =\frac{22}{7}$]

Two dice are thrown at the same time. Determine the probability that the (i) sum of the numbers on the two dice is 5, and (ii) difference of the numbers on the two dice is 3.

Aarush bought 2 pencils and 3 chocolates for ₹ 11 and Tanish bought 1 pencil and 2 chocolates for ₹ 7 from the same shop. Represent this situation in the form of a pair of linear equations. Find the price of 1 pencil and 1 chocolate, graphically.

A person on a tour has ₹ 4,200 for expenses. If he extends his tour for 3 days, he has to cut down his daily expenses by ₹ 70. Find the original duration of the tour.

The area of a right-angled triangle is 600 $c{m}^2$. If the base of the triangle exceeds the altitude by 10 cm, find all the three dimensions of the triangle.

Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

As shown in the given figure, a girl of height 90 cm is walking away from the base of a lamp post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds.

An SBI health insurance agent found the following data for distribution of ages of 100 policy holders. The health insurance policies are given to persons of age 15 years and onwards, but less than 60 years.

Find the modal age and median age of the policy holders.

Your elder brother wants to buy a car and plans to take a loan from a bank for his car. He repays his total loan of ₹ 1,18,000 by paying every month, starting with the first instalment of ₹ 1,000 and he increases the instalment by ₹ 100 every month.
Based on the information given above, answer the following questions :
(i) Find the amount paid by him in the $30^{th}$ instalment. (1)
(ii) If the total number of instalments is 40, what is the amount paid in the last instalment ? (1)
(iii) (a) What amount does he still have to pay after the $30^{th}$ instalment ? (2)
OR
(iii) (b) Find the ratio of the tenth instalment to the last instalment. (2)

Tejas is standing at the top of a building and observes a car at an angle of depression of $30^{\circ }$ as it approaches the base of the building at a uniform speed. 6 seconds later, the angle of depression increases to $60^{\circ }$, and at that moment, the car is 25 m away from the building.
Based on the information given above, answer the following questions :
(i) What is the height of the building ? (1)
(ii) What is the distance between the two positions of the car ? (1)
(iii) (a) What would be the total time taken by the car to reach the foot of the building from the starting point ? (2)
OR
(iii) (b) What is the distance of the observer from the car when it makes an angle of $60^{\circ }$ ? (2)

On a Sunday your parents took you to a fair. You could see lot of toys displayed and you wanted them to buy a Rubik's cube and a strawberry ice-cream for you.
Based on the information given above, answer the following questions :
(i) Find the length of the diagonal of Rubik's cube if each edge measures 6 cm. (1)
(ii) Find the volume of Rubik's cube if the length of the edge is 7 cm. (1)
(iii) (a) What is the curved surface area of hemisphere (ice-cream) if the base radius is 7 cm ? (2)
OR
(iii) (b) If two cubes of edges 4 cm are joined end-to-end, then find the surface area of the resulting cuboid. (2)