10th Maths Half Yearly Exam 2023 Question Paper with Answers

Analyze authentic past board-level trends with the official 10th Maths Half Yearly Exam 2023 Question Paper. The state-wide half-yearly examination acts as a crucial preliminary board test, as it covers the entire 100-mark full syllabus under strict public exam conditions. This resource provides the complete 2023 question paper, highlighting critical 2-mark and 5-mark compulsory questions, challenging algebraic matrices, and essential 8-mark practical geometry and graph problems. Practicing with this past paper is one of the most effective ways to evaluate your time-management skills and master the official evaluation blueprint!

COMMON HALF YEARLY EXAMINATION - 2023

Standard X Reg No: __________________ MATHEMATICS

Time: 3.00 hours Marks: 100

Part - I: Choose the correct answer 14 × 1 = 14

1. A = {a, b, p}, B = {2, 3}, C = {p, q, r, s} then n[(A ∪ C) × B] is
a) 8 b) 20 c) 12 d) 16

2. If f: A → B is a constant function, then the range of f will have _______ elements.
a) 2 b) 0 c) 1 d) none of these

3. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is
a) 2025 b) 5220 c) 5025 d) 2520

4. The value of (1³ + 2³ + 3³ + ... + 15³) - (1 + 2 + 3 + ... + 15) is
a) 14400 b) 14200 c) 14280 d) 14520

5. Graph of a linear equation is a
a) straight line b) circle c) parabola d) hyperbola

6. Find the matrix X if 2X +

1	3
5	7

5	7
9	5

-2	-2
2	-1

2	2
2	-1

1	2
2	2

2	1
2	2

7. In a ΔABC, AD is the bisector of ∠BAC. If AB = 8 cm, BD = 6 cm and DC = 3 cm, the length of the side AC is
a) 6 cm b) 4 cm c) 3 cm d) 8 cm

8. The slope of the line joining (12, 3), (4, a) is ¹/₈. The value of 'a' is
a) 1 b) 4 c) -5 d) 2

9. (sec θ + tan θ)(sec θ - tan θ) is
a) -1 b) sec²θ c) tan²θ d) 1

10. The angle of elevation of a cloud from a point h metres above a lake is β. The angle of depression of its reflection in the lake is 45°. The height of location of the cloud from the lake is
a) ^{h(1 + tan β)}/_{(1 - tan β)} b) ^{h(1 - tan β)}/_{(1 + tan β)} c) h tan(45° - β) d) none of these

11. The height of a right circular cone whose radius is 5 cm and slant height is 13 cm will be
a) 12 cm b) 10 cm c) 13 cm d) 5 cm

12. The ratio of the volumes of a cylinder, a cone and a sphere, if each has the same diameter and same height is
a) 1:2:3 b) 2:1:3 c) 1:3:2 d) 3:1:2

13. The mean of 100 observations is 40 and their standard deviation is 3. The sum of squares of all observations is
a) 40000 b) 160900 c) 160000 d) 30000

14. Which of the following is incorrect?
a) P(A) > 1 b) 0 ≤ P(A) ≤ 1 c) P(φ) = 0 d) P(A) + P(Ā) = 1

PART - II: Answer any 10 questions (Q.No. 28 is compulsory) 10 × 2 = 20

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

16. If the ordered pairs (x² - 3x, y² + 4y) and (-2, 5) are equal, then find x and y.

17. What is the time 100 hours after 7 a.m.?

18. Find the 19th term of an A.P. -11, -15, -19, ...

19. Simplify: ^5t³/_{(4t - 8)} × ^{(6t - 12)}/_10t.

20. If A =

7	8	6
1	3	9
-4	3	-1

, B =

4	11	3
-1	2	4
7	5	0

, then find 2A + B.

21. Find the length of the tangent drawn from a point whose distance from the centre of a circle is 5 cm and the radius of the circle is 3 cm.

22. Check whether the given lines are parallel or perpendicular: 5x + 23y + 14 = 0 and 23x - 5y + 9 = 0.

23. Show that ¹/_{(1 + sin θ)} + ¹/_{(1 - sin θ)} = 2 sec²θ.

24. From the top of a rock 50√3 m high, the angle of depression of a car on the ground is observed to be 30°. Find the distance of the car from the rock.

25. Find the diameter of a sphere whose surface area is 154 m².

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

27. Two coins are tossed together. What is the probability of getting different faces on the coins?

28. Find the slope of a line joining the given points (-6, 1) and (-3, 2). (Compulsory)

PART - III: Answer any 10 questions (Q.No. 42 is compulsory) 10 × 5 = 50

29. Let f: A → B be a function defined by f(x) = ^x/₂ - 1, where A = {2, 4, 6, 10, 12}, B = {0, 1, 2, 4, 5, 9}. Represent f by:
i) Set of ordered pairs
ii) a table
iii) an arrow diagram
iv) a graph

30. If f(x) = x², g(x) = 2x and h(x) = x + 4, prove that f ∘ (g ∘ h) = (f ∘ g) ∘ h.

31. If nine times ninth term is equal to fifteen times fifteenth term, show that six times twenty fourth term is zero.

32. Find the sum of the following series: 10³ + 11³ + 12³ + ... + 20³.

33. Find the square root of 64x⁴ - 16x³ + 17x² - 2x + 1.

34. State and prove Thales theorem.

35. Find the value of k, if the area of a quadrilateral is 28 sq.units, whose vertices are taken in order (-4, -2), (-3, k), (3, -2) and (2, 3).

36. If the vertices of ΔABC are A(6, 2), B(-5, -1) and C(1, 9), find the equation of median through A.

37. From the top of a tree of height 13 m the angle of elevation and depression of the top and bottom of another tree are 45° and 30° respectively. Find the height of the second tree. (√3 = 1.732)

38. If the radii of the circular ends of a frustum which is 45 cm high are 28 cm and 7 cm, find the volume of the frustum.

39. Find the number of spherical lead shots, each of diameter 6 cm that can be made from a solid cuboid of lead having dimensions 24 cm × 22 cm × 12 cm.

40. The following table gives the values of mean and variance of heights and weights of the 10th standard students of a school. Which is more varying than the other?

	Height	Weight
Mean	155 cm	46.50 kg
Variance	72.25 cm²	28.09 kg²

41. Two dice are rolled once. Find the probability of getting an even number on the first die or a total of face sum 8.

42. If A =

3	1
-1	2

, show that A² - 5A + 7I₂ = 0. (Compulsory)

PART - IV: Answer all the questions 2 × 8 = 16

43. (a) Construct a triangle similar to a given triangle LMN with its side equal to ⁴/₅ of the corresponding sides of the triangle LMN (scale factor ⁴/₅ < 1).

(b) Draw a circle of diameter 6 cm from a point P, which is 8 cm away from its centre. Draw the two tangents PA and PB to the circle and measure their lengths.

44. (a) Nishanth is the winner in a Marathon race of 12 km distance. He ran at a uniform speed of 12 km/hr and reached the destination in 1 hour. He was followed by Aradhana, Jayanth, Sathya and Swetha with their respective speed of 6 km/hr, 4 km/hr, 3 km/hr and 2 km/hr. And, they covered the distance in 2 hours, 3 hours, 4 hours and 6 hours respectively.
Draw the speed-time graph and use it to find the time taken of Kaushik with his speed of 2.4 km/hr.

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