CBSE Class 10 Maths Basic 2025 Question Paper 430/3/1 with Solutions

If $p=2^3\times 3^2\times 5$ and $q=2^2\times 3^3$, then the LCM of $p$ and $q$ is :

1. (A)$2^3\times 3^3$
2. (B)$2^2\times 3^2$
3. (C)$2^2\times 3^2\times 5$
4. (D)$2^3\times 3^3\times 5$

$3^n$, where $n$ is a natural number, cannot end with the digit :

1. (A)3
2. (B)5
3. (C)7
4. (D)9

A prime number has :

1. (A)exactly two prime factors
2. (B)exactly one prime factor
3. (C)at least one prime factor
4. (D)at least two prime factors

For what value(s) of $k$ does the system of equations $kx+2y=3$ and $2x+y=5$ have a unique solution ?

1. (A)$k$ = a real number
2. (B)$k\ne 8$
3. (C)$k\ne 4$
4. (D)$k=4$

If $x=\sqrt{x}$, $(x\ne 0)$ is expressed as a quadratic equation in the form $a{x}^2+bx+c=0$, then the value of $a+b+c$ is :

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

If point $(1,2)$ is the mid-point of the line segment joining the points $(3,5)$ and $(2a,b)$, then $(a,b)=$

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

The distance of point $(-3,4)$ from y-axis is :

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

The value of $\frac{{\sec }^{2}30^{\circ }+{\tan }^{2}30^{\circ }}{{\sin }^{2}45^{\circ }+{\cos }^{2}45^{\circ }}$ is :

1. (A)1
2. (B)$\frac{5}{3}$
3. (C)$\frac{13}{3}$
4. (D)7

Which types of triangles are always similar ?

1. (A)Right-angled triangles
2. (B)Acute-angled triangles
3. (C)Isosceles triangles
4. (D)Equilateral triangles

What values of $x$ and $y$ will make $△ABC$ similar to $△QRP$ in the figures given below ?

1. (A)$x=6$, $y=5$
2. (B)$x=5$, $y=6$
3. (C)$x=6$, $y=6$
4. (D)$x=12$, $y=3.2$

In the given figure, from which point can the tangent not be drawn to the circle with radius $r$ ?

1. (A)X
2. (B)P
3. (C)Q
4. (D)Y

In the given figure, the angle of elevation of point A from point C is :

1. (A)$30^{\circ }$
2. (B)$45^{\circ }$
3. (C)$60^{\circ }$
4. (D)Cannot be determined

The angle of the sector of a circle whose area is one-eighth of the area of the circle is :

1. (A)$22{\frac{1}{2}}^{\circ }$
2. (B)$45^{\circ }$
3. (C)$60^{\circ }$
4. (D)$90^{\circ }$

The perimeter of a quadrant of a circle of circumference 22 cm is :

1. (A)29 cm
2. (B)22 cm
3. (C)12.5 cm
4. (D)5.5 cm

A cone and cylinder have same height and same radius. The volume of the cone and the volume of the cylinder are in the ratio :

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

The following table shows the marks scored by 23 students of a class.
Marks: 0 – 10, 10 – 20, 20 – 30, 30 – 40, 40 – 50
Number of Students: 5, 3, 4, 8, 3
The lower limit of the modal class is :

1. (A)10
2. (B)20
3. (C)30
4. (D)40

For a distribution, if mean = 15 and mode = 12, then its median is :

1. (A)12
2. (B)13
3. (C)14
4. (D)15

A pair of dice is thrown simultaneously. Let E denote the event that “The sum of numbers obtained on both dice is at least 9.” The number of outcomes in favour of event E is :

1. (A)4
2. (B)6
3. (C)10
4. (D)26

Assertion (A) : For $\sin \theta =1$, $\cos \theta$ must be 0.
Reason (R) : ${\sin }^2\theta -{\cos }^2\theta =1$.

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) : Every quadratic equation has two real roots.
Reason (R) : A quadratic polynomial can have at most two zeroes.

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.

Solve the following system of equations graphically :
$2x-3y=-6$ and $x=3$

A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a shadow of 45 m on the ground. Find the height of the tower.

In the given figure, $\frac{QR}{QS}=\frac{QT}{PR}$ and $\angle 1=\angle 2$. Prove that $△PQS\sim △TQR$.

In the given figure, TP and TQ are two tangents. If $\angle PTQ=50^{\circ }$, then find the measure of $\angle OPQ$.

If $\sin 3A=1$, then find the value of $\cos 2A-{\tan }^{2}45^{\circ }$.

