10th Maths Quarterly Exam 2026 Question Paper & Answer Key | Dharmapuri District (EM)

QUARTERLY EXAMINATION - 2026

Instructions:

• Write clearly and legibly without mistakes and overwriting utilising the maximum time allotted for the exam
• Answers should be in your own style without changing the main core concept.
• Use only black or blue ink pen to write the exam.
• Draw clear diagrams wherever necessary.

1. If n(A × B) = 6 and A = {1, 3} then n(B) is
a) 1      b) 2      c) 3      d) 6

2. If {(a, 8), (6, b)} represents an identity function, then the value of a and b are respectively
a) (8, 6)      b) (8, 8)      c) (6, 8)      d) (6, 6)

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

4. Let n(A) = m and n(B) = n then the total number of functions that can be defined from A to B is
a) mⁿ      b) n^m      c) 2^mn − 1      d) 2^m

5. The sum of the exponents of the prime factors in the prime factorization of 1729 is
a) 1      b) 2      c) 3      d) 4

6. Given F₁ = 1, F₂ = 3 and F_n = F_n−1 + F_n−2 then F₅ is
a) 3      b) 5      c) 8      d) 11

7. If 6 times of 6^th term of an A.P is equal to 7 times the 7^th term then the 13^th term of the A.P is
a) 0      b) 6      c) 7      d) 13

8. The Value of (1³ + 2³ + 3³ + ... + 15³) − (1 + 2 + 3 + ... + 15) is
a) 14400      b) 14200      c) 14280      d) 14520

9. The square root of (256 x⁸ y⁴ z¹⁰) / (25 x⁶ y⁴ z⁸) is equal to
a) ¹⁶/₅ |(x² z²) / y²|      b) 16 |y² / (x² z⁴)|      c) ¹⁶/₅ |y / x²|      d) ¹⁶/₅ |(x z²) / y|

10. Which of the following should be added to make x⁴ + 64 a perfect square
a) 4x²      b) 16x²      c) x²      d) −8x²

11. Graph of a linear equation is a
a) Straight line      b) Circle      c) Parabola      d) None of the above

12. In ΔLMN, ∠L = 60°, ∠M = 50°. If ΔLMN ~ ΔPQR then the value of ∠R is
a) 40°      b) 70°      c) 30°      d) 110°

13. The area of triangle formed by the points (−5, 0) (0, −5) and (5, 0) is
a) 0 Sq.units      b) 25 Sq.units      c) 5 Sq.units      d) None of these

14. If (5, 7), (3, P) and (6, 6) are collinear, then find the value of P is
a) 3      b) 6      c) 9      d) 12

15. Let A = {1, 2, 3} and B = {x | x is a prime number less than 10}. Find A × B and B × A

16. Show that the function f: N → N defined by f(m) = m² + m + 3 is one - one function.

17. Find k if f∁f(k) = 5, Where f(k) = 2k − 1

18. Find all positive integers, when divided by 3 leaves remainder 2.

19. Find the least number that is divisible by the first ten natural numbers.

20. Compute x, such that 10⁴ ≡ x (mod 19).

21. If 3 + k, 18 − k, 5k + 1 are in A.P then find k

22. How many terms of the series 1 + 5 + 9 + ... must be taken so that their sum is 190?

23. Find the area of the triangle formed by point (1, −1) (−4, 6) and (−3, −5).

24. Simplify: [(4x²y) / (2z²)] × [(6xz³) / (20y⁴)]

25. Find the sum and Product of the roots of x² + 8x − 65 = 0

26. If ΔABC is similar to ΔDEF such that BC = 3 Cm, EF = 4 Cm and area of ΔABC = 54 Cm², find the area of ΔDEF

27. Find the equation of a line through the given pair of points (2, 3) and (−7, −1)

28. If 13824 = 2^a × 3^b then find a and b.

29. Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime number. Verify that (A ∩ B) × C = (A × C) ∩ (B × C)

30. 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

32. Consider the functions f(x) = x − 4, g(x) = x² and h(x) = 3x − 5 show that (f ∁ g) ∁ h = f ∁ (g ∁ h)

33. If p₁^x₁ × p₂^x₂ × p₃^x₃ × p₄^x₄ = 113400 where p₁, p₂, p₃, p₄ are primes in ascending order and x₁, x₂, x₃, x₄ are integers, find the value of p₁, p₂, p₃, p₄ and x₁, x₂, x₃, x₄.

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

35. Find the sum of n terms of the series: 5 + 55 + 555 + ...

36. Rekha has 15 Square colour papers of sizes 10 cm, 11 cm, 12 cm, ..., 24 cm. How much area can be decorated with these colour papers?

