CBSE CLASS 10 MATHEMATICS ANSWER KEY 2025

 

General Instructions:

• This question paper contains 38 questions. All questions are compulsory.
• This question paper is divided into five Sections – A, B, C, D, and E.
• In Section A, Questions no. 1 to 18 are multiple-choice questions (MCQs) and questions number 19 and 20 are Assertion-Reason based questions of 1 mark each.
• In Section B, Questions no. 21 to 25 are very short answer (VSA) type questions, carrying 2 marks each.
• In Section C, Questions no. 26 to 31 are short answer (SA) type questions, carrying 3 marks each.
• In Section D, Questions no. 32 to 35 are long answer (LA) type questions carrying 5 marks each.
• In Section E, Questions no. 36 to 38 are case study-based questions carrying 4 marks each. Internal choice is provided in 2 marks questions in each case study.
• There is no overall choice. However, an internal choice has been provided in:
  • 2 questions in Section B
  • 2 questions in Section C
  • 2 questions in Section D
  • 3 questions in Section E
• Draw neat diagrams wherever required. Take π = ²²⁄₇ wherever required, if not stated.
• Use of calculator is not allowed.

SECTION A

This section has 20 Multiple Choice Questions (MCQs) carrying 1 mark each.

Question 1:

If α and β are the zeroes of the polynomial 3x² + 6x + k such that α + β + αβ = -2/3, then the value of k is:

• (A) -8
• (B) 8
• (C) -4
• (D) 4

Answer: (D) 4

Question 2:

If x = 1 and y = 2 is a solution of the pair of linear equations 2x - 3y + a = 0 and 2x + 3y - b = 0, then:

• (A) a = 2b
• (B) 2a = b
• (C) a + 2b = 0
• (D) 2a + b = 0

Answer: (B) 2a = b

Question 3:

The mid-point of the line segment joining the points P(-4, 5) and Q(4, 6) lies on:

• (A) x-axis
• (B) y-axis
• (C) origin
• (D) neither x-axis nor y-axis

Answer: (B) y-axis

Question 4:

If θ is an acute angle and 7 + 4 sin θ = 9, then the value of θ is:

• (A) 90°
• (B) 30°
• (C) 45°
• (D) 60°

Answer: (B) 30°

Question 5:

The value of tan²θ - (1/cosθ × sec θ) is:

• (A) 1
• (B) 0
• (C) -1
• (D) 2

Answer:  (C) -1

Question 6:

If HCF(98, 28) = m and LCM(98, 28) = n, then the value of n - 7m is:

• (A) 0
• (B) 28
• (C) 98
• (D) 198

Answer: (C) 98

7. The tangents drawn at the extremities of the diameter of a circle are always:
(A) parallel✅
(B) perpendicular 
(C) equal
(D) intersecting
Answer: The tangents at the endpoints of a diameter are always Parallel to each other.

8. In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3 DE. Then, the two triangles are:
(A) congruent but not similar
(B) congruent as well as similar
(C) neither congruent nor similar
(D) similar but not congruent ✅
Answer: The triangles are similar as their corresponding angles are equal, but they are not congruent as their sides are in a 3:1 ratio.

9. If (-1)ⁿ + (-1)⁸ = 0, then n is:
(A) any positive integer
(B) any negative integer
(C) any odd number ✅
(D) any even number
Answer: Since (-1)⁸ = 1, for the sum to be 0, (-1)ⁿ must be -1, which happens when n is an odd number.

10. Two polynomials are shown in the graph below. The number of distinct zeroes of both the polynomials is:

(A) 3
(B) 5
(C) 2✅
(D) 4 
Answer: By analyzing the graph, there are 2  distinct x-intercepts, meaning there are 2 distinct zeroes.
11. If the sum of the first m terms of an AP is 2m² + 3m, then its second term is:
(A) 10
(B) 9 ✅
(C) 12
(D) 4
Answer: The first term (a) is found by putting m=1, and the second term (a + d) by putting m=2, leading to a second term of 9.

12. Mode and Mean of a data are 15x and 18x, respectively. Then the median of the data is:
(A) x
(B) 11x
(C) 17x ✅
(D) 34x
Answer: Using the empirical relation: Mode = 3(Median) - 2(Mean), we solve for Median = 17x.

13. A card is selected at random from a deck of 52 playing cards. The probability of it being a red face card is:
(A) 3/13 ✅
(B) 2/13
(C) 1/2
(D) 3/26
Answer: There are 6 red face cards (King, Queen, Jack of Hearts and Diamonds). So, Probability = 6/52 = 3/26.

14. Which of the following is a rational number between √3 and √5?
(A) 1.4142387954012...
(B) 2326 
(C) π
(D) 1.857142✅
Answer: A rational number is one that can be expressed as p/q, where p and q are integers. 
15. If a sector of a circle has an area of 40π sq. units and a central angle of 72°, the radius of the circle is:
(A) 200 units
(B) 100 units
(C) 20 units 
(D) 10√2 units✅
Answer: The formula for the area of a sector is (θ/360) × πr². Solving for r, we get r = 10√2 units units.

16. In the given figure, PA is a tangent from an external point P to a circle with center O. If ∠POB = 115°, then ∠APO is equal to:

(A) 25° ✅
(B) 65°
(C) 90°
(D) 35°
Answer: Since PA is a tangent, ∠APO is the complement of half of ∠POB, which results in 25°.

17. A kite is flying at a height of 150 m from the ground. It is attached to a string inclined at an angle of 30° to the horizontal. The length of the string is:
(A) 100√3 m
(B) 300 m
(C) 150√2 m
(D) 150√3 m ✅
Answer: Using the sine formula: length of string = height/sin(30°) = 150/0.5 = 150√3 m.
18. A piece of wire 20 cm long is bent into the form of an arc of a circle of radius 60/π cm. The angle subtended by the arc at the centre of the circle is:
(A) 30°
(B) 60° ✅
(C) 90°
(D) 50°
Answer: The angle subtended at the center is given by θ = (Length of arc / Radius). Using given values, θ = (20 / (60/π)) × 180/π = 60°.

