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

QUARTERLY EXAMINATION

1. If A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8} then state which of the following statement is true.
a) (A × C) ⊂ (B × D)      b) (B × D) ⊂ (A × C)
c) (A × B) ⊂ (A × D)      d) (D × A) ⊂ (B × A)

2. The Range of the relation R = {(x, x²) | x is a prime number less than 13} is
a) {2, 3, 5, 7}      b) {2, 3, 5, 7, 11}      c) {4, 9, 25, 49, 121}      d) {1, 4, 9, 25, 49, 121}

3. Let A = {1, 2, 3, 4} and B = {4, 8, 9, 10}. A function f: A → B given by f = {(1, 4), (2, 8), (3, 9), (4, 10)} is a
a) Many-one function      b) Identity function      c) One-to-one function      d) Into function

4. Using Euclid's division lemma, if the cube of any positive integer is divided by 9 then the possible remainders are
a) 0, 1, 8      b) 1, 4, 8      c) 0, 1, 3      d) 1, 3, 5

5. If t_n is the n^th term of an A.P then t_2n − t_n is
a) nd      b) 2nd      c) 2d      d) 3nd

6. The next term of the sequence ³/₁₆, ¹/₈, ¹/₁₂, ¹/₁₈, ... is
a) ¹/₂₄      b) ¹/₂₇      c) ²/₃      d) ¹/₈₁

7. The solution of the system x + y − 3z = −6, −7y + 7z = 7, 3z = 9 is
a) x = 1, y = 2, z = 3      b) x = −1, y = 2, z = 3
c) x = −1, y = −2, z = 3      d) x = 1, y = −2, z = 3

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

9. If ax² + bx + c = 0 has equal roots, then c is equal to
a) b² / (2a)      b) b² / (4a)      c) −b² / (2a)      d) −b² / (4a)

10. If in triangle ABC and EDF, AB / DE = BC / FD then they will be similar, when
a) ∠B = ∠E      b) ∠A = ∠D      c) ∠B = ∠D      d) ∠A = ∠F

11. The straight line given by the equation x = 11 is
a) Parallel to X axis      b) passing through the origin
c) parallel to Y axis      d) passing through the point (0, 11)

12. If the slope of the line PQ is 1 / √3 then slope of the perpendicular bisector of PQ is
a) √3      b) −√3      c) 1 / √3      d) 0

13. When proving that a quadrilateral is a trapezium, it is necessary to show
a) Two sides are parallel      b) Two parallel and two non-parallel sides
c) Opposite sides are parallel      d) All sides are of equal length

14. The value of sin²θ + 1 / tan²θ is equal to
a) tan²θ      b) 1      c) cot²θ      d) 0

15. If A = {m, n}, B = ∅ then find A × B and A × A.

16. Let X = {1, 2, 3, 4} and Y = {2, 4, 6, 8, 10} and R = {(1, 2), (2, 4), (3, 6), (4, 8)}. Show that R is a function and find its domain, co-domain and range?

17. If f(x) = x² − 1, g(x) = x − 2 find a, if g ∁ f(a) = 1.

18. Solve: 5x ≡ 4 (mod 6).

19. Find the number of terms in the A.P. 3, 6, 9, 12, ..., 111.

20. Find the sum of 2 + 4 + 6 + ... + 80.

21. Find the excluded values of the expression (7p + 2) / (8p² + 13p + 5).

22. Find the LCM of 9a³b², 12a²b²c.

23. 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.

24. Find the slope of a line joining the points (5, √5) with the origin.

25. Find the equation of a line joining the points (3, −4) and having slope −⁵/₇.

26. If the straight lines 12y = −(p + 3)x + 12 and 12x − 7y = 16 are perpendicular then find 'p'.

27. Prove: √[(1 + sin θ) / (1 − sin θ)] = sec θ + tan θ.

28. If P = x / (x + y) and Q = y / (x + y) then find 1 / (P² − Q²).

29. Given A = {1, 2, 3}, B = {2, 3, 5}, C = {3, 4} and D = {1, 3, 5}, check if (A ∩ C) × (B ∩ D) = (A × B) ∩ (C × D) is true?

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

33. 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?

