10th Maths Full Test Question Paper 2 with Answers

Advance your board exam conditioning with the 10th Maths Full Test Question Paper 2. As the second installment in our full-syllabus mock series, this 100-mark paper introduces an entirely new set of high-yield challenges to test your problem-solving depth. Structured strictly according to the latest SSLC blueprint, it features critical textbook examples, tricky compulsory-type equations, and essential 8-mark graph and geometry constructions. Use this test paper as a timed, realistic home rehearsal to systematically spot lingering conceptual gaps, master your rough-work time allocation, and lock in your step-by-step presentation for a perfect centum!

SSLC - 10TH

MATHEMATICS FULL PAPER - 2

TIME: 3 hrs MAXIMUM MARKS: 100

PART - I: CHOOSE THE MOST SUITABLE ANSWER [14 × 1 = 14]

Note: (i) Answer all the questions.
(ii) Choose the most appropriate answer from the given four alternatives and write the option code and the corresponding answer.

1. If n(A × B) = 6 and A = {1, 3} then n(B) is:
(1) 1 (2) 2 (3) 3 (4) 6

2. If f: A → B is a bijective function and if n(B) = 7 then n(A) is equal to:
(1) 7 (2) 49 (3) 1 (4) 14

3. If the HCF of 65 and 117 is expressible in the form 65m - 117, then the value of m is:
(1) 4 (2) 2 (3) 1 (4) 3

4. If 1 + 2 + 3 + ... + n = k then 1³ + 2³ + 3³ + ... + n³ is equal to:
(1) k² (2) k³ (3) ^{k(k + 1)}/₂ (4) (k + 1)³

5. If (x - 6) is the HCF of x² - 2x - 24 and x² - kx - 6, then the value of k is:
(1) 3 (2) 4 (3) 6 (4) 8

6. If A is a 2 × 3 matrix and B is a 3 × 4 matrix, how many columns does AB have?
(1) 3 (2) 4 (3) 2 (4) 5

7. If ^a³/_{(a - b)} is added with ^b³/_{(b - a)}, then the new expression is:
(1) a² + ab + b² (2) a² - ab + b² (3) a³ + b³ (4) a² - b²

8. The lengths of the sides of two similar triangles ΔABC and ΔPQR are 36 cm and 24 cm respectively. If PQ = 10 cm, then the length of AB is:
(1) ⁵/₃ cm (2) ^10√6/₃ cm (3) ⁶⁶/₂ cm (4) 15 cm

9. The area of triangle formed by the points (-5, 0), (0, -5) and (5, 0) is:
(1) 0 sq. units (2) 25 sq. units (3) 5 sq. units (4) 50 sq. units

10. If sin θ = cos θ, then 2 tan²θ + sin²θ - 1 is equal to:
(1) ³/₂ (2) -³/₂ (3) ²/₃ (4) -²/₃

11. If the ratio of the height of a tower and the length of its shadow is √3 : 1, then the angle of elevation of the sun has measure:
(1) 45° (2) 30° (3) 90° (4) 60°

12. The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be:
(1) 12 cm (2) 10 cm (3) 13 cm (4) 5 cm

13. The range of the data 8, 8, 8, 8, ..., 8 is:
(1) 0 (2) 1 (3) 8 (4) 3

14. The probability that a student will score a centum in mathematics is ⁴/₅. The probability that he will not score a centum is:
(1) ¹/₅ (2) ²/₅ (3) ³/₅ (4) ⁴/₅

PART - II: ANSWER ANY 10 QUESTIONS [10 × 2 = 20]

(Question No. 28 is compulsory. Select any 9 questions from the first 13 questions.)

15. A relation R is given by R = {(x, y) | y = x + 3, x ∈ {0, 1, 2, 3, 4, 5}}. Determine its domain and range.

16. Find k if f(k) = 5, where f(x) = 2x - 1.

17. If 13824 = 2^x × 3^y then find the values of x and y.

18. In a G.P. 729, 243, 81, ..., find a, r, and a₅.

19. Find the excluded values of ^t/_{(t² - 5t + 6)}.

20. If A =

5	2	2
√7	0.5	√2
8	3	1

, then verify that (A^T)^T = A.