19. Assertion (A): The probability of selecting a number at random from the numbers 1 to 20 is 1.
Reason (R): For any event E, if P(E) = 1, then E is called a sure event.
(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. ✅
Answer: The probability of selecting a number randomly from 1 to 20 is 1/20, not 1. Hence, Assertion (A) is false, but Reason (R) is true.

20. Assertion (A): If we join two hemispheres of the same radius along their bases, then we get a sphere.
Reason (R): Total Surface Area of a sphere of radius r is 3πr².
(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.
Answer: The total surface area of a sphere is 4πr², not 3πr². So, Assertion (A) is true, but Reason (R) is false.
SECTION B
This section has 5 Very Short Answer (VSA) type questions carrying 2 marks each.
5 × 2 = 10

21.
(a) If x cos 60° + y cos 0° + sin 30° - cot 45° = 5, then find the value of x + 2y.
OR
(b) Evaluate: (tan² 60°) / (sin² 60° + cos² 30°).

22. Find the zeroes of the polynomial p(x) = x² + (4/3)x - 4/3.

23. The coordinates of the centre of a circle are (2a, a - 7). Find the value(s) of ‘a’ if the circle passes through the point (11, -9) and has a diameter of 10√2 units.

24.
(a) If △ABC ~ △PQR in which AB = 6 cm, BC = 4 cm, AC = 8 cm, and PR = 6 cm, then find the length of (PQ + QR).
OR
(b) In the given figure, QR/QS = QT/PR and ∠1 = ∠2, show that △PQS ~ △TQR.

25. A person is standing at P outside a circular ground at a distance of 26 m from the centre of the ground. He found that his distances from the points A and B on the ground are 10 m (PA and PB are tangents to the circle). Find the radius of the circular ground.

SECTION C
This section has 6 Short Answer (SA) type questions carrying 3 marks each.
6 × 3 = 18

26.
(a) In the given figure, O is the centre of the circle and BCD is tangent to it at C. Prove that ∠BAC + ∠ACD = 90°.
OR

(b) Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

27. Prove that:
(a) 1 - tan θ / 1 + tan θ = 1 - sec 2θ / 1 + sec 2θ
(b) sin A . cos A + sin A . cos A / sin A . cos A - sin A . cos A = 2 / 2 sin A - 1

28. Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also, find the point of intersection.

29. Prove that 1 / √5 is an irrational number.

30. A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains 1408 m³ of air, find the height of the cylindrical part. (Use π = 22/7).

31. Two dice are thrown at the same time. Determine the probability that the difference of the numbers on the two dice is 2.

SECTION D
This section has 4 Long Answer (LA) type questions carrying 5 marks each.
4 × 5 = 20

32. Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. He received ₹1,580 as the total annual interest. However, had he interchanged the amount of investments in the two schemes, he would have received ₹20 more as annual interest. How much money did he invest in each scheme?
33. (a) The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that DF × EF = FB × FA.
OR
(b) In ΔABC, if AD ⊥ BC and AD² = BD × DC, then prove that ∠BAC = 90°.

34. (a) The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of the other two sides of the triangle.
OR
(b) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.

35. Find the missing frequency ‘f’ in the following table, if the mean of the given data is 18. Hence find the mode.

Daily Allowance	Number of Children
11 – 13	7
13 – 15	6
15 – 17	9
17 – 19	13
19 – 21	f
21 – 23	5
23 – 25	4

SECTION E
This section has 3 case study-based questions carrying 4 marks each. 3×4=12

Case Study - 1
36. A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being 300 metres. To make the event more challenging and engaging, the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres, and so on. The total number of rounds planned is 10.

Based on the information given above, answer the following questions:
(i) Write the fourth, fifth, and sixth terms of the Arithmetic Progression so formed. (1 mark)

(ii) Determine the distance of the 8th round. (1 mark)

(iii) (a) Find the total distance run after completing all 10 rounds. (2 marks)

OR

(iii) (b) If a runner completes only the first 6 rounds, what is the total distance run by the runner? (2 marks)
Case Study - 2
37. A brooch is a decorative piece often worn on clothing like jackets, blouses, or dresses to add elegance. Made from precious metals and decorated with gemstones, brooches come in many shapes and designs.

One such brooch is made with silver wire in the form of a circle with a diameter of 35 mm. The wire is also used in making 5 diameters, which divide the circle into 10 equal sectors as shown in the figure.

Based on the above given information, answer the following questions:
(i) Find the central angle of each sector. (1 mark)

(ii) Find the length of the arc ACB. (1 mark)

(iii) (a) Find the area of each sector of the brooch. (2 marks)

OR

(iii) (b) Find the total length of the silver wire used. (2 marks)
Case Study - 3
38. Amrita stood near the base of a lighthouse, gazing up at its towering height. She measured the angle of elevation to the top and found it to be 60°. Then, she climbed a nearby observation deck, 40 metres higher than her original position, and noticed the angle of elevation to the top of the lighthouse to be 45°.

Based on the above given information, answer the following questions:
(i) If CD is h metres, find the distance BD in terms of 'h'. (1 mark)

(ii) Find distance BC in terms of 'h'. (1 mark)

(iii) (a) Find the height CE of the lighthouse [Use √3 = 1.73]. (2 marks)

OR

(iii) (b) Find distance AE, if AC = 100 m. (2 marks)

CBSE CLASS 10 MATHEMATICS QUESTION PAPER 2025 PDF