10th Maths Important 5 Mark Questions Chapter-Wise List

Maximize your score in Section-C with this definitive compilation of 10th Maths Important 5 Mark Questions. Long-answer problems carry the highest weightage in the mathematics blueprint, making them the ultimate deciding factor for securing a perfect centum. This expert-curated guide brings you the most frequently asked textbook examples, exercise problems, and compulsory-type questions organized chapter-by-chapter. By breaking down complex equations in Algebra, theorems in Geometry, proofs in Trigonometry, and formulas in Mensuration, this resource ensures you master the exact step-by-step working required to score full marks.

10th Maths Public Exam 2027 Expected 5 Mark Questions

Important 5-Marks Questions

1. Let A = {x ∈ ℕ | 1 < x < 4}, B = {x ∈ W | 0 ≤ x < 2} and C = {x ∈ ℕ | x < 3}. Then verify that
(i) A × (B ∪ C) = (A × B) ∪ (A × C)
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)

2. Let A = {x ∈ W | x < 2}, B = {x ∈ ℕ | 1 < x ≤ 4} and C = {3, 5}. Verify that
(i) A × (B ∪ C) = (A × B) ∪ (A × C)
(ii) A × (B ∩ C) = (A × B) ∩ (A × C)
(iii) (A ∪ B) × C = (A × C) ∪ (B × C)

3. 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 numbers. Verify that
(i) (A ∩ B) × C = (A × C) ∩ (B × C)
(ii) A × (B - C) = (A × B) - (A × C)

4. Let A = {1, 2, 3, 4} and B = {2, 5, 8, 11, 14} be two sets. Let f : A → B be a function given by f(x) = 3x - 1. Represent this function
(i) by arrow diagram
(ii) in a table form
(iii) as a set of ordered pairs
(iv) graphical form

5. 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

6. Show that (f ∘ g) ∘ h = f ∘ (g ∘ h) in each case:
(i) f(x) = x - 1, g(x) = 3x + 1, h(x) = x²
ii) f(x) = x - 4, g(x) = x², h(x) = 3x - 5
iii) f(x) = 2x + 3, g(x) = 1 - 2x and h(x) = 3x
iv) f(x) = x², g(x) = 3x and h(x) = x - 2

7. Find the sum of series:
i) 3 + 33 + 333 + ...
ii) 5 + 55 + 555 + ...

8. Find the sum of:
i) 15² + 16² + 17² + ... + 28²
ii) 6² + 7² + 8² + ... + 21²
iii) 9³ + 10³ + 11³ + ... + 21³
iv) 10³ + 11³ + 12³ + ... + 20³

9. 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?

10. Show that in a triangle, the medians are concurrent.

11. Find the area of the quadrilateral whose vertices are at:
i) (-9, -2), (-8, -4), (2, 2) and (1, -3)
ii) (-9, 0), (-8, 6), (-1, -2) and (-6, -3)
iii) (8, 6), (5, 11), (-5, 12), (-4, 3)

12. (i) Find the value of k, if the area of a quadrilateral is 28 sq.units, whose vertices are taken in the order (-4, -2), (-3, k), (3, -2) and (2, 3).
(ii) If the area of the triangle formed by the vertices A(-1, 2), B(k, -2) and C(7, 4) area is 22 sq. units, find the value of k.

13. Let A(3, -4), B(9, -4), C(5, 7) and D(7, 7). Show that ABCD is a trapezium.

14. A(-3, 0), B(10, -2) and C(12, 3) are the vertices of ΔABC. Find the equation of the altitude through A and B.

15. Two ships are sailing in the sea on either sides of a lighthouse. The angle of elevation of the top of the lighthouse as observed from the ships are 30° and 45° respectively. If the lighthouse is 200 m high, find the distance between the two ships. (√3 = 1.732)

16. A metallic sphere of radius 16 cm is melted and recast into small spheres each of radius 2 cm. How many small spheres can be obtained?

17. A cone of height 24 cm is made up of modeling clay. A child reshapes it in the form of a cylinder of the same radius as the cone. Find the height of the cylinder.

18. 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.

19. Find the variance and standard deviation of the wages of 9 workers given below:
₹310, ₹290, ₹320, ₹280, ₹300, ₹290, ₹320, ₹310, ₹280.

20. Find the coefficient of variation of 24, 26, 33, 37, 29, 31.

21. Two unbiased dice are rolled once. Find the probability of getting:
(i) a doublet (equal numbers on both dice)
(ii) the product as a prime number
(iii) the sum as a prime number
(iv) the sum as 1

22. Two dice are rolled. Find the probability that the sum of outcomes is:
(i) equal to 4
(ii) greater than 10
(iii) less than 13

23. Three unbiased coins are tossed once. Find the probability of getting atmost 2 tails or atleast 2 heads.

24. If two dice are rolled, then find the probability of getting the product of face values 6 or the difference of face values 5.

25. In a class of 50 students, 28 opted for NCC, 30 opted for NSS and 18 opted both NCC and NSS. One of the students is selected at random. Find the probability that
(i) The student opted for NCC but not NSS.
(ii) The student opted for NSS but not NCC.

26. Three fair coins are tossed together. Find the probability of getting:
(i) all heads
(ii) at least one tail
(iii) atmost one head
(iv) atmost two tails

27. A coin is tossed thrice. Find the probability of getting exactly two heads or at least one tail or two consecutive heads.

28. Find the square root of:
(i) x⁴ - 12x³ + 42x² - 36x + 9
(ii) 37x² - 28x³ + 4x⁴ + 42x + 9
(iii) 16x⁴ + 8x² + 1
(iv) 121x⁴ - 198x³ - 183x² + 216x + 144
(v) 64x⁴ - 16x³ + 17x² - 2x + 1

29. Find the values of a and b if the following polynomials are perfect squares:
i) 4x⁴ - 12x³ + 37x² + bx + a
ii) ax⁴ + bx³ + 361x² + 220x + 100
iii) 9x⁴ + 12x³ + 28x² + ax + b
iv) 36x⁴ - 60x³ + 61x² - mx + n
v) x⁴ - 8x³ + mx² + nx + 16