CBSE Class 10 2025 Mathematics Set – 1 (30/6/1) Question Paper (Soon): Download Solution PDF With Answer Key

Question 1:

\(\sqrt{0.4}\) is a/an

• (A) natural number
• (B) integer
• (C) rational number
• (D) irrational number

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

• (A) 4
• (B) 2
• (C) 0
• (D) 6

Which of the following quadratic equations has real and equal roots?

• (A) \((x + 1)^2 = 2x + 1\)
• (B) \(x^2 + x = 0\)
• (C) \(x^2 - 4 = 0\)
• (D) \(x^2 + x + 1 = 0\)

If the zeroes of the polynomial \(ax^2 + bx + \dfrac{2a}{b}\) are reciprocal of each other, then the value of \(b\) is

• (A) 2
• (B) \(\dfrac{1}{2}\)
• (C) -2
• (D) \(-\dfrac{1}{2}\)

The distance of the point A(\(-3\), \(-4\)) from x-axis is

• (A) 3
• (B) 4
• (C) 5
• (D) 7

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

• (A) \(90^\circ\)
• (B) \(120^\circ\)
• (C) \(150^\circ\)
• (D) \(180^\circ\)

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

• (A) \(\dfrac{1}{4}\)
• (B) \(\dfrac{1}{2}\)
• (C) \(\dfrac{3}{4}\)
• (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

• (A) \(3\sqrt{2}\) cm
• (B) \(6\sqrt{2}\) cm
• (C) 4 cm
• (D) 6 cm

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

• (A) \(3\sqrt{2}\) cm
• (B) \(6\sqrt{2}\) cm
• (C) 3 cm
• (D) 6 cm

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 10 cm.

Q: Distance between every pair of parallel tangents must be between 5 cm and 10 cm.

R: Distance between every pair of parallel tangents is 5 cm.

S: There does not exist a point outside the circle from where length of tangent is 5 cm.

• (A) P
• (B) Q
• (C) R
• (D) S

In the adjoining figure, TS is a tangent to a circle with centre O. The value of \(2x^\circ\) is

• (A) 22.5
• (B) 45
• (C) 67.5
• (D) 90

If \(\dfrac{2\tan30^\circ}{1+\tan^2 30^\circ} = \dfrac{2\tan30^\circ}{\sqrt{1 - \tan^2 30^\circ}}\), then \(x : y =\)

• (A) 1 : 1
• (B) 1 : 2
• (C) 2 : 1
• (D) 4 : 1

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

• (A) \(30^\circ\)
• (B) \(45^\circ\)
• (C) \(60^\circ\)
• (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

• (A) 1 : 2
• (B) 1 : 3
• (C) 2 : 1
• (D) 3 : 1

The system of equations \(2x + 1 = 0\) and \(3y - 5 = 0\) has

• (A) unique solution
• (B) two solutions
• (C) no solution
• (D) infinite number of solutions

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

• (A) 4
• (B) \(\dfrac{\sqrt{15}}{4}\)
• (C) \(\sqrt{15}\)
• (D) \(\dfrac{4}{\sqrt{15}}\)

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

• (A) 12 and 13
• (B) 13 and 12
• (C) 10 and 15
• (D) 15 and 10

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

• (A) 52 is the mean of the data.
• (B) 52 is the median of the data.
• (C) 52 is the mode of the data.
• (D) 52 is the range of the data.

Assertion (A): For any two prime numbers \(p\) and \(q\), their HCF is 1 and LCM is \(p + q\).

Reason (R): For any two natural numbers, HCF × LCM = product of numbers.

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

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 \(P(E) + P(F) = 1\).

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

(a) Solve the following pair of equations algebraically:
\[ 101x + 102y = 304 \quad (i)
102x + 101y = 305 \quad (ii) \]

(b) 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.

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

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

Prove that the 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.

P is a point on the side BC of \(\triangle ABC\) such that \(\angle APC = \angle BAC\). Prove that \(AC^2 = BC \cdot CP\).

The number of red balls in a bag is three more than the number of black balls. If the probability of drawing a red ball at random from the given bag is \(\dfrac{12}{23}\), find the total number of balls in the given bag.

(a) Prove that \(\sqrt{5}\) is an irrational number.

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

(i) \(pqr + 1\) is a composite number

(ii) \(pqr + 1\) is a prime number

Find the zeroes of the polynomial \(p(x) = 3x^2 - 4x - 4\). Hence, write a polynomial whose each of the zeroes is 2 more than the zeroes of \(p(x)\).

Check whether the following pair of equations is consistent or not. If consistent, solve graphically:
\[ x + 3y = 6
3y - 2x = -12 \]

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.

(a) Prove that: \[ \frac{\cos\theta - 2\cos^3\theta}{\sin\theta - 2\sin^3\theta} + \cot\theta = 0 \]

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

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 \(\sin 2\theta\).

(a) 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.

(b) 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.

(a) 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.

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

From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and surface area of the remaining solid.

(Use \(\pi = \dfrac{22{7}, \sqrt{5} = 2.2\))

The following distribution shows the marks of 230 students in a particular subject. If the median marks are 46, then find the values of \(x\) and \(y\).

Marks  Number of Students

10 -- 20   12

20 -- 30   30

30 -- 40   \(x\)

40 -- 50   65

50 -- 60   \(y\)

60 -- 70   25

70 -- 80   18
 

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 the 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 the given information, answer the following questions:


[(i)] What is the measure of \(\angle POA\)?
[(ii)] Find the length of wire needed to fence the entire piece of land.
[(iii)]

[(a)] Find the area of the region in which saplings of Mango tree are planted.

OR

[(b)] Find the length of wire needed to fence the region III.

Question 37:

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?
[(ii)] How much longer is the 8^th lane than the 4^th lane?
[(iii)]
[(a)] While practicing for a race, a student took one round each in the first six lanes. Find the total distance covered by the student.

OR

[(b)] A student took one round each in lanes 4 to 8. Find the total distance covered by the student.

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 the Statue of Unity using it.
He noted the following observations from two places:

Situation – I:
The angle of elevation of the top of the 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 the Statue from a Place B which is 40 m above the ground is found to be \(30^\circ\) and the 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.
[(ii)] Represent the Situation – II with the help of a diagram.
[(iii)] Calculate the height of the Statue excluding the base and also find the height including the base with the help of Situation – I.

OR

[(iv)] 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 the top of base of the Statue from point B.