10th Maths Unit 2 Important 5 Mark Questions

Secure your top scores in board exams by practicing the 10th Maths Unit 2 Important 5 Marks questions. The Numbers and Sequences chapter carries significant weight, frequently featuring complex long-answer questions on Arithmetic Progressions (AP), Geometric Progressions (GP), and special series. This curated list focuses on the most repeated and high-yielding 5-mark problems, complete with clear breakdowns to help you master the core formulas and logical steps needed to achieve a perfect score.

UNIT - 2 (NUMBERS AND SEQUENCES)

IMPORTANT FIVE MARKS

1. Find the HCF of 396, 504, 636.

2. When the positive integers a, b and c are divided by 13, the respective remainders are 9, 7 and 10. Show that a + b + c is divisible by 13.

3. If d is the highest common factor of 32 and 60, find x and y satisfying d = 32x + 60y.

4. 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.

5. Find the greatest number consisting of 6 digits which is exactly divisible by 24, 15, 36?

6. Find the first five terms of the following sequence: a₁ = 1, a₂ = 1, a_n = ^a_n-1/_{(a_n-2 + 3)}, n ≥ 3, n ∈ ℕ.

7. If a₁ = 1, a₂ = 1 and a_n = 2a_n-1 + a_n-2, n ≥ 3, then find the first six terms of the sequence.

8. If l^th, m^th and n^th terms of an A.P. are x, y, z respectively, then show that
(i) x(m - n) + y(n - l) + z(l - m) = 0
(ii) (x - y)n + (y - z)l + (z - x)m = 0

9. In an A.P., sum of three consecutive terms is 27 and their product is 288. Find the numbers.

10. In an A.P., sum of four consecutive terms is 28 and the sum of their squares is 276. Find the four numbers.

11. A mother divides ₹207 into three parts such that the amounts are in A.P. and gives it to her three children. The product of the two least amounts that the children had is ₹4623. Find the amount received by each child.

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

13. The ratio of the 6th and 8th term of an A.P. is 7:9. Find the ratio of 9th term to 13th term.

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

15. The 13th term of an A.P. is 3 and the sum of first 13 terms is 234. Find the common difference and the sum of the first 21 terms.

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

17. The sum of n, 2n, 3n terms of an A.P. are S₁, S₂, and S₃ respectively. Prove that S₃ = 3(S₂ - S₁).

18. The 104^th term and 4th term of an A.P. are 125 and 0. Find the sum of first 35 terms.

19. If S₁, S₂, S₃, ... S_m are the sums of n terms of m A.P.'s whose first terms are 1, 2, 3, ... m and whose common differences are 1, 3, 5, ... (2m - 1) respectively, then show that S₁ + S₂ + S₃ + ... + S_m = ¹/₂ mn(mn + 1).

20. If a, b, c are three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. then prove that x^{b - c} × y^{c - a} × z^{a - b} = 1.

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

22. Find the sum to n terms of the series 3 + 33 + 333 + ...

23. Find the sum to n terms of the series 0.4 + 0.44 + 0.444 + ... to n terms.

24. S_n = (x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ... n terms, then prove that (x - y)S_n = [^{x²(xⁿ - 1)}/_{(x - 1)} - ^{y²(yⁿ - 1)}/_{(y - 1)}].