37. Find the area of the quadrilateral formed by the points (8, 6) (5, 11) (−5, 12), (−4, 3)

38. Solve: 3x − 2y + z = 2, 2x + 3y − z = 5, x + y + z = 6

39. Simplify: [1 / (x² − 5x + 6)] + [1 / (x² − 3x + 2)] − [1 / (x² − 8x + 15)]

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

41. State and Prove Thales Theorem.

42. Find the equation of a straight line passing through the point (6, −2) and perpendicular to the line joining the points (6, 7) and (2, −3).

43. a) Construct a triangle similar to a given triangle PQR with its sides equal to ⁷/₄ of the corresponding sides of the triangle PQR (Scale factor ⁷/₄ > 1).
(OR) b) Construct a ΔPQR in which the base PQ = 4.5 Cm, ∠R = 35° and the median RG from R to PQ is 6 Cm.

44. 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 will it travel in 90 minutes?
    iii) the time required to cover a distance of 300 km from the graph.
(OR) b) Draw the graph of xy = 24, x, y > 0. From the graph find,
    (i) y when x = 3 and (ii) x when y = 6.

1. c) 3

2. a) (8, 6)

3. a) 7

4. b) n^m

5. c) 3

6. d) 11

7. a) 0

8. c) 14280

9. d) ¹⁶/₅ |(x z²) / y|

10. b) 16x²

11. a) Straight line

12. b) 70°

13. b) 25 sq.u

14. c) 9

15. A = {1, 2, 3}, B = {2, 3, 5, 7}
A × B = {(1, 2), (1, 3), (1, 5), (1, 7), (2, 2), (2, 3), (2, 5), (2, 7), (3, 2), (3, 3), (3, 5), (3, 7)}
B × A = {(2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (5, 1), (5, 2), (5, 3), (7, 1), (7, 2), (7, 3)}

16. f(m) = m² + m + 3
f(1) = 1 + 1 + 3 = 5
f(2) = 4 + 2 + 3 = 9
Different elements in N have different images in N.
It is one-one function.

17. f∁f(k) = 5
2(2k − 1) − 1 = 5
4k − 2 − 1 = 5
4k = 8 ⇒ k = 2

18. a = bq + r, 0 ≤ r < b
2 = 3(0) + 2
5 = 3(1) + 2
Ans: 2, 5, 8, 11, ...

19. LCM of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Ans: 2520

20. 10⁴ ≡ x (mod 19)
(10)² = 100 ≡ 5 (mod 19)
(10)⁴ ≡ 25 ≡ 6 (mod 19)
10⁴ ≡ 6 (mod 19) ⇒ x = 6

21. t₂ − t₁ = t₃ − t₂
(18 − k) − (3 + k) = (5k + 1) − (18 − k)
15 − 2k = 6k − 17
8k = 32 ⇒ k = 4

22. S_n = (n / 2) [2a + (n − 1)d]
a = 1, d = 4
(n / 2) [2(1) + (n − 1)4] = 190
(n / 2) [4n − 2] = 190
2n² − n − 190 = 0
(n − 10)(2n + 19) = 0
n = 10, n = −¹⁹/₂ (not possible)

23. Area of ΔABC
= ¹/₂ | [−3, 1, −4, −3] over [−5, −1, 6, −5] |
= ¹/₂ [(3 + 6 + 20) − (−5 + 4 − 18)]
= ¹/₂ (48) = 24 sq.u

24. [(4x²y) / (2z²)] × [(6xz³) / (20y⁴)] = (3x³z) / (5y³)

25. x² + 8x − 65 = 0
Sum = −b / a = −8 / 1 = −8
Product = c / a = −65

26. Area ΔDEF = (16 × 54) / 9 = 96 cm²

27. (y − y₁) / (y₂ − y₁) = (x − x₁) / (x₂ − x₁)
(y − 3) / (−1 − 3) = (x − 2) / (−7 − 2)
(y − 3) / (−4) = (x − 2) / (−9)
4x − 9y + 19 = 0

28. 13824 = 2^a × 3^b
13824 = 2⁹ × 3³ ⇒ a = 9, b = 3

29. A = {1, 2, 3, 4, 5, 6, 7}
B = {2, 3, 5, 7}
C = {2}
A ∩ B = {2, 3, 5, 7}
(A ∩ B) × C = {(2, 2), (3, 2), (5, 2), (7, 2)}
A × C = {(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (7, 2)}
B × C = {(2, 2), (3, 2), (5, 2), (7, 2)}
(A × C) ∩ (B × C) = {(2, 2), (3, 2), (5, 2), (7, 2)}

31.
f(4) = 10
f(−2) = 2
f(4) + 2f(1) = 10 + 2(−1) = 8
[f(1) − 3f(4)] / f(−3) = [−1 − 3(10)] / 1 = −31

32.
f∁g(x) = f[g(x)] = f(x²) = x² − 4
(f∁g)∁h(x) = f∁g(3x − 5) = (3x − 5)² − 4 — (1)
g∁h(x) = g(h(x)) = g(3x − 5) = (3x − 5)²
f∁(g∁h)(x) = (3x − 5)² − 4 — (2)
(1) = (2) proved.

