CBSE Class 10 Maths Standard 2025 Question Paper 30/6/3 with Solutions

For a circle with centre O and radius 5 cm, which of the following statements is true ?
P : Distance between every pair of parallel tangents is 5 cm.
Q : Distance between every pair of parallel tangents is 10 cm.
R : Distance between every pair of parallel tangents must be between 5 cm and 10 cm.
S : There does not exist a point outside the circle from where length of tangent is 5 cm.

1. (A)P
2. (B)Q
3. (C)R
4. (D)S

In the adjoining figure, AP and AQ are tangents to the circle with centre O. If reflex $\angle POQ=210^{\circ }$, the value of $2x$ is

1. (A)$30^{\circ }$
2. (B)$60^{\circ }$
3. (C)$120^{\circ }$
4. (D)$300^{\circ }$

If $x=2\sin 60^{\circ }\cos 60^{\circ }$ and $y={\sin }^{2}30^{\circ }-{\cos }^{2}30^{\circ }$ and $x^2=k{y}^2$, the value of $k$ is

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

A peacock sitting on the top of a tree of height 10 m observes a snake moving on the ground. If the snake is $10\sqrt{3}$ m away from the base of the tree, then angle of depression of the snake from the eye of the peacock is

1. (A)$30^{\circ }$
2. (B)$45^{\circ }$
3. (C)$60^{\circ }$
4. (D)$90^{\circ }$

If a cone of greatest possible volume is hollowed out from a solid wooden cylinder, then the ratio of the volume of remaining wood to the volume of cone hollowed out is

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

If the mode of some observations is 10 and sum of mean and median is 25, then the mean and median respectively are

1. (A)12 and 13
2. (B)13 and 12
3. (C)10 and 15
4. (D)15 and 10

If the maximum number of students has obtained 52 marks out of 80, then

1. (A)52 is the mean of the data.
2. (B)52 is the median of the data.
3. (C)52 is the mode of the data.
4. (D)52 is the range of the data.

The system of equations $y+a=0$ and $2x=b$ has

1. (A)No solution
2. (B)$\left(-a,\frac{b}{2}\right)$ as its solution
3. (C)$\left(\frac{b}{2},-a\right)$ as its solution
4. (D)Infinite solutions

In a right triangle ABC, right-angled at A, if $\sin B=\frac{1}{4}$, then the value of $\sec B$ is

1. (A)4
2. (B)$\frac{\sqrt{15}}{4}$
3. (C)$\sqrt{15}$
4. (D)$\frac{4}{\sqrt{15}}$

$\sqrt{0.4}$ is a/an

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

Which of the following cannot be the unit digit of $8^n$, where $n$ is a natural number ?

1. (A)4
2. (B)2
3. (C)0
4. (D)6

Which of the following equations does not have a real root ?

1. (A)$x^2=0$
2. (B)$2x-1=3$
3. (C)$x^2+1=0$
4. (D)$x^3+x^2=0$

If the zeroes of the polynomial $a{x}^2+bx+\frac{2a}{b}$ are reciprocal of each other, then the value of $b$ is

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

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

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

In the adjoining figure, $PQ\parallel XY\parallel BC$, $AP=2$ cm, $PX=1.5$ cm and $BX=4$ cm. If $QY=0.75$ cm, then $AQ+CY=$

1. (A)6 cm
2. (B)4.5 cm
3. (C)3 cm
4. (D)5.25 cm

Given $△ABC\sim △PQR$, $\angle A=30^{\circ }$ and $\angle Q=90^{\circ }$. The value of $(\angle R+\angle B)$ is

1. (A)$90^{\circ }$
2. (B)$120^{\circ }$
3. (C)$150^{\circ }$
4. (D)$180^{\circ }$

Two coins are tossed simultaneously. The probability of getting atleast one head is

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

In the adjoining figure, PA and PB are tangents to a circle with centre O such that $\angle P=90^{\circ }$. If $AB=3\sqrt{2}$ cm, then the diameter of the circle is

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

In an experiment of throwing a die,
Assertion (A) : Event $E_1$ : getting a number less than 3 and Event $E_2$ : getting a number greater than 3 are complementary events.
Reason (R) : If two events E and F are complementary events, then $P(E)+P(F)=1$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is 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) : For two odd prime numbers $x$ and $y$, ($x\ne y$), LCM$(2x,4y)=4xy$
Reason (R) : LCM$(x,y)$ is a multiple of HCF$(x,y)$.

1. (A)Both Assertion (A) and Reason (R) are true and Reason (R) is 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.

If $a\sec \theta +b\tan \theta =m$ and $b\sec \theta +a\tan \theta =n$, prove that $a^2+n^2=b^2+m^2$

Use the identity : ${\sin }^{2}A+{\cos }^{2}A=1$ to prove that ${\tan }^{2}A+1={\sec }^{2}A$. Hence, find the value of $\tan A$, when $\sec A=\frac{5}{3}$, where A is an acute angle.

Prove that abscissa of a point P which is equidistant from points with coordinates $A(7,1)$ and $B(3,5)$ is 2 more than its ordinate.

In the adjoining figure, $AP=1$ cm, $BP=2$ cm, $AQ=1.5$ cm and $AC=4.5$ cm.
Prove that $△APQ\sim △ABC$. Hence find the length of PQ, if $BC=3.6$ cm.

A bag contains balls numbered 2 to 91 such that each ball bears a different number. A ball is drawn at random from the bag. Find the probability that
(i) it bears a 2– digit number
(ii) it bears a multiple of 1.