21. In ΔABC if DE || BC, AD = x, DB = x - 2, AE = x + 2 and EC = x - 1, then find the lengths of the sides AB and AC.

22. If the three points (3, -1), (a, 3), and (1, -3) are collinear, find the value of a.

23. Find the equation of a straight line which is parallel to the line 3x - 7y = 12 and passing through the point (6, 4).

24. Prove that √[^{(1 + cos θ)}/_{(1 - cos θ)}] = cosec θ + cot θ.

25. If the total surface area of a cone of radius 7 cm is 704 cm², then find its slant height.

26. A right circular cylinder just encloses a sphere of radius r units. Find the curved surface area of the cylinder.

27. Find the standard deviation of first 21 natural numbers.

28. In a two children family, find the probability that there is at least one girl in the family. (Compulsory)

PART - III: ANSWER ANY 10 QUESTIONS [10 × 5 = 50]

(Q.No. 42 is compulsory. Select any 9 questions from the first 13 questions.)

29. Let A = {x ∈ ℕ | 1 < x < 4}, B = {x ∈ W | 0 ≤ x < 2} and C = {x ∈ ℕ | x < 3}. Verify that A × (B ∪ C) = (A × B) ∪ (A × C).

30. If the function f is defined by:

f(x) =
  • x + 2, x > 1
  • 2, -1 ≤ x ≤ 1
  • x - 1, -3 < x < -1

find the values of: i) f(3) ii) f(0) iii) f(-1.5) iv) f(2) + f(-2)

31. The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms.

32. Find the sum to n terms of the series 5 + 55 + 555 + ...

33. If 9x⁴ + 12x³ + 28x² + ax + b is a perfect square, find the values of a and b.

34. If A =

a	b
c	d

and I =

1	0
0	1

, show that A² - (a + d)A = (bc - ad)I₂.

35. State and prove Pythagoras theorem.

36. Find the area of the quadrilateral formed by the points (-9, -2), (-8, -4), (2, 2), and (1, -3).

37. Find the equation of the altitude of ΔABC through A, where the vertices are A(6, 2), B(-5, -1) and C(1, 9).

38. From the top of a lighthouse, the angle of depression of two ships on opposite sides of it are observed to be 30° and 60°. If the height of the lighthouse is h, find the distance between the ships.

39. A container open at the top is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container at the rate of ₹40 per litre.

40. A right circular cylindrical container of base radius 6 cm and height 15 cm is full of ice cream. The ice cream is to be filled in cones of height 9 cm and base radius 3 cm, having a hemispherical cap. Find the number of cones needed to empty the container.

41. Two dice are rolled one after another. Find the probability of getting an even number on the first die or a total score of 8.

42. Simplify: ^{(¹/_p + ¹/_q + ¹/_r)}/_{(¹/_p - ¹/_q + ¹/_r)} × [1 + ^{(q² + r² - p²)}/_(2qr)]. (Compulsory)

SECTION - IV: ANSWER BOTH QUESTIONS [2 × 8 = 16]

Answer both the questions choosing either of the alternatives:

43. a) A bus is travelling at a uniform speed of 50 km/hr. Draw the distance-time graph and hence find:
i) the constant of variation
ii) how far it will travel in 1¹/₂ hours
iii) the time required to cover a distance of 300 km from the graph

(OR)

b) Draw the graph of y = x² + 3x - 4 and hence solve x² + 3x - 4 = 0.

44. a) Construct a ΔPQR such that QR = 6.5 cm, ∠P = 60°, and the altitude from P to QR is 4.5 cm.

b) Take a point which is 11 cm away from the centre of a circle of radius 4 cm and draw the two tangents to the circle from the point.