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

If the HCF of two positive integers $a$ and $b$ is 1, then their LCM is :

1. (A)$a+b$
2. (B)$a$
3. (C)$b$
4. (D)$ab$

The number $3+\sqrt{2}$ is :

1. (A)a rational number
2. (B)an irrational number
3. (C)an integer
4. (D)a natural number

The discriminant of the quadratic equation $x^2-3x-2=0$ is :

1. (A)1
2. (B)17
3. (C)$\sqrt{17}$
4. (D)$-\sqrt{17}$

The equation $x+\frac{1}{x}=3$ $(x\ne 0)$ is expressed as a quadratic equation in the form of $a{x}^2+bx+c=0$. The value of $a-b+c$ is :

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

For a point $(3,-5)$, the value of (abscissa $-$ ordinate) is :

1. (A)$-8$
2. (B)$-2$
3. (C)2
4. (D)8

The mid-point of a line segment divides the line segment in the ratio :

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

Which of the following is not the criterion for similarity of triangles ?

1. (A)AAA
2. (B)SSS
3. (C)SAS
4. (D)RHS

From the figures given below, which of the following is true about the measure of $\angle P$ ?

1. (A)$\angle P=60^{\circ }$
2. (B)$\angle P=80^{\circ }$
3. (C)$\angle P=40^{\circ }$
4. (D)The measure of $\angle P$ cannot be determined

In the given figure, PA is a tangent to a circle with centre O. If OP = 10 cm, then the length of AP is :

1. (A)$10\sqrt{3}$ cm
2. (B)20 cm
3. (C)5 cm
4. (D)$5\sqrt{3}$ cm

Which of the following statements is false ?

1. (A)$\tan 45^{\circ }=\cot 45^{\circ }$
2. (B)$\sin 90^{\circ }=\tan 45^{\circ }$
3. (C)$\sin 30^{\circ }=\cos 30^{\circ }$
4. (D)$\sin 45^{\circ }=\cos 45^{\circ }$

The value of $\left({\tan }^{2}A-\frac{1}{{\cos }^{2}A}\right)$ is :

1. (A)more than 1
2. (B)1
3. (C)0
4. (D)$-1$

In the given figure, which of the following angles represents the angle of depression ?

1. (A)$x$
2. (B)$y$
3. (C)$z$
4. (D)$a$

The perimeter of the shaded region in the given figure is :

1. (A)$l$
2. (B)$l+a$
3. (C)$l+2r$
4. (D)$l+2r+a$

The ratio of the area of a quadrant of a circle to the area of the same circle is :

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

For which of the following solids is the lateral/curved surface area and total surface area the same ?

1. (A)Cube
2. (B)Cuboid
3. (C)Hemisphere
4. (D)Sphere

The class mark of the median class of the following data is :
Class Interval: 10 – 25, 25 – 40, 40 – 55, 55 – 70, 70 – 85, 85 – 100
Frequency: 2, 3, 7, 6, 6, 6

1. (A)40
2. (B)55
3. (C)47.5
4. (D)62.5

The following distribution shows the number of runs scored by some batsmen in test matches :
Runs Scored: 3000 – 4000, 4000 – 5000, 5000 – 6000, 6000 – 7000
Number of Batsmen: 5, 10, 9, 8
The lower limit of the modal class is :

1. (A)3000
2. (B)4000
3. (C)5000
4. (D)6000

In a random experiment of throwing a die, which of the following is a sure event ?

1. (A)Getting a number between 1 and 6
2. (B)Getting an odd number < 7
3. (C)Getting an even number < 7
4. (D)Getting a natural number < 7

Assertion (A) : For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b.
Reason (R) : HCF of any two natural numbers divides both the numbers.

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) : The value of p for which the system of equations $4x+py+8=0$ and $2x+2y+2=0$ is consistent is 4.
Reason (R) : The system of equations $a_{1}x+b_{1}y=c_1$ and $a_{2}x+b_{2}y=c_2$ is consistent with infinitely many solutions, if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$.

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 for $x$ and $y$ :
$\frac{x}{2}+\frac{2y}{3}=-1$ and $x-\frac{y}{3}=3$

In the given figure, if PQ $\parallel$ RS, then prove that $△POQ\sim △SOR$.

In the given figure, $△OSR\sim △OQP$, $\angle ROQ=125^{\circ }$ and $\angle ORS=70^{\circ }$. Find the measures of $\angle OSR$ and $\angle OQP$.

Two concentric circles are of radii 6 cm and 10 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Find the values of A and B $(0\le A<90^{\circ },0\le B<90^{\circ })$, if $\tan (A+B)=1$ and $\tan (A-B)=\frac{1}{\sqrt{3}}$.