If $(\sec A+\tan A)(1-\sin A)=k\cos A$, then find the value of $k$.

In the given figure, OPRQ is a quadrant of a circle with centre O and radius 10 cm. If OP = 10 cm and OS = 6 cm, find the area of the shaded region. (Use $\pi =3.14$)

Prove that $\sqrt{2}$ is an irrational number.

Find which among the following numbers a, b and c is/are composite numbers.
$a=7\times 11\times 13+13$
$b=6\times 5\times 4+4$
$c=7\times 13+6$

Find the zeroes of the polynomial $9s^2-6s+1$ and verify the relationship between the zeroes and the coefficients of the given polynomial.

A fraction becomes $\frac{1}{3}$, when 1 is subtracted from the numerator and it becomes $\frac{1}{4}$, when 8 is added to its denominator. Find the fraction.

Find the value of $k$ for which the following pair of linear equations will have infinitely many solutions :
$kx+3y-(k-3)=0$ and $12x+ky-k=0$
Hence, find any two solutions of the given pair of equations.

Prove that :
$\frac{1+\text{cosec }A}{\text{cosec }A}=\frac{{\cos }^{2}A}{1-\sin A}$

A box contains 6 blue, 4 white and 8 red marbles. A marble is drawn at random from this box. Find the probability that the marble so drawn is :
(i) white
(ii) white or red
(iii) not red

In the given figure, a circle is inscribed in a quadrilateral ABCD which touches the sides AB, BC, CD and DA at P, Q, R and S respectively. Prove that $\angle AOB+\angle COD=180^{\circ }$.

Find two consecutive odd integers, sum of whose squares is 290.

A charity trust decides to build a rectangular hall having an area of 300 ${\text{m}}^2$. The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.

State and Prove “Basic Proportionality Theorem”.

In the given figure, CM and RN are respectively, the medians of $△ABC$ and $△PQR$. If $△ABC\sim △PQR$, prove that :
(i) $△AMC\sim △PNR$
(ii) $\angle BCM=\angle QRN$
(iii) $△BMC\sim △QNR$

A perfume bottle is in the form of a cylinder but the bottom of the bottle has a hemispherical raised portion to reduce the capacity of the bottle. The inner diameter of the bottle is 5 cm and the height of the bottle is 10 cm. Find the capacity of the perfume bottle in mL. (Use $\pi =3.14$ and 1 ${\text{cm}}^3$ = 1 mL)

The weights of 30 students of a class are given in the following distribution table :
Weight (in kg): 40 – 45, 45 – 50, 50 – 55, 55 – 60, 60 – 65, 65 – 70
Number of students: 2, 5, 8, 6, 6, 3
Find the median weight of the students.

In a society, a yoga instructor was hired to train the people of the society to live a healthy lifestyle. Yoga sessions were held daily from 5 p.m. to 7 p.m. in the society park. On day one, 5 people joined the yoga session, on day two, 3 more people joined, on day three, another 3 people joined and in this manner every next day, 3 more people kept on joining.
Based on the given information, answer the following questions :
(i) On which day did 59 people join the yoga session ? (1)
(ii) How many people joined the yoga session on the 31st day ? (1)
(iii) (a) The yoga instructor was paid ₹100 for each person attending the yoga session. On which day would he earn ₹5,000 ? (2)
OR
(b) What was the total amount earned by the yoga instructor in 16 days ? (2)

Two motorboats A and B are waiting at the opposite banks of a river in order to reach the opposite side. From a point P on the bridge, 20 m above the river, the angles of depression of the boats are $30^{\circ }$ and $45^{\circ }$ respectively, as shown in the figure given below. Both the boats leave at the same time at the speed of 10 m/s and 5 m/s, respectively
Based on the above information, answer the following questions :
(i) Find the distance travelled by boat A to reach point D in the river, vertically below the point P. (Use $\sqrt{3}=1.73$) (1)
(ii) What is the width of the river ? (1)
(iii) (a) Which boat will reach point D first, and how much earlier, than the other boat ? (2)
OR
(b) What is the distance between the two boats after 3 seconds ? (2)

There is a semicircular park in Aman’s society. He wishes to plant saplings along the boundary of the park. There is a borewell at the centre O of the park along the diameter AB as shown in the figure below.
Based on the above information, answer the following questions :
(i) Find the coordinates of point O. (1)
(ii) Find the radius of the semicircular park. (1)
(iii) (a) One sapling is kept at point C(12, y). Find the coordinates of C. (2)
OR
(b) One sapling is kept at point P along AB so that PA = $\frac{1}{3}$ PB. Find the coordinates of P. (2)