33.
113400 = 2³ × 3⁴ × 5² × 7¹
p₁ = 2, x₁ = 3
p₂ = 3, x₂ = 4
p₃ = 5, x₃ = 2
p₄ = 7, x₄ = 1

34.
(a − d) + a + (a + d) = 27 ⇒ 3a = 27 ⇒ a = 9
(a − d) × a × (a + d) = 288
(a² − d²) × a = 288
(9² − d²) × 9 = 288
81 − d² = 32 ⇒ d² = 49 ⇒ d = ±7
Terms: 2, 9, 16 or 16, 9, 2

35. S_n = 5 + 55 + 555 + ... n terms
= 5 [1 + 11 + 111 + ... n terms]
= ⁵/₉ [9 + 99 + 999 + ... n terms]
= ⁵/₉ [(10 + 100 + 1000 + ...) − (1 + 1 + 1 + ...)]
= ⁵/₉ [ {10(10ⁿ − 1) / (10 − 1)} − n ]
= ⁵⁰/₈₁ (10ⁿ − 1) − ⁵ⁿ/₉

36. Area = 10² + 11² + ... + 24²
= (1² + 2² + ... + 24²) − (1² + 2² + ... + 9²)
= [ {24(24 + 1)(2 × 24 + 1)} / 6 ] − [ {9(9 + 1)(2 × 9 + 1)} / 6 ]
= [ (24 × 25 × 49) / 6 ] − [ (9 × 10 × 19) / 6 ]
= 4900 − 285 = 4615 cm²

37. Area of quadrilateral ABCD
= ¹/₂ | [8, 5, −5, −4, 8] over [6, 11, 12, 3, 6] |
= ¹/₂ [(109) − (−49)] = ¹/₂ (158) = 79 sq. u

38.
3x − 2y + z = 2 — (1)
2x + 3y − z = 5 — (2)
(1) + (2) ⇒ 5x + y = 7 — (4)

2x + 3y − z = 5 — (2)
x + y + z = 6 — (3)
(2) + (3) ⇒ 3x + 4y = 11 — (5)

(4) × 4 ⇒ 20x + 4y = 28
− (5) ⇒ 3x + 4y = 11
17x = 17 ⇒ x = 1
sub in (4): 5(1) + y = 7 ⇒ y = 2
sub in (3): 1 + 2 + z = 6 ⇒ z = 3

39.
[1 / (x² − 5x + 6)] + [1 / (x² − 3x + 2)] − [1 / (x² − 8x + 15)]
= [1 / ((x − 2)(x − 3))] + [1 / ((x − 2)(x − 1))] − [1 / ((x − 5)(x − 3))]
= [ (x − 1)(x − 5) + (x − 3)(x − 5) − (x − 1)(x − 2) ] / [ (x − 1)(x − 2)(x − 3)(x − 5) ]
= (x² − 11x + 18) / [ (x − 1)(x − 2)(x − 3)(x − 5) ]
= [ (x − 9)(x − 2) ] / [ (x − 1)(x − 2)(x − 3)(x − 5) ]
= (x − 9) / [ (x − 1)(x − 3)(x − 5) ]

40.
Square root of 9x⁴ + 12x³ + 28x² + ax + b:
Quotient line: 3x² + 2x + 4
• 3x² | 9x⁴ → 9x⁴ (subtract)
• 6x² + 2x | 12x³ + 28x² → 12x³ + 4x² (subtract)
• 6x² + 4x + 4 | 24x² + ax + b → 24x² + 16x + 16 (subtract)
Remainder = 0
Since it is a perfect square: a = 16, b = 16

41. Refer Book

42.
Given point (x₁, y₁) = (6, −2)
Points: A(6, 7) and B(2, −3)
Equation of line joining A and B:
(y − y₁) / (y₂ − y₁) = (x − x₁) / (x₂ − x₁)
(y − 7) / (−3 − 7) = (x − 6) / (2 − 6)
(y − 7) / (−10) = (x − 6) / (−4)
(y − 7) / 5 = (x − 6) / 2
2y − 14 = 5x − 30
5x − 2y − 16 = 0

Equation of perpendicular line is of the form:
2x + 5y + k = 0
Passes through (6, −2):
2(6) + 5(−2) + k = 0
12 − 10 + k = 0
2 + k = 0 ⇒ k = −2
Required equation: 2x + 5y − 2 = 0