Prove that $\tan 45^{\circ }=1$ geometrically.

A chord of a circle of diameter 20 cm subtends an angle of $60^{\circ }$ at the centre of the circle. Find the area of the corresponding minor segment of the circle. (Use $\pi =3.14$ and $\sqrt{3}=1.73$)

Prove that $\sqrt{3}$ is an irrational number.

The factor tree of a number $x$ is shown below :
Find the values of $x$, $y$, $a$ and $b$. Hence, write the product of the prime factors of the number $x$ so obtained.

Find a quadratic polynomial whose sum and product of zeroes are 0 and $-9$, respectively. Also, find the zeroes of the polynomial so obtained.

Solve the following system of equations graphically :
$x+3y=6$; $2x-3y=12$

$x$ and $y$ are complementary angles such that $x:y=1:2$. Express the given information as a system of linear equations in two variables and hence solve it.

Prove that a rectangle circumscribing a circle is a square.

Prove that :
$\frac{1+{\cot }^{2}A}{1+{\tan }^{2}A}={\left(\frac{1-\cot A}{1-\tan A}\right)}^2$

A lot consists of 200 pens of which 180 are good and the rest are defective. A customer will buy a pen if it is not defective. The shopkeeper draws a pen at random and gives it to the customer. What is the probability that the customer will not buy it ? Another lot of 100 pens containing 80 good pens is mixed with the previous lot of 200 pens. The shopkeeper now draws one pen at random from the entire lot and gives it to the customer. What is the probability that the customer will buy the pen ?

The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.

Find the value(s) of k for which the equation $2x^2+kx+3=0$ has real and equal roots. Hence, find the roots of the equations so obtained.

State “Basic Proportionality Theorem” and use it to prove the following :
In a quadrilateral ABCD, diagonals AC and BD intersect each other at O such that $\frac{AO}{BO}=\frac{CO}{DO}$ as shown in the given figure. Prove that ABCD is a trapezium.

A toy is in the form of a cone surmounted on a hemisphere. The cone and hemisphere have the same radii. The height of the conical part of the toy is equal to the diameter of its base. If the radius of the conical part is 5 cm, find the volume of the toy.

A cubical block is surmounted by a hemisphere of radius 3.5 cm. What is the smallest possible length of the edge of the cube so that the hemisphere can totally lie on the cube ? Find the total surface area of the solid so formed.

The following data gives the information on the observed lifetime (in hours) of 200 electrical components :
Lifetime (in hours): 0 – 20, 20 – 40, 40 – 60, 60 – 80, 80 – 100, 100 – 120
Number of electrical components: 10, 35, 50, 60, 30, 15
Find the mean lifetime (in hours) of the electrical components.

An injured bird was found on the roof of a building. The building is 15 m high. A fireman was called to rescue the bird. The fireman used an adjustable ladder to reach the roof. He placed the ladder in such a way that the ladder makes an angle of $60^{\circ }$ with the ground in order to reach the roof.
Based on the above information, answer the following questions :
(i) Find the length of the ladder used by the fireman to reach the roof. (1)
(ii) Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building. (1)
(iii) In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of $30^{\circ }$ with the ground.
(a) Draw a neat diagram to represent the above situation and hence find the width of the road between the building and the wall. (2)
OR
(b) Find the length of the ladder used by the fireman in this case. (2)

In a garden, saplings of rose flowers were planted at equal intervals to form a spiral pattern. The spiral is made up of successive semicircles, with centres alternatively at A and B, starting with centre at A, of radii 50 cm, 100 cm, 150 cm, ....... as shown in the figure given below. Spiral 1 has 10 flowers, Spiral 2 has 20 flowers, Spiral 3 has 30 flowers and so on.
Based on the above information, answer the following questions :
(i) What is the radius of the $13^{th}$ spiral ? (1)
(ii) If the radius of the $n^{th}$ spiral is 500 cm, find the value of $n$. (1)
(iii) (a) Find the total number of saplings till the $11^{th}$ spiral. (2)
OR
(b) Till which spiral, will there be a total of 450 saplings ? (2)

In a society, there is a circular park having two gates. The gates are placed at points A(10, 20) and B(50, 50), as shown in the figure below. Two fountains are installed at points P and Q on AB such that AP = PQ = QB.
Based on the above information, answer the following questions :
(i) Find the coordinates of the centre C. (1)
(ii) Find the radius of the circular park. (1)
(iii) (a) Find the coordinates of the point P. (2)
OR
(b) Find the distance of the fountain at Q from gate A. (2)