CBSE Class 10 Maths Basic 2025 Question Paper 430/4/2 with Solutions

The $20^{th}$ term of the A.P. : $10\sqrt{2},6\sqrt{2},2\sqrt{2},....$ is :

1. (A)$-76+10\sqrt{2}$
2. (B)$-62\sqrt{2}$
3. (C)$-66\sqrt{2}$
4. (D)$86\sqrt{2}$

If $\sec \theta -\tan \theta =2$, then $\sec \theta +\tan \theta$ is equal to :

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

PQ and PR are tangents to the circle of radius 3 cm and centre O. If length of each tangent is 4 cm, then perimeter of $△$ OQP is :

1. (A)5 cm
2. (B)12 cm
3. (C)9 cm
4. (D)8 cm

The value of $m$ for which lines $14x+my=20$ and $-3x+2y=16$ are parallel, is :

1. (A)$-\frac{3}{14}$
2. (B)$-\frac{7}{3}$
3. (C)$-\frac{28}{3}$
4. (D)$-\frac{3}{28}$

If $n^{th}$ term of an A.P. is $5n-6$, then its common difference is :

1. (A)$-6$
2. (B)$5n$
3. (C)5
4. (D)6

The distance between the points $(2,-7)$ and $(-2,-1)$ is :

1. (A)10
2. (B)$2\sqrt{13}$
3. (C)8
4. (D)$4\sqrt{13}$

The length of arc subtending an angle of $210^{\circ }$ at the centre of the circle, is $\frac{44}{3}$ cm. The radius of the circle is :

1. (A)$2\sqrt{2}$ cm
2. (B)4 cm
3. (C)8 cm
4. (D)$\frac{1}{4}$ cm

A card is drawn at random from a well shuffled deck of 52 playing cards. The probability that drawn card shows number ‘9’ is :

1. (A)$\frac{1}{26}$
2. (B)$\frac{4}{13}$
3. (C)$\frac{1}{52}$
4. (D)$\frac{1}{13}$

In $△$ ABC, PQ $\parallel$ BC. It is given that AP = 2.4 cm, PB = 3.6 cm and BC = 5.4 cm. PQ is equal to :

1. (A)2.7 cm
2. (B)1.8 cm
3. (C)3.6 cm
4. (D)2.16 cm

PA and PB are tangents to a circle with centre O. If $\angle AOB=105^{\circ }$ then $\angle OAP+\angle APB$ is equal to :

1. (A)$75^{\circ }$
2. (B)$175^{\circ }$
3. (C)$180^{\circ }$
4. (D)$165^{\circ }$

Two dice are rolled together. The probability that at least one of them shows a six, is :

1. (A)$\frac{12}{36}$
2. (B)$\frac{5}{36}$
3. (C)$\frac{11}{36}$
4. (D)$\frac{6}{36}$

The curved surface area of a cone with base radius 7 cm, is 550 cm${}^2$. The slant height of the cone is :

1. (A)25 cm
2. (B)14 cm
3. (C)20 cm
4. (D)24 cm

If $\tan A=\frac{1}{2}$, then $\sin A$ is equal to :

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

If $\sqrt{2}\sin \theta =1$, then $\cot \theta \times \text{cosec}\theta$ is equal to :

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

Three coins are tossed together. The probability that only one coin shows tail, is :

1. (A)$\frac{1}{2}$
2. (B)$\frac{3}{8}$
3. (C)$\frac{7}{8}$
4. (D)1

In the given figure, graph of $p(x)$ is shown. Number of distinct zeroes of $p(x)$ is :

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

Two right circular cylinders of equal volumes have their heights in the ratio 1:2. The ratio of their radii is :

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

$\alpha ,\beta$ are zeroes of the polynomial $2x^2+5x+1$. The value of $\left(\frac{1}{\alpha }+\frac{1}{\beta }\right)$ is :

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

Assertion (A) : If E is an event such that $P(E)=\frac{1}{999}$, then $P(\overline{E})=0.001$.
Reason (R) : $P(E)+P(\overline{E})=1$

1. (A)Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. (B)Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of 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) : Median marks of students in a class test is 16. It means half of the class got marks less than 16.
Reason (R) : Median divides the distribution in two equal parts.

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

Two chords BA and CD intersect at point P outside the circle. Prove that $△PDA\sim △PBC$

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Find the ratio in which point P $(-1,m)$ divides the line segment joining the points A$(2,5)$ and B$(-5,-2)$. Hence, find the value of $m$.

