CBSE Class 10 Mathematics 2025 — Set 4

Questions in this 2025 Mathematics paper (Set 4)

All 38 questions from this set, exactly as asked. Try each one, then open the question-paper PDF above for the complete paper.

1. 1.
   If x = ab³ and y = a³b, where a and b are prime numbers, then [HCF(x, y) – LCM(x, y)] is equal to:

   • (a) 1 – a³b³
   • (b) ab(1 – ab)
   • (c) ab – a³b³
   • (d) ab(1 – ab)(1 + ab)
2. 2.
   (1 + √3)² – (1 – √3)² is:

   • (a) a positive rational number
   • (b) a negative integer
   • (c) a positive irrational number
   • (d) a negative irrational number
3. 3.
   The value of 'a' for which ax² + x + a = 0 has equal and positive roots is:

   • (a) 2
   • (b) –2
   • (c) 1/2
   • (d) –1/2
4. 4.
   The distance of a point A from x-axis is 3 units. Which of the following cannot be the coordinates of the point A?

   • (a) (1, 3)
   • (b) (–3, –3)
   • (c) (–3, 3)
   • (d) (3, 1)
5. 5.
   The number of red balls in a bag is 10 more than the number of black balls. If the probability of drawing a red ball at random from this bag is 5/8, then the total number of balls in the bag is:

   • (a) 50
   • (b) 60
   • (c) 80
   • (d) 40
6. 6.
   The value of 'p' for which the equations px + 3y = p – 3, 12x + py = p has infinitely many solutions is:

   • (a) –6 only
   • (b) 6 only
   • (c) ±6
   • (d) Any real number except ±6
7. 7.
   ΔABC and ΔPQR are shown in the adjoining figures. In ΔABC: AB = 6 cm, BC = 6 cm, with a segment of 3.8 cm and 3√3 cm. In ΔPQR: PQ = 7.6 cm, QR = 12 cm, PR = 6√3 cm, ∠Q = 60°. The measure of ∠C is:

   • (a) 140°
   • (b) 80°
   • (c) 60°
   • (d) 40°
8. 8.
   tan 2A = 3 tan A is true, when the measure of ∠A is:

   • (a) 90°
   • (b) 60°
   • (c) 45°
   • (d) 30°
9. 9.
   Which of the following statements is true?

   • (a) sin 20° > sin 70°
   • (b) sin 20° > cos 20°
   • (c) cos 20° > cos 70°
   • (d) tan 20° > tan 70°
10. 10.
    A 30 m long rope is tightly stretched and tied from the top of a pole to the ground. If the rope makes an angle of 60° with the ground, the height of the pole is:

    • (a) 10√3 m
    • (b) 30√3 m
    • (c) 15 m
    • (d) 15√3 m
11. 11.
    On the top face of a wooden cube of side 7 cm, hemispherical depressions of radius 0.35 cm are to be formed by taking out the wood. The maximum number of depressions that can be formed is:

    • (a) 400
    • (b) 100
    • (c) 20
    • (d) 10
12. 12.
    The cumulative frequency for calculating median is obtained by adding the frequencies of all the:

    • (a) classes up to the median class
    • (b) classes following the median class
    • (c) classes preceding the median class
    • (d) all classes
13. 13.
    If mean and median of a given set of observations are 10 and 11 respectively, then the value of mode is:

    • (a) 10.5
    • (b) 8
    • (c) 13
    • (d) 21
14. 14.
    In the adjoining figure, AB is the chord of the larger circle touching the smaller circle. The centre of both the circles is O. If AB = 2r and OP = r, then the radius of the larger circle is:

    • (a) 2r
    • (b) 3r
    • (c) 2√2 r
    • (d) √2 r
15. 15.
    A circle is inscribed in a parallelogram. If one side of the parallelogram is 5 cm, then the perimeter of the parallelogram is:

    • (a) 20 cm
    • (b) less than 20 cm
    • (c) more than 20 cm but less than 40 cm
    • (d) 40 cm
16. 16.
    E and F are points on the sides AB and AC respectively of a ΔABC such that AE/EB = AF/FC = 1/2. Which of the following relation is true?

