CBSE Class 10 Maths Basic Half Yearly Sample Paper 2025 with Solutions

1. The zeroes of a quadratic polynomial 16x² – 9 are:

a) 3/4, 3/4 b) -3/4, 3/4 c) 3/2, 2/3 d) -3/4, -3/4

2. The sum of the exponents of the prime numbers in the factorisation of 196 is

a) 1 b) 2 c) 4 d) 6

3. If y = 1 is a common root of ay² + ay + 3 = 0 and y² + y + b = 0, then ab=

a) 3 b) -7/2 c) 6 d) -3

4. The value of k for which the system of equations kx + 2y = 5 and 3x + 4y = 1 have no solutions, is

a) k = 3/2 b) k ≠ 3/2 c) k ≠ 2/3 d) k = 15

5. If α and β are the zeroes of the polynomial x² – 1, then the value of (α + β) is:

a) 2 b) 1 c) -1 d) 0

6. The distance of the point (-6, 8) from the origin is:

a) 6 b) -6 c) 8 d) 10

7. In the given figure, ∆ABC ~ ∆QRP. If AC = 6 cm, BC = 5 cm, QR = 3 cm and PR = x; then the value of x is:

a) 3.6 cm b) 2.5 cm c) 10 cm d) 3.2 cm

8. The pair of linear equations 2x = 5y + 6 and 15y = 6x – 18 represents two lines which are:

a) intersecting b) parallel c) coincident d) either intersecting or parallel

9. In the given figure, PQ and PR are tangents drawn from P to the circle with centre O such that ∠QPR = 65°. The measure of ∠QOR is:

a) 65° b) 125° c) 115° d) 90°

10. If sin A = cos A, then A is equal to:

a) 30° b) 45° c) 60° d) 90°

11. In ∆ABC, right angled at C, if tan A = 8/7, then the value of cot B is

a) 7/8 b) 8/7 c) 7/√113 d) 8/√113

12. Area of a quadrant of a circle of radius 7 cm is:

a) 154 cm² b) 77 cm² c) 77/2 cm² d) 77/4 cm²

13. In the given figure, the perimeter of ∆ABC is:

a) 30 cm b) 15 cm c) 45 cm d) 60 cm

14. If a pole 6m height casts a shadow 2√3 m long on the ground, then the sun’s elevation is:

a) 15° b) 30° c) 45° d) 60°

15. Cards are marked with numbers 1 to 50 are placed in the box and mixed thoroughly. One card is drawn at random from the box. What is the probability of getting a multiple of 5?

a) 1 b) 0 c) 1/25 d) 1/5

16. The value of p for which 2p+1, 10 and 3p+4 are three consecutive terms of an AP is,

a) -1 b) 3 c) 1 d) 2

17. The radius of a spherical balloon increases from 7cm to 14cm when air is pumped into it. The ratio of the surface area in both cases is:

a) 4 : 1 b) 1 : 4 c) 3 : 1 d) 1 : 3

18. If the mean and median of a distribution are 32 and 30, respectively, then its mode is:

a) 36 b) 26 c) 30 d) 20

ASSERTION AND REASONING PRACTICE QUESTIONS

Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:

(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.

19. Assertion (A): The mid-point of a line segment divides the line in the ratio 1: 1.

Reason (R): The ratio in which the point (–3, 3) divides the line segment joining the points (– 5, 4) and (–2, 3) is 1:2.

20. Assertion: The value of sin 60° cos 30° + sin 30° cos 60° is 1.

Reason: sin 90° = 1 and cos 90° = 0.

SECTION B

(Very short answer type questions)

21. Find the number of solutions that the following pair of linear equations has. x + 2y = 6, 2x + 4y = 12

22. a) In ∆ABC, if D is the mid-point of the sides AB and DE || BC, show that AE = EC.

OR

b) In the given figure, ABC and AMP are two right triangles, right-angled at B and M, respectively. Prove that ∆ABC ~ ∆AMP.

23. Prove that tangents drawn at the ends of a diameter are parallel.

24. a) If the HCF (152, 95) is 19, find their LCM.

OR

b) Emily wants to make flower bouquets using two types of flowers. She has 36 roses and 48 lilies. She wants to make the largest bouquets possible with the same number of roses and lilies in each bouquet. How many bouquets can she make?