Show that $45^n$ can not end with the digit 0, $n$ being a natural number. Write the prime number ‘$a$’ which on multiplying with $45^n$ makes the product end with the digit 0.

A coin is dropped at random on the rectangular region shown in the figure. What is the probability that it will land inside the circle with radius 0.7 m ?

A die is thrown twice. What is the probability that (i) difference between two numbers obtained is 3 ? (ii) sum of the numbers obtained is 8 ?

If points A$(-5,y)$, B$(2,-2)$, C$(8,4)$ and D$(x,5)$ taken in order, form a parallelogram ABCD, then find the values of $x$ and $y$. Hence, find lengths of sides of the parallelogram.

A$(6,-3)$, B$(0,5)$ and C$(-2,1)$ are vertices of $△$ABC. Points P$(3,1)$ and Q$(2,-1)$ lie on sides AB and AC respectively. Check whether $\frac{AP}{PB}=\frac{AQ}{QC}$.

A chord of a circle of radius 10 cm subtends an angle of $60^{\circ }$ at the centre O. Find the area of the shaded region.
(Use $\sqrt{3}=1.73,\sqrt{2}=1.41$ and $\pi =3.14$)

Prove that the parallelogram circumscribing a circle is a rhombus.

Find the sum of the A.P. $7,10\frac{1}{2},14,.....84$.

If the sum of first $n$ terms of an A.P. is given by $S_n=\frac{n}{2}(2n+8)$. Then, find its first term and common difference. Hence, find its $15^{th}$ term.

Find the zeroes of the polynomial $p(x)=4x^2-4x-3$ and verify the relationship between zeroes and its coefficients.

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

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

In a $△$ ABC, P and Q are points on AB and AC respectively such that PQ $\parallel$ BC. Prove that the median AD, drawn from A to BC, bisects PQ.

As observed from the top of a 70 m high lighthouse from the sea level, the angles of depression of two ships are $30^{\circ }$ and $45^{\circ }$. If one ship is exactly behind the other on the same sides of the lighthouse, find the distance between the two ships. (Use $\sqrt{3}=1.73$)

It is given that $p^2{x}^2+(p^2-q^2)x-q^2=0$ ; $(p\ne 0)$
(i) Show that the discriminant (D) of above equation is a perfect square.
(ii) Find the roots of the equation.

Three consecutive positive integers are such that the sum of the square of smallest and product of other two is 67. Find the numbers, using quadratic equation.

Find ‘median’ and ‘mode’ of the following data :
Class: 100-105, 105-110, 110-115, 115-120, 120-125, 125-130
Frequency: 6, 8, 10, 4, 9, 3

Rahim and Nadeem are two friends whose plots are adjacent to each other. Rahim’s son made a drawing of the plots with necessary details.
It is decided that Rahim will fence the triangular plot ABC and Nadeem will fence along the sides AF, FE and BE.
Observe the diagram carefully and answer the following questions :
(Use $\sqrt{2}=1.41$ and $\sqrt{3}=1.73$)
(i) Find length BC. (1)
(ii) Find length AG. (1)
(iii) (a) Calculate perimeter of $△$ABC. (2)
OR
(b) Calculate length of (AF + FE + EB). (2)

Playing in a ball pool is good entertainment for kids. Suhana bought 600 new balls of diameter 7 cm to fill in the pool for her kids. The cuboidal box containing 600 balls has dimensions 42 cm × 91 cm × 50 cm ($l\times b\times h$).
Based on above information, answer the following questions :
(i) Find the volume of one ball. (1)
(ii) 10 balls are painted with neon colours. Determine the area of painted surface. (1)
(iii) (a) Find the volume of empty space in the box. (2)
OR
(b) The lowermost layer of the balls covers the base of the box edge to edge when balls are placed evenly adjacent to each other. (A) How much area is covered by one ball? (B) How many balls are there in lowermost layer? (2)

A telecommunication company came up with two plans– plan A and plan B for its customers.
The plans are represented by linear equations where ‘$t$’ represents the time (in minutes) bought and ‘C’ represents the cost. The equations are :
Plan A : $3C=20t$
Plan B : $3C=10t+300$
Based on above information, answer the following questions :
(i) If you purchase plan B, how much initial amount you have to pay ? (1)
(ii) Charu purchased plan A. How many minutes she bought for ₹ 250 ? (1)
(iii) (a) At how many minutes, do both the plans charge the same amount? What is that amount? (2)
OR
(b) Which plan is better if you want to buy 60 minutes? Give reason for your answer. (2)