    • (a) EF = 2BC
    • (b) BC = 2EF
    • (c) EF = 3BC
    • (d) BC = 3EF
17. 17.
    Which of the following statements is true for a polynomial p(x) of degree 3?

    • (a) p(x) has at most two distinct zeroes.
    • (b) p(x) has at least two distinct zeroes.
    • (c) p(x) has exactly three distinct zeroes.
    • (d) p(x) has at most three distinct zeroes.
18. 18.
    A pair of dice is thrown. The probability that sum of numbers appearing on top faces is at most 10 is:

    • (a) 1/11
    • (b) 10/11
    • (c) 5/6
    • (d) 11/12
19. 19.
    Assertion (A): 4ⁿ ends with digit 0 for some natural number n. Reason (R): For a number 'x' having 2 and 5 as its prime factors, xⁿ always ends with digit 0 for every natural number n.

    • (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…
20. 20.
    Assertion (A): Tangents drawn at the end points of a diameter of a circle are always parallel to each other. Reason (R): The lengths of tangents drawn to a circle from a point outside the circle are always equal.

    • (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…
21. 21.
    Solve the following system of equations algebraically: 30x + 44y = 10; 40x + 55y = 13
22. 22 (a).
    A 1.5 m tall boy is walking away from the base of a lamp post which is 12 m high, at the speed of 2.5 m/sec. Find the length of his shadow after 3 seconds.
23. 22 (b).
    OR (B) In parallelogram ABCD, side AD is produced to a point E and BE, CD intersect at F. Prove that ΔABE ~ ΔCFB.
24. 23.
    Find the coordinates of the point C which lies on the line AB produced such that AC = 2BC, where coordinates of points A and B are (–1, 7) and (4, –3) respectively.
25. 24 (a).
    Find the value of x for which (sin A + cosec A)² + (cos A + sec A)² = x + tan²A + cot²A
26. 24 (b).
    OR (B) Evaluate the following: (3 sin30° – 4 sin³30°) / (2 sin²50° + 2 cos²50°)
27. 25.
    Two friends Anil and Ashraf were born in the December month in the year 2010. Find the probability that:

    • (i) they have same date of birth.
    • (ii) they have different dates of birth.
28. 26 (a).
    Prove that √2 is an irrational number.
29. 26 (b).
    OR (B) Let x and y be two distinct prime numbers and p = x²y³, q = xy³, r = x³y². Find the HCF and LCM of p, q, and r. Further check if HCF(p, q, r) × LCM(p, q, r) = p × q × r or not.
30. 27.
    The monthly incomes of two persons are in the ratio 9 : 7 and their monthly expenditures are in the ratio 4 : 3. If each saved ₹5,000, express the given situation algebraically as a system of linear equations in two variables. Hence, find their respective monthly incomes.
31. 28.
    P(x, y), Q(–2, –3) and R(2, 3) are the vertices of a right triangle PQR right angled at P. Find the relationship between x and y. Hence, find all possible values of x for which y = 2.
32. 29 (a).
    Prove that: (cosA + sinA – 1)/(cosA – sinA + 1) = cosecA – cotA
33. 29 (b).
    OR (B) If cotθ + cosθ = p and cotθ – cosθ = q, prove that p² – q² = 4√(pq).
34. 30.
    α and β are zeroes of a quadratic polynomial px² + qx + 1. Form a quadratic polynomial whose zeroes are 2/α and 2/β.
35. 31.
    Rectangle ABCD circumscribes the circle of radius 10 cm. Prove that ABCD is a square. Hence, find the perimeter of ABCD.
36. 32 (a).
    The sides of a right triangle are such that the longest side is 4 m more than the shortest side and the third side is 2 m less than the longest side. Find the length of each side of the triangle. Also, find the difference between the numerical values of the area and the perimeter of the given right triangle.
37. 32 (b).
    OR (B) Express the equation (x–2)/(x–3) + (x–4)/(x–5) = 10/3; (x ≠ 3, 5) as a quadratic equation in standard form. Hence, find the roots of the equation so formed.
38. 33 (a).
    The corresponding sides of ΔABC and ΔPQR are in the ratio 3 : 5. AD⊥BC and PS⊥QR as shown in the figures.

