CBSE Class 10 Maths Standard 2025 Question Paper 30/5/3 with Solutions

Which of the following statements is false ?

1. (A)Infinite number of tangents can be drawn to a circle.
2. (B)Infinite number of tangents can be drawn to a circle from a point outside the circle.
3. (C)Infinite number of secants can be drawn to a circle from a point outside the circle.
4. (D)Angle between tangent and diameter at point of contact is $90^{\circ }$.

In the adjoining figure, PA and PB are tangents to a circle with centre O. The measure of angle APB is

1. (A)$210^{\circ }$
2. (B)$150^{\circ }$
3. (C)$105^{\circ }$
4. (D)$30^{\circ }$

The value of $(1-2{\sin }^{2}60^{\circ })$ is same as that of

1. (A)$\sin 30^{\circ }$
2. (B)$-\sin 30^{\circ }$
3. (C)$\cos 60^{\circ }$
4. (D)$-\cos 30^{\circ }$

An observer 1.8 m tall stands away from a chimney at a distance of 38.2 m along the ground. The angle of elevation of top of chimney from the eyes of observer is $45^{\circ }$. The height of chimney above the ground is

1. (A)38.2 m
2. (B)36.4 m
3. (C)40 m
4. (D)$(38.2)\sqrt{2}$ m

In the adjoining figure, the sum of radii of two concentric circles is 16 cm. The length of chord AB which touches the inner circle at P is 16 cm. The difference of the radii of the given circles is

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

A cone of height 12 cm and slant height 13 cm is surmounted on a hemisphere having radius equal to that of cone. The entire height of the solid is

1. (A)17 cm
2. (B)18 cm
3. (C)22 cm
4. (D)23 cm

If $x$ median + $y$ mean = $z$ mode; is the empirical relationship between mean, median and mode, then the value of $x+y+z$ is

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

Following data shows the marks obtained by 100 students in a class test :
Marks obtained: 20, 29, 28, 33, 42, 38, 43, 25
Number of students: 6, 28, 24, 15, 2, 4, 1, 20
The median will be the average of which two observations ?

1. (A)29 and 33
2. (B)25 and 28
3. (C)28 and 29
4. (D)33 and 38

The probability of getting a composite number greater than 3 on throwing a die is

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

$(\sqrt{3}+2{)}^2+(\sqrt{3}-2{)}^2$ is a/an

1. (A)positive rational number
2. (B)negative rational number
3. (C)positive irrational number
4. (D)negative irrational number

Let $p=x^2{y}^3{z}^n$ and $q=x^3{y}^m{z}^2$, where $x,y,z$ are prime numbers. If LCM $(p,q)=x^3{y}^4{z}^3$, then the value of $(2m+3n)$ is

1. (A)18
2. (B)17
3. (C)15
4. (D)14

For any prime number $p$, if $p$ divides $a^2$, where $a$ is any real number then $p$ also divides

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

Which of the following equation is a quadratic equation ?

1. (A)${\left(x+\frac{1}{x}\right)}^2=2$
2. (B)$(x-\sqrt{x}{)}^2+2x\sqrt{x}=0$
3. (C)$(x+1{)}^3=(1-x{)}^3$
4. (D)$(\sqrt{x}+1{)}^2=x^2$

If $x^2+bx+b=0$ has two real and distinct roots, then the value of $b$ can be

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

In the following figure, P and Q are points of trisection of line segment AB :
the value of $\frac{AB}{PB}=$

1. (A)1
2. (B)1.5
3. (C)$\frac{2}{3}$
4. (D)2

A bag contains red coloured, blue coloured and green coloured balls in the ratio 2 : 3 : 4. A ball is drawn at random from the given bag. The probability that the ball so drawn being not of blue colour is

1. (A)$\frac{1}{9}$
2. (B)$\frac{1}{3}$
3. (C)$\frac{2}{3}$
4. (D)$\frac{8}{9}$

Which of the following statements is false ?

1. (A)Two right triangles are always similar.
2. (B)Two squares are always similar.
3. (C)Two equilateral triangles are always similar.
4. (D)Two circles are always similar.

In the adjoining figure, ABCD is a trapezium in which $XY\parallel AB\parallel CD$. If $AX=\frac{2}{3}AD$, then $CY:YB=$

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

Assertion (A) : For an acute angle $\theta$, value of $\text{cosec}\theta$ cannot be $\frac{1}{\sqrt{2}}$.
Reason (R) : $\text{cosec}\theta \ge 1$ for $0^{\circ }\le \theta \le 90^{\circ }$

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 (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) : For an A.P., 3,6,9, ..., 198, $10^{\text{th}}$ term from the end is 168.
Reason (R) : If 'a' and '$l$' are the first term and last term of an A.P. with common difference 'd', then $n^{\text{th}}$ term from the end of the given A.P. is $l-(n-1)d$.

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 (A).
3. (C)Assertion (A) is true, but Reason (R) is false.
4. (D)Assertion (A) is false, but Reason (R) is true.

The coordinates of the end points of the line segment AB are $A(-2,-2)$ and $B(2,-4)$. P is the point on AB such that $BP=\frac{4}{7}AB$. Find the coordinates of point P.

It is given that $\sin (A-B)=\sin A\cos B-\cos A\sin B$. Use it to find the value of $\sin 15^{\circ }$.

If $\sin A=y$, then express $\cos A$ and $\tan A$ in terms of $y$.

In $△ABC$ and $△PQR$, AD and PS are altitudes such that $△ABD\sim △PQS$ and $△ACD\sim △PRS$. Prove that $△ABC\sim △PQR$.