34. Find the sum of all natural numbers between 300 and 600 which are divisible by 7.

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

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

37. If α, β are the roots of the equation 3x² + 7x − 2 = 0, find the values of:
(i) (α / β) + (β / α)      (ii) (α² / β) + (β² / α)

38. State and prove basic proportionality theorem.

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

40. Find the equation of the perpendicular bisector of the line joining the points A(−4, 2) and B(6, −4).

41. Prove: [(sin³A + cos³A) / (sin A + cos A)] + [(sin³A − cos³A) / (sin A − cos A)] = 2.

42. A line makes positive intercepts on coordinate axes whose sum is 7 and it passes through (−3, 8). Find its equation.

43. a) Construct a triangle similar to a given triangle LMN with its sides equal to ⁴/₅ of the corresponding sides of the triangle LMN (Scale factor ⁴/₅ < 1).
(OR) b) Construct a ΔPQR in which PQ = 8 cm, ∠R = 60°, and the median RG from R to PQ is 5.8 cm. Find the length of the altitude from R to PQ.

44. a) A bus is travelling at a uniform speed of 50 km/hr. Draw the time-distance 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, where x, y > 0. Using the graph, find:
    (i) y when x = 3 and
    (ii) x when y = 6.

1. a) (A × C) ⊂ (B × D)

2. c) {4, 9, 25, 49, 121}

3. c) One-to-one function

4. a) 0, 1, 8

5. a) nd

6. b) ¹/₂₇

7. a) x = 1, y = 2, z = 3

8. b) 16x²

9. b) b² / (4a)

10. c) ∠B = ∠D

11. c) parallel to Y axis

12. b) −√3

13. b) Two parallel and two non-parallel sides

14. b) 1

15. A × B = {m, n} × ∅ = ∅
A × A = {(m, m), (m, n), (n, m), (n, n)}

16. Domain = {1, 2, 3, 4}
Co-domain = {2, 4, 6, 8, 10}
Range = {2, 4, 6, 8}

17. f(x) = x² − 1, g(x) = x − 2
g ∁ f(a) = f(a) − 2 = a² − 1 − 2 = a² − 3
a² − 3 = 1 ⇒ a² = 4 ⇒ a = ±2

18. 5x ≡ 4 (mod 6)
5x − 4 = 6n ⇒ x = 2, 8, 14, ...

19. a = 3, d = 3, l = 111
n = [(l − a) / d] + 1 = [(111 − 3) / 3] + 1 = (108 / 3) + 1 = 36 + 1 = 37

20. Sum = 2 + 4 + 6 + ... + 80
= 2(1 + 2 + 3 + ... + 40) = 2 × [(40 × 41) / 2] = 1640

21. (7p + 2) / (8p² + 13p + 5) = (7p + 2) / [(p + 1)(8p + 5)]
(p + 1)(8p + 5) = 0 ⇒ p = −1 or p = −⁵/₈
Excluded values are −1, −⁵/₈

22. LCM of 9a³b², 12a²b²c = 36a³b²c

23. Area(ΔABC) / Area(ΔDEF) = BC² / EF²
54 / Area(ΔDEF) = 3² / 4² = 9 / 16
Area(ΔDEF) = (54 × 16) / 9 = 96 cm²

24. Slope m = (y₂ − y₁) / (x₂ − x₁) = (0 − √5) / (0 − 5) = √5 / 5 = 1 / √5

25. y − y₁ = m(x − x₁) ⇒ y + 4 = −⁵/₇ (x − 3)
7y + 28 = −5x + 15 ⇒ 5x + 7y + 13 = 0

26. m₁ = −(p + 3) / 12,   m₂ = −12 / (−7) = ¹²/₇
m₁ × m₂ = −1 ⇒ [−(p + 3) / 12] × (12 / 7) = −1
p + 3 = 7 ⇒ p = 4

27. LHS = √[ (1 + sin θ) / (1 − sin θ) × (1 + sin θ) / (1 + sin θ) ]
= √[ (1 + sin θ)² / (1 − sin²θ) ] = √[ (1 + sin θ)² / cos²θ ]
= (1 + sin θ) / cos θ = (1 / cos θ) + (sin θ / cos θ) = sec θ + tan θ = RHS. Hence proved.

28. P = x / (x + y),   Q = y / (x + y)
P² − Q² = (x² − y²) / (x + y)² = [(x + y)(x − y)] / (x + y)² = (x − y) / (x + y)
1 / (P² − Q²) = (x + y) / (x − y)

29. LHS: (A ∩ C) = {3},   (B ∩ D) = {3, 5}
(A ∩ C) × (B ∩ D) = {(3, 3), (3, 5)} — (1)
RHS: A × B = {(1, 2), (1, 3), (1, 5), (2, 2), (2, 3), (2, 5), (3, 2), (3, 3), (3, 5)}
C × D = {(3, 1), (3, 3), (3, 5), (4, 1), (4, 3), (4, 5)}
(A × B) ∩ (C × D) = {(3, 3), (3, 5)} — (2)
From (1) and (2), (A ∩ C) × (B ∩ D) = (A × B) ∩ (C × D) is True.

