MOCK BOARD EXAMINATION
MATHEMATICS — CLASS X (STANDARD)
Time Allowed: 3 Hours | Maximum Marks: 80
GENERAL INSTRUCTIONS
- This question paper contains 38 questions divided into five sections — A, B, C, D and E.
- Section A: Q1–Q18 are MCQs and Q19–Q20 are Assertion-Reason questions. Each carries 1 mark.
- Section B: Q21–Q25 are Very Short Answer type questions. Each carries 2 marks.
- Section C: Q26–Q31 are Short Answer type questions. Each carries 3 marks.
- Section D: Q32–Q35 are Long Answer type questions. Each carries 5 marks.
- Section E: Q36–Q38 are Case Study based questions. Each carries 4 marks.
- There is no overall choice. However, internal choices are provided in some questions.
- Use of calculator is not permitted.
SECTION — A (20 Marks)
All questions carry 1 mark each. Choose the correct option (Q1–Q18).
Q1. The HCF of 96 and 404 is: (a) 4 (b) 8 (c) 12 (d) 16
Q2. If the zeroes of the quadratic polynomial x² + (a + 1)x + b are 2 and −3, then: (a) a = −7, b = −1 (b) a = 5, b = −1 (c) a = 2, b = −6 (d) a = 0, b = −6
Q3. The pair of linear equations 2x + 3y = 7 and 4x + 6y = 14 has: (a) No solution (b) Unique solution (c) Infinitely many solutions (d) Exactly two solutions
Q4. The roots of the quadratic equation 2x² − 5x + 3 = 0 are: (a) 1 and 3/2 (b) −1 and 3/2 (c) 1 and −3/2 (d) 3 and 1/2
Q5. The 10th term of the AP: 3, 8, 13, 18, ... is: (a) 43 (b) 48 (c) 53 (d) 58
Q6. In ΔABC, DE ∥ BC. If AD = 4 cm, DB = 6 cm and AE = 3 cm, then EC = (a) 4 cm (b) 4.5 cm (c) 5 cm (d) 5.5 cm
Q7. If sin A = 3/5, then cos A = (a) 4/5 (b) 3/4 (c) 5/4 (d) 5/3
Q8. The value of (sin 30° + cos 60°) − (sin 60° + cos 30°) is: (a) 0 (b) 1 (c) √3 − 1 (d) 1 − √3
Q9. The angle of elevation of the top of a tower from a point 30 m away is 60°. The height of the tower is: (a) 10√3 m (b) 30 m (c) 30√3 m (d) 60 m
Q10. The distance between the points (2, −3) and (−4, 5) is: (a) 10 units (b) √100 units (c) √28 units (d) Both a and b
Q11. The coordinates of the mid-point of the segment joining A(2, 5) and B(6, 3) are: (a) (4, 4) (b) (8, 8) (c) (3, 4) (d) (4, 3)
Q12. The area of a circle inscribed in a square of side 10 cm is: (a) 25π cm² (b) 50π cm² (c) 100π cm² (d) 5π cm²
Q13. The volume of a cylinder with radius 7 cm and height 10 cm is: (a) 1540 cm³ (b) 2156 cm³ (c) 3080 cm³ (d) 4312 cm³
Q14. A die is thrown once. The probability of getting a prime number is: (a) 1/6 (b) 1/3 (c) 1/2 (d) 2/3
Q15. If the mean of 5 observations x, x+2, x+4, x+6, x+8 is 11, then the value of x is: (a) 7 (b) 9 (c) 11 (d) 13
Q16. The tangent at any point of a circle is __________ to the radius at the point of contact. (a) Parallel (b) Perpendicular (c) Equal (d) Bisected
Q17. The sum of the first 20 natural numbers is: (a) 190 (b) 200 (c) 210 (d) 220
Q18. If tan θ = 1/√3, then the value of (sin θ + cos θ) is: (a) √3/2 (b) (√3 + 1)/2 (c) 1 (d) (1 + √3)/√3
For Q19–Q20, choose the correct option: (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.
Q19. Assertion (A): √5 is an irrational number. Reason (R): The decimal expansion of an irrational number is non-terminating and non-repeating.