From a pack of 52 cards, all aces and all kings are removed. A card is drawn at random from the remaining cards. Find the probability that the card so drawn is
(i) a face card.
(ii) a card of red colour.

The cost of 2 kg apples and 1 kg of grapes on a day was found to be ₹ 320. The cost of 4 kg apples and 2 kg grapes was found to be ₹ 600. If cost of 1 kg of apples and 1 kg of grapes is ₹ $x$ and ₹ $y$ respectively, represent the given situation algebraically as a system of equations and check whether the system so obtained is consistent or not.

Solve for $x$ and $y$ :
$\sqrt{2}x+\sqrt{3}y=5$ and
$\sqrt{3}x-\sqrt{8}y=-\sqrt{6}$

Find the zeroes of the polynomial $p(x)=6x^2-5x-1$. Hence, obtain a polynomial each of whose zeroes is three times the zeroes of $p(x)$.

Find a relation between $x$ and $y$ such that $P(x,y)$ is equidistant from the points $A(3,5)$ and $B(7,1)$. Hence, write the coordinates of the points on $x$-axis and y-axis which are equidistant from points A and B.

Prove the following trigonometric identity :
$\frac{1+\text{cosec}A}{\text{cosec}A}=\frac{{\cos }^{2}A}{1-\sin A}$

Let $2A+B$ and $A+2B$ be acute angles such that $\sin (2A+B)=\frac{\sqrt{3}}{2}$ and $\tan (A+2B)=1$. Find the value of $\cot (4A-7B)$.

In the adjoining figure, $XY$ and $X^{\prime }{Y}^{\prime }$ are parallel tangents to a circle with centre O. Another tangent AB touches the circle at C intersecting $XY$ at A and $X^{\prime }{Y}^{\prime }$ at B. Prove that AB subtends right angle at the centre of the circle; or $\angle AOB=90^{\circ }$.

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

State true or false for each of the following statements and justify in each case :
(i) $2\times 3\times 5\times 7+7$ is a composite number.
(ii) $2\times 3\times 5\times 7+1$ is a composite number.

Solve the following system of equations graphically :
$2x+3y=6$
$x+y-1=0$
Also, find the sum of ordinates of the points where given lines meet y axis.

State the basic proportionality theorem.
Use the theorem to do the following :
In $△ABC$, AD is the angle bisector of angle A. BA is produced to E such that $CE\parallel AD$. Prove that $\frac{BD}{DC}=\frac{BA}{AC}$.

From one of the faces of a solid wooden cube of side 14 cm, maximum number of hemispheres of diameter 1.4 cm are scooped out. Find the total number of hemispheres that can be scooped out. Also, find the total surface area of the remaining solid.

From a solid cylinder of height 24 cm and radius 5 cm, two cones of height 12 cm and radius 5 cm are hollowed out. Find the volume and surface area of the remaining solid.

The following table gives the daily income of 50 cab drivers of a particular city :
Income (₹): 500 - 600, 600 - 700, 700 - 800, 800 - 900, 900 - 1000
No. of Drivers: 12, 14, 8, 6, 10
Find the mean income and the modal income.

A 2-digit number is seven times the sum of its digits and two (2) more than 5 times the product of its digits. Find the number.

Find the value(s) of $p$ for which the quadratic equation given as $(p+4)x^2-(p+1)x+1=0$ has real and equal roots. Also, find the roots of the equation(s) so obtained.

Passenger boarding stairs, sometimes referred to as boarding ramps, stair cars or aircraft steps, provide a mobile means to travel between the aircraft doors and the ground. Larger aircraft have door sills 5 to 20 feet (1 foot = 30 cm) high. Stairs facilitate safe boarding and de-boarding.
An aircraft has a door sill at a height of 15 feet above the ground. A stair car is placed at a horizontal distance of 15 feet from the plane.
Based on given information, answer the questions given in part (i) and (ii).
(i) Find the angle at which stairs are inclined to reach the door sill 15 feet high above the ground. (1)
(ii) Find the length of stairs used to reach the door sill. (1)
Further, answer any one of the following questions :
(iii) (a) If the 20 feet long stairs is inclined at an angle of $60^{\circ }$ to reach the door sill, then find the height of the door sill above the ground. (use $\sqrt{3}=1.732$) (2)
OR (iii) (b) What should be the shortest possible length of stairs to reach the door sill of the plane 20 feet above the ground, if the angle of elevation cannot exceed $30^{\circ }$ ? Also, find the horizontal distance of base of stair car from the plane. (2)

A farmer has a circular piece of land. He wishes to construct his house in the form of largest possible square within the land as shown below.
The radius of circular piece of land is 35 m.
Based on given information, answer the following questions :
(i) Find the length of wire needed to fence the entire land. (1)
(ii) Find the length of each side of the square land on which house will be constructed. (1)
(iii) (a) The farmer wishes to grow grass on the shaded region around the house. Find the cost of growing the grass at the rate of ₹ 50 per square metre. (2)
OR (iii) (b) Find the ratio of area of land on which house is built to remaining area of circular piece of land. (2)

In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown in the figure below, such that there is one triangle in the first row, three triangles in the second row, five triangles in the third row and so on.
Based on given information, answer the following questions using Arithmetic Progression.
(i) How many triangles will be there in bottom most row ? (1)
(ii) How many triangles will be there in fourth row from the bottom ? (1)
(iii) (a) Find the total number of triangles of side 1 cm each till $8^{\text{th}}$ row. (2)
OR (iii) (b) How many more number of triangles are there from $5^{\text{th}}$ row to $10^{\text{th}}$ row than in first 4 rows ? Show working. (2)