CBSE Class 10 Mathematics 2025 — Set 5

Questions in this 2025 Mathematics paper (Set 5)

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.
   (√3 + 2)² + (√3 − 2)² is a/an

   • (a) positive rational number
   • (b) negative rational number
   • (c) positive irrational number
   • (d) negative irrational number
2. 2.
   Let x = a² b³ c⁴ and y = a³ bᵐ c², where a, b, c are prime numbers. If LCM of x and y is a³ b⁴ c⁴, then the value of m + n is

   • (a) 10
   • (b) 7
   • (c) 6
   • (d) 5
3. 3.
   For any prime number p, if p divides a², where a is any real number, then p also divides

   • (a) a
   • (b) a^(1/2)
   • (c) a^(3/2)
   • (d) a^(1/8)
4. 4.
   Which of the following equations is a quadratic equation?

   • (a) x² + 1 = (x − 1)²
   • (b) (x + √x)² = 2x√x
   • (c) x³ + 3x² = (x + 1)³
   • (d) (x + 1)(x − 1) = (x + 1)²
5. 5.
   If x² + bx + b = 0 has two real and distinct roots, then the value of b can be

   • (a) 0
   • (b) 4
   • (c) 3
   • (d) −3
6. 6.
   In the figure given below, points P, Q, R divide the line segment AB into four equal parts. The point Q divides PB in the ratio

   • (a) 1 : 3
   • (b) 2 : 3
   • (c) 1 : 2
   • (d) 1 : 1
7. 7.
   A bag contains red balls and black balls in the ratio 3 : 7. A ball is drawn at random. The probability that the ball so drawn is black in colour, is

   • (a) 3/7
   • (b) 0.3
   • (c) 0.7
   • (d) 1/7
8. 8.
   Which of the following statements is false?

   • (a) Two right triangles are always similar.
   • (b) Two squares are always similar.
   • (c) Two equilateral triangles are always similar.
   • (d) Two circles are always similar.
9. 9.
   In the adjoining figure, ABCD is a trapezium in which XY ∥ AB ∥ CD. If AX = (2/3) AD, then CY : YB =

   • (a) 2 : 3
   • (b) 3 : 2
   • (c) 1 : 3
   • (d) 1 : 2
10. 10.
    Which of the following statements is false?

    • (a) Infinite number of tangents can be drawn to a circle.
    • (b) Infinite number of tangents can be drawn to a circle from a point outside the circle.
    • (c) Infinite number of secants can be drawn to a circle from a point outside the circle.
    • (d) Angle between tangent and diameter at point of contact is 90°.
11. 11.
    In the adjoining figure, PA and PB are tangents to a circle with centre O. The measure of angle APB is

    • (a) 210°
    • (b) 150°
    • (c) 105°
    • (d) 30°
12. 12.
    (1 − tan² 30°)/(1 + tan² 30°) is equal to

    • (a) sin 60°
    • (b) cos 60°
    • (c) tan 60°
    • (d) sec 60°
13. 13.
    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°. The height of chimney above the ground is

    • (a) 38.2 m
    • (b) 36.4 m
    • (c) 40 m
    • (d) (38.2)√2 m
14. 14.
    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

    • (a) 8 cm
    • (b) 4 cm
    • (c) 2 cm
    • (d) 3 cm
15. 15.
    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

    • (a) 17 cm
    • (b) 18 cm
    • (c) 22 cm
    • (d) 23 cm
16. 16.
    If x median + y mean = z mode; is the empirical relationship between mean, median and mode, then the value of x + y + z is

    • (a) 6
    • (b) 3
    • (c) 2
    • (d) 1
17. 17.
    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?

    • (a) 29 and 33
    • (b) 25 and 28
    • (c) 28 and 29
    • (d) 33 and 38
18. 18.
    The probability of getting a composite number greater than 3 on throwing a die is

    • (a) 1/6
    • (b) 1/3
    • (c) 1/2
    • (d) 2/3
19. 19.
    Assertion (A): For an acute angle θ, sin θ = 3/5 ⇒ cos θ = −4/5. Reason (R): For any value of θ, (0° ≤ θ ≤ 90°), sin²θ + cos²θ = 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…
20. 20.
    Assertion (A): For an A.P. 3, 6, 9, ..., 198, the 10th 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 the nᵗʰ term from the end of the given A.P. is l − (n − 1)d.