    • (i) Prove that ΔADC ~ ΔPSR.
    • (ii) If AD = 4 cm, find the length of PS.
    • (iii) Using
    • (ii) find ar(ΔABC) : ar(ΔPQR).
39. 33 (b).
    OR (B) State Basic Proportionality Theorem. Use it to prove the following: If three parallel lines l, m, n are intersected by transversals q and s as shown in the adjoining figure, then AB/BC = DE/EF.
40. 34.
    A wooden cubical die is formed by forming hemispherical depressions on each face of the cube such that face 1 has one depression, face 2 has two depressions and so on. The sum of number of hemispherical depressions on opposite faces is always 7. If the edge of the cubical die measures 5 cm and each hemispherical depression is of diameter 1.4 cm, find the total surface area of the die so formed.
41. 35.
    The following table shows the number of patients of different age group who were discharged from the hospital in a particular month: Age (in years) | Number of Patients Discharged 5-15 | 6 15-25 | 11 25-35 | 21 35-45 | 23 45-55 | 14 55-65 | 5 Total | 80 Find the 'mean' and the 'mode' of the above data.
42. 36 (a).
    Case Study: The Olympic symbol comprising five interlocking rings represents the union of the five continents. Students made 5 circular rings in the school lawn using ropes. Each circular ring required 44 m of rope. ΔOAB is an equilateral triangle and all unshaded regions are congruent. (i) Find the radius of each circular ring.
43. 36 (b).
    Case Study (continued): (ii) What is the measure of ∠AOB?
44. 36 (c).
    Case Study (continued): (iii)(a) Find the area of shaded region R₁.
45. 37 (a).
    Case Study: Cable cars at hill stations are a major tourist attraction. The cable car ride from base to top is 5000 m. Poles are installed at equal intervals. Distance of first pole from base = 200 m, subsequent poles at intervals of 150 m, and distance of last pole from top = 300 m. (i) Find the distance of 10th pole from the base.
46. 37 (b).
    Case Study (continued): (ii) Find the distance between 15th pole and 25th pole.
47. 37 (c).
    Case Study (continued): (iii)(a) Find the time taken by cable car to reach 15th pole from the top if it is moving at the speed of 5 m/sec and coming from top.
48. 38 (a).
    Case Study: A drone was used to facilitate movement of an ambulance on a straight highway to a point P on the ground where there was an accident. The ambulance was travelling at 60 km/h. The drone stopped at point Q, 100 m vertically above point P. The angle of depression of the ambulance was 30° at a particular instant. (i) Represent the above situation with the help of a diagram.
49. 38 (b).
    Case Study (continued): (ii) Find the distance between the ambulance and the site of accident (P) at the particular instant. (Use √3 = 1.73)
50. 38 (c).
    Case Study (continued): (iii)(a) Find the time (in seconds) in which the angle of depression changes from 30° to 45°.

Class 10 Mathematics exam pattern (80 marks)

The theory paper carries 80 marks over 3 hours (38 questions, with internal choice in some). Section-wise structure:

Structure per the CBSE 2023-24 sample-paper design; question wording varies by set.

Prepare with

How to use these papers

1. 1Start chapter-wise early in the year — solve only the Mathematics questions from a chapter you have just finished.
2. 2Switch to full timed papers 2–3 months before the exam: one complete set in the real time limit, no notes.
3. 3Self-mark against the marking scheme, then fix every mistake with our free NCERT solutions.
4. 4Re-attempt your weakest chapters until the recurring question types feel routine.

CBSE Class 10 Mathematics 2025 paper — FAQ