Solve the following pair of equations algebraically :
$101x+102y=304$
$102x+101y=305$

In a pair of supplementary angles, the greater angle exceeds the smaller by $50^{\circ }$. Express the given situation as a system of linear equations in two variables and hence obtain the measure of each angle.

Check whether the given system of equations is consistent or not. If consistent, solve graphically.
$x-2y=0$
$2x+y=0$

If the points $A(6,1)$, $B(p,2)$, $C(9,4)$ and $D(7,q)$ are the vertices of a parallelogram ABCD, then find the values of $p$ and $q$. Hence, check whether ABCD is a rectangle or not.

Prove that : $\frac{\cos \theta -2{\cos }^3\theta }{\sin \theta -2{\sin }^3\theta }+\cot \theta =0$.

Given that $\sin \theta +\cos \theta =x$, prove that ${\sin }^4\theta +{\cos }^4\theta =\frac{2-(x^2-1{)}^2}{2}$.

In the adjoining figure, TP and TQ are tangents drawn to a circle with centre O. If $\angle OPQ=15^{\circ }$ and $\angle PTQ=\theta$, then find the value of $\sin 2\theta$.

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

Let $p$, $q$ and $r$ be three distinct prime numbers.
Check whether $p\cdot q\cdot r+q$ is a composite number or not.
Further, give an example for 3 distinct primes $p$, $q$, $r$ such that
(i) $p\cdot q\cdot r+1$ is a composite number.
(ii) $p\cdot q\cdot r+1$ is a prime number.

Find the zeroes of the polynomial $r(x)=4x^2+3x-1$. Hence, write a polynomial whose zeroes are reciprocal of the zeroes of polynomial $r(x)$.

If a line drawn parallel to one side of triangle intersecting the other two sides in distinct points divides the two sides in the same ratio, then it is parallel to third side.
State and prove the converse of the above statement.

In the adjoining figure, $△CAB$ is a right triangle, right angled at A and $AD\perp BC$. Prove that $△ADB\sim △CDA$. Further, if $BC=10$ cm and $CD=2$ cm, find the length of AD.

Fermentation tanks are designed in the form of cylinder mounted on a cone as shown below :
The total height of the tank is 3.3 m and height of conical part is 1.2 m. The diameter of the cylindrical as well as conical part is 1 m. Find the capacity of the tank. If the level of liquid in the tank is 0.7 m from the top, find the surface area of the tank in contact with liquid.

The population of lions was noted in different regions across the world in the following table :
Number of lions: 0 – 100, 100 – 200, 200 – 300, 300 – 400, 400 – 500, 500 – 600, 600 – 700, 700 – 800, 800 – 900, 900 – 1000
Number of regions: 2, 5, 9, 12, $x$, 20, 15, 9, y, 2
Total: 100
If the median of the given data is 525, find the values of $x$ and y.

There is a circular park of diameter 65 m as shown in the following figure, where AB is a diameter.
An entry gate is to be constructed at a point P on the boundary of the park such that distance of P from A is 35 m more than the distance of P from B. Find distance of point P from A and B respectively.

Find the smallest value of $p$ for which the quadratic equation $x^2-2(p+1)x+p^2=0$ has real roots. Hence, find the roots of the equation so obtained.

The Statue of Unity situated in Gujarat is the world's largest Statue which stands over a 58 m high base. As part of the project, a student constructed an inclinometer and wishes to find the height of Statue of Unity using it.
He noted following observations from two places :
Situation – I :
The angle of elevation of the top of Statue from Place A which is $80\sqrt{3}$ m away from the base of the Statue is found to be $60^{\circ }$.
Situation – II :
The angle of elevation of the top of Statue from a Place B which is 40 m above the ground is found to be $30^{\circ }$ and entire height of the Statue including the base is found to be 240 m.
Based on given information, answer the following questions :
(i) Represent the Situation – I with the help of a diagram. (1)
(ii) Represent the Situation – II with the help of a diagram. (1)
(iii) (a) Calculate the height of Statue excluding the base and also find the height including the base with the help of Situation – I. (2)
OR (iii) (b) Find the horizontal distance of point B (Situation – II) from the Statue and the value of $\tan \alpha$, where $\alpha$ is the angle of elevation of top of base of the Statue from point B. (2)

Anurag purchased a farmhouse which is in the form of a semicircle of diameter 70 m. He divides it into three parts by taking a point P on the semicircle in such a way that $\angle PAB=30^{\circ }$ as shown in the following figure, where O is the centre of semicircle.
In part I, he planted saplings of Mango tree, in part II, he grew tomatoes and in part III, he grew oranges. Based on given information, answer the following questions.
(i) What is the measure of $\angle POA$ ? (1)
(ii) Find the length of wire needed to fence entire piece of land. (1)
(iii) (a) Find the area of region in which saplings of Mango tree are planted. (2)
OR (iii) (b) Find the length of wire needed to fence the region III. (2)

In order to organise, Annual Sports Day, a school prepared an eight lane running track with an integrated football field inside the track area as shown below :
The length of innermost lane of the track is 400 m and each subsequent lane is 7.6 m longer than the preceding lane.
Based on given information, answer the following questions, using concept of Arithmetic Progression.
(i) What is the length of the $6^{th}$ lane ? (1)
(ii) How long is the $8^{th}$ lane than that of $4^{th}$ lane ? (1)
(iii) (a) While practicing for a race, a student took one round each in first six lanes. Find the total distance covered by the student. (2)
OR (iii) (b) A student took one round each in lane 4 to lane 8. Find the total distance covered by the student. (2)