31.
(i) f(4) = 3(4) − 2 = 12 − 2 = 10
(ii) f(−2) = (−2)² − 2 = 4 − 2 = 2
(iii) f(4) + 2f(1) = 10 + 2(−1) = 10 − 2 = 8
(iv) [f(1) − 3f(4)] / f(−3) = [−1 − 30] / [2(−3) + 7] = −31 / 1 = −31

32. Sum = (a − d) + a + (a + d) = 27 ⇒ 3a = 27 ⇒ a = 9
Product = (a − d)a(a + d) = 288 ⇒ (a² − d²)a = 288
(81 − d²)9 = 288 ⇒ 81 − d² = 32 ⇒ d² = 49 ⇒ d = ±7
Three terms are: 2, 9, 16 or 16, 9, 2

33. Total Area = 10² + 11² + 12² + ... + 24²
= (1² + 2² + ... + 24²) − (1² + 2² + ... + 9²)
= [(24 × 25 × 49) / 6] − [(9 × 10 × 19) / 6] = 4900 − 285 = 4615 cm²

34. a = 301, d = 7, l = 595
n = [(l − a) / d] + 1 = [(595 − 301) / 7] + 1 = (294 / 7) + 1 = 42 + 1 = 43
S₄₃ = (43 / 2) [301 + 595] = (43 / 2) × 896 = 43 × 448 = 19264

35. Square root of 64x⁴ − 16x³ + 17x² − 2x + 1 by division method gives quotient 8x² − x + 1
√(64x⁴ − 16x³ + 17x² − 2x + 1) = |8x² − x + 1|

36. [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 − 6x + 5 + x² − 8x + 15 − x² + 3x − 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)]

37. 3x² + 7x − 2 = 0 ⇒ α + β = −⁷/₃,   αβ = −²/₃
(i) (α / β) + (β / α) = (α² + β²) / αβ = [(α + β)² − 2αβ] / αβ = [(−⁷/₃)² − 2(−²/₃)] / (−²/₃) = (⁴⁹/₉ + ⁴/₃) × (−³/₂) = (⁶¹/₉) × (−³/₂) = −⁶¹/₆
(ii) (α² / β) + (β² / α) = (α³ + β³) / αβ = [(α + β)³ − 3αβ(α + β)] / αβ = [(−⁷/₃)³ − 3(−²/₃)(−⁷/₃)] / (−²/₃) = ⁴⁶⁹/₁₈

38. State and Prove Basic Proportionality Theorem (Thales Theorem) — Refer Textbook.

39. Area of quadrilateral = ¹/₂ [(x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) − (x₂y₁ + x₃y₂ + x₄y₃ + x₁y₄)]
Points: (8, 6), (5, 11), (−5, 12), (−4, 3)
= ¹/₂ [(88 + 60 − 15 − 24) − (30 − 55 − 48 + 24)] = ¹/₂ [109 − (−49)] = ¹/₂ (158) = 79 sq. units

40. Midpoint of AB, D = [(−4 + 6) / 2, (2 − 4) / 2] = (1, −1)
Slope of AB = (−4 − 2) / [6 − (−4)] = −6 / 10 = −³/₅
Slope of perpendicular bisector m = ⁵/₃
Equation: y − y₁ = m(x − x₁) ⇒ y + 1 = ⁵/₃ (x − 1)
3y + 3 = 5x − 5 ⇒ 5x − 3y − 8 = 0

41. LHS = [(sin A + cos A)(sin²A − sin A cos A + cos²A)] / (sin A + cos A) + [(sin A − cos A)(sin²A + sin A cos A + cos²A)] / (sin A − cos A)
= (1 − sin A cos A) + (1 + sin A cos A) = 1 + 1 = 2 = RHS. Hence proved.

42. Sum of intercepts = 7 ⇒ a + b = 7 ⇒ b = 7 − a
Equation: (x / a) + [y / (7 − a)] = 1
Passes through (−3, 8): [−3 / a] + [8 / (7 − a)] = 1 ⇒ −3(7 − a) + 8a = a(7 − a)
−21 + 3a + 8a = 7a − a² ⇒ a² + 4a − 21 = 0 ⇒ (a + 7)(a − 3) = 0
Since intercepts are positive, a = 3 ⇒ b = 7 − 3 = 4
Equation: (x / 3) + (y / 4) = 1 ⇒ 4x + 3y − 12 = 0

43. a) Refer Textbook • b) Refer Textbook