25. Two players, Sangeeta and Reshma, play a tennis match. It is known that the probability of Sangeeta winning the match is 0.62. What is the probability of Reshma winning the match?

SECTION C

(Short answer type questions) 

26. Prove that √5 is irrational

27. Find the zeroes of x² -25 and verify the relationship between the zeroes and their coefficients. 

28. a) In the given figure, O is the centre of the circle. AB and AC are tangents drawn to the circle from point A. If ∠BAC = 65°, then find the measure of ∠BOC. Also find ∠OBC.

OR

b) In the given figure, O is the centre of the circle and QPR is a tangent to it at P. Prove that ∠QAP + ∠APR = 90°.

29. A chord of a circle, of radius 14 cm, subtends and angle of 90° at the centre. Find the area of the minor segment.

30. a) Prove that cosA/1+sinA + 1+ sinA/cos A =2secA.

OR 

(b) Prove that (sinA+cosecA)2+(cosA+secA)2=7+tan2A+cot2A

31. A pair of dice is thrown together. Find the probability of getting 

i) A doublet

ii) The sum of the numbers on two dice is 10. 

iii) Prime numbers on both dice. 

SECTION D

(Long answer type questions)

32. a) The diagonals of a rectangular field are 25 m more than the shorter side. If the longer side is 23 m more than the shorter side, find the length of the sides of the field.

OR

b) The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages (in years) was 124. Determine their present ages.

33. Find the solution of the following pair of linear equations in two variables graphically: x + 3y = 6, 2x – 3y =12. Also, find the area of the triangle formed by the two lines and the y-axis.

34. A vessel in the form of a hemispherical bowl is surmounted by a hollow cylinder of the same diameter. The diameter of the hemispherical bowl is 14 cm, and the total height of the vessel is 13 cm. Find the inner surface area of the vessel. Also, find the volume of the vessel.

35. a) From the top of a building 60 m high, the angles of depression of the top and bottom of a tower are observed to be 30° and 60°, respectively. Find the height of the tower. Also, find the distance between the building and the tower. (Use √3 = 1.732)

OR

b) The angle of elevation of the top of a building from a point A on the ground is 30°. On moving a distance of 30m towards its base to the point B, the angle of elevation changes to 45°. Find the height of the building and the distance of its base from point A. (Use √3 = 1.732)

SECTION E

(Case Study Questions)

36. Nayan got his name registered for a sprint race. The race is scheduled for a month later than the time he registered for the race. He started practising for the race. His current run time is 51 sec for the distance to be covered in the race. He decided to gradually decrease the time by 2 seconds every day. He wants to reduce his time to 31 seconds.

i) What is a suitable AP for the above situation?

ii) If Nayan is able to achieve his target, then in how many days will he be able to achieve it?

iii) a) On which day will he be able to complete the race in 41 seconds? OR

b) What is the nth term of this AP?

37. A student drew a quadrilateral ABCD on a graph sheet (Cartesian plane). Observe the figure and answer the questions.

i) Find the length of side AB.

ii) Find the coordinates of the midpoint of the diagonal BD.

iii) a) Find the ratio in which the x-axis divides the side CD.

OR

b) Find the coordinate of the point on the x-axis equidistant from B and D.

38. In a Vidyalaya, there are three sections: A, B and C. 30 students are there in section A, in section B, there are 41 students and 29 students are in section C. A periodic test was conducted to assess the performance of students, thereafter analyse and plan the teaching learning process accordingly. The marks obtained out of 40 are given below in the table: 

i) How many students have obtained more than or equal to 35 marks?

ii) Arrange the given data in a class interval and write the lower limit of the modal class.

iii) Find the mode of the data.

Q. No

Answers

1

b) -3/4, 3/4

2

c) 4

3

a) 3

4

a) 3/2

5

d) 0

6

d) 10

7

b) 2.5 cm

8

b) parallel

9

c) 115°

10

b) 45°

11

b) 8/7

12

c) 77/2 cm2

13

a) 30 cm

14

d) 60°

15

d) 1/5

16

b) 3

17

b) 1 : 4

18

b) 26

19

(c) Assertion (A) is true, but reason (R) is false.

20

(b) Both assertion (A) and reason (R) are true, but reason (R) is not the correct explanation of assertion (A).

CBSE Class 10 Maths Basic Half Yearly Sample Paper Answer Key 2025 with Solutions