    • (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…
21. 21.
    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.
22. 22.
    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 = (4/7) AB. Find the coordinates of point P.
23. 23.
    It is given that sin(A − B) = sin A cos B − cos A sin B. Use it to find the value of sin 15°.
24. 24.
    AD and PS are medians of triangles ABC and PQR respectively such that △ABD ~ △PQS. Prove that △ABC ~ △PQR.
25. 25.
    While shuffling a pack of 52 cards, one card was accidently dropped. Find the probability that the dropped card

    • (i) is not a face card.
    • (ii) is a black king.
26. 26.
    Prove that √3 is an irrational number.
27. 27.
    Obtain the zeroes of the polynomial 7x² + 18x − 9. Hence, write a polynomial each of whose zeroes is twice the zeroes of the given polynomial.
28. 28.
    Solve the following system of equations graphically: 2x − y − 2 = 0 −4x + y + 4 = 0 Also, find the absolute difference between the ordinates of the points where given lines cut the y-axis.
29. 29.
    Find a relation between x and y such that the point 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.
30. 30.
    Prove the following trigonometric identity: (1 + cosec A) / cosec A = cos² A / (1 − sin A)
31. 31.
    In the adjoining figure, XY and X'Y' are parallel tangents to a circle with centre O. Another tangent AB touches the circle at C intersecting XY at A and X'Y' at B. Prove that AB subtends right angle at the centre of the circle; or ∠AOB = 90°.
32. 32.
    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.
33. 33.
    If a line is drawn parallel to one side of a triangle intersecting the other two sides in distinct points, then it divides the two sides in the same ratio. Prove it. Also, state the converse of the above statement.
34. 34.
    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.
35. 35.
    Medical check-up was carried out for 35 students of a class and their weights were recorded as follows: Weight (in kg): 38-40, 40-42, 42-44, 44-46, 46-48, 48-50, 50-52 Number of Students: 3, 2, 4, 5, 14, 4, 3 Find the difference between the mean weight and the median weight.
36. 36 (i).
    [Case Study] 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. (i) Find the length of wire needed to fence the entire land.
37. 36 (ii).
    [Case Study] A farmer has a circular piece of land of radius 35 m. He wishes to construct his house in the form of largest possible square within the land. (ii) Find the length of each side of the square land on which house will be constructed.
38. 36 (iii)(a).
    [Case Study] A farmer has a circular piece of land of radius 35 m with a largest possible square house inscribed in it. (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.
39. 36 (iii)(b).
    [Case Study] (OR) A farmer has a circular piece of land of radius 35 m with a largest possible square house inscribed in it. (iii)(b) Find the ratio of area of land on which house is built to remaining area of circular piece of land.
40. 37 (i).
    [Case Study] In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed as shown, 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. (i) How many triangles will be there in the bottom most row?
41. 37 (ii).
    [Case Study] In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed in rows: 1, 3, 5, 7, ... (ii) How many triangles will be there in fourth row from the bottom?
42. 37 (iii)(a).
    [Case Study] In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed in rows: 1, 3, 5, 7, ... (iii)(a) Find the total number of triangles of side 1 cm each till the 8th row.
43. 37 (iii)(b).
    [Case Study] (OR) In an equilateral triangle of side 10 cm, equilateral triangles of side 1 cm are formed in rows: 1, 3, 5, 7, ... (iii)(b) How many more number of triangles are there from 5th row to 10th row than in first 4 rows? Show working.
44. 38 (i).
    [Case Study] Passenger boarding stairs (stair cars/aircraft steps) provide a mobile means to travel between aircraft doors and the ground. Larger aircraft have door sills 5 to 20 feet (1 foot = 30 cm) high. 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. (i) Find the angle at which stairs are inclined…
45. 38 (ii).
    [Case Study] An aircraft has a door sill at 15 feet height. A stair car is placed 15 feet away horizontally. (ii) Find the length of stairs used to reach the door sill.
46. 38 (iii)(a).
    [Case Study] An aircraft door sill is at height 15 feet. A stair car is placed 15 feet away. (iii)(a) If the 20 feet long stairs is inclined at an angle of 60° to reach the door sill, then find the height of the door sill above the ground. (Use √3 = 1.732)
47. 38 (iii)(b).
    [Case Study] (OR) The door sill of a plane is 20 feet above the ground. (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°? Also, find the horizontal distance of base of stair car from the plane.

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