Q20. Assertion (A): If the discriminant of a quadratic equation is zero, the equation has two distinct real roots. Reason (R): Discriminant D = b² − 4ac.
SECTION — B (10 Marks)
All questions carry 2 marks each.
Q21. Find the LCM and HCF of 510 and 92 using prime factorisation.
Q22. Find the zeroes of the polynomial p(x) = x² − 3 and verify the relationship between zeroes and coefficients.
Q23. In the figure, DE ∥ BC. If AD = 2.4 cm, AB = 6 cm and AC = 5 cm, find AE and EC.
Q24. Evaluate: (sin 45° + cos 45°) / tan 45°
OR
Prove that: tan 60° / (sin 30° × cos 30°) = 4/√3
Q25. Two tangents TP and TQ are drawn from an external point T to a circle with centre O. Prove that ∠PTQ + ∠POQ = 180°.
SECTION — C (18 Marks)
All questions carry 3 marks each.
Q26. Prove that 3 + 2√5 is irrational.
Q27. Solve the following pair of linear equations by the substitution method: x + y = 14 and x − y = 4
Q28. Find the roots of the equation: 3x² − 2√6x + 2 = 0
OR
A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. Find the speed of the train.
Q29. If the sum of the first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.
Q30. Prove that the lengths of tangents drawn from an external point to a circle are equal.
Q31. The following data gives the observed lifetimes (in hours) of 225 electrical components. Find the modal lifetime.
| Lifetime (hours) | No. of Components |
|---|---|
| 0 – 20 | 10 |
| 20 – 40 | 35 |
| 40 – 60 | 52 |
| 60 – 80 | 61 |
| 80 – 100 | 38 |
| 100 – 120 | 29 |
SECTION — D (20 Marks)
All questions carry 5 marks each.
Q32. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.
OR
The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 15 seconds, the angle of elevation becomes 30°. If the aeroplane is flying at a constant height of 1500√3 m, find the speed of the aeroplane.
Q33. Prove that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). Using this theorem, find the length of the altitude drawn from the right-angle vertex to the hypotenuse in a right triangle with sides 6 cm, 8 cm and 10 cm.
Q34. A well of diameter 3 m is dug 14 m deep. The earth taken out has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment. (Use π = 22/7)
OR
A solid toy is in the form of a hemisphere surmounted by a right circular cone of height 2 cm and radius 4 cm. Find the total volume, total surface area, and slant height of the toy. (Use π = 22/7)
Q35. The following frequency distribution gives the monthly consumption of electricity of 68 consumers. Find the median, mean and mode of the data and compare all three measures.
| Monthly Consumption (units) | No. of Consumers |
|---|---|
| 65 – 85 | 4 |
| 85 – 105 | 5 |
| 105 – 125 | 13 |
| 125 – 145 | 20 |
| 145 – 165 | 14 |
| 165 – 185 | 8 |
| 185 – 205 | 4 |
SECTION — E (12 Marks)
Each Case Study question carries 4 marks.
Q36. CASE STUDY – 1: Real Numbers A mathematics teacher asked students to find the HCF and LCM of 12, 18 and 24 using prime factorisation.
(i) Write the prime factorisation of 12, 18 and 24. [1 mark] (ii) Find HCF (12, 18, 24). [1 mark] (iii) Find LCM (12, 18, 24). [2 marks]
Q37. CASE STUDY – 2: Coordinate Geometry A city park is in the form of a quadrilateral ABCD with vertices A(1, 2), B(4, 6), C(7, 2) and D(4, −2).
(i) Find the length of diagonal AC. [1 mark] (ii) Find the length of diagonal BD. [1 mark] (iii) Find the mid-point of both diagonals. What do you observe? What type of quadrilateral is ABCD? [2 marks]
Q38. CASE STUDY – 3: Statistics and Probability During a survey, 200 students were asked about their favourite sport:
| Sport | No. of Students |
|---|---|
| Cricket | 80 |
| Football | 50 |
| Badminton | 40 |
| Basketball | 20 |
| Others | 10 |
(i) What is the probability that a randomly selected student likes Cricket? [1 mark] (ii) What is the probability that a student likes Football or Basketball? [1 mark] (iii) What is the probability that a student does NOT like Badminton? [2 marks]
— END OF QUESTION PAPER — All the best!
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