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

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

There are two sections A and B of Grade X. There are 28 students in Section A and 30 students in Section B. What is the minimum number of books you will acquire for the class library so that they can be distributed equally among students of Section A or Section B ?

1. (A)144
2. (B)2
3. (C)420
4. (D)272

The pair of linear equations $\frac{3x}{2}+\frac{5y}{3}=7$ and $9x+10y=14$, is :

1. (A)consistent
2. (B)inconsistent
3. (C)consistent with one solution
4. (D)consistent with many solutions

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.

The value of x for which 2x, (x + 10) and (3x + 2) are the three consecutive terms of an A.P. is :

1. (A)6
2. (B)$-6$
3. (C)18
4. (D)$-18$

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

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

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)$-\frac{1}{3}$

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

If the distance between the points (4, p) and (1, 0) is 5, then p is equal to :

1. (A)$\pm 4$
2. (B)4
3. (C)$-4$
4. (D)0

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)$\frac{364}{3}\pi$

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)$\frac{5}{3}$

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)$\frac{2\pi r}{4}-r^2\sin 60^{\circ }$

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

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)$10^{\circ }$

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

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)$\frac{9}{25}$

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)$90^{\circ }$

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.

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.

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.

Evaluate : $\frac{5{\cos }^{2}60^{\circ }+4{\sec }^{2}30^{\circ }-{\tan }^{2}45^{\circ }}{{\sin }^{2}30^{\circ }+{\cos }^{2}30^{\circ }}$

Prove that : $1+\frac{{\cot }^2\alpha }{1+cosec\alpha }=cosec\alpha$

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

Find the value of p, for which one zero of the quadratic polynomial $p{x}^2-14x+8$ is 6 times the other.

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.

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 $(p^2-q^2){\sec }^2\alpha =p^2$.

Find the area of the segment AYB shown in the figure, if the radius of the circle is 21 cm and $\angle AOB=120^{\circ }$. [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.

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 the tangent at any point of a circle is perpendicular to the radius through the point of contact.

If a regular hexagon ABCDEF circumscribes a circle, then prove that AB + CD + EF = BC + DE + FA.

If the median of the following distribution is 32.5, then find the values of x and y.

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.

In a flight of 600 km, an aircraft slowed down its speed due to bad weather. Its average speed for the trip reduced by 200 km/h from its usual speed and time of flight increased by 30 minutes. Find the scheduled duration of the flight.

Two pipes are used to fill a swimming pool. If the pipe of the larger diameter is used for 4 hours and the pipe of the smaller diameter for 9 hours, only half of the pool can be filled. Find how long it would take for each pipe to fill the pool, separately, if the pipe of smaller diameter takes 10 hours more than the pipe of larger diameter to fill the pool.

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.

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)

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)