LIWA

Real Numbers

CBSE Class 10 · Maths · Chapter 1

Past board exam questions

CBSE 2027 sample paper · 1 mark

If HCF (2520, 6600) = 40 and LCM (2520, 6600) = 252 × k, then the value of k is

  1. 165
  2. 1600
  3. 1625
  4. 1650
Show answer
1650

CBSE 2027 sample paper · 1 mark

The greatest number which divides both 134 and 188, leaving remainders 4 and 6 respectively, is:

  1. 13
  2. 26
  3. 39
  4. 65
Show answer
26

CBSE 2027 sample paper · 2 marks

A hall has a length of 9.75 m, breadth of 6.75 m and height of 5.25 m. What is the length of the longest unmarked ruler that can exactly measure the dimensions of the hall?

Show answer
Length = 9.75 m = 975 cm, breadth = 6.75 m = 675 cm, height = 5.25 m = 525 cm. Length of the longest ruler = HCF (975, 675, 525). 975 = 3 × 52 × 13, 675 = 33 × 52, 525 = 3 × 52 × 7. HCF = 3 × 52 = 75. The longest unmarked ruler is 75 cm (0.75 m) long.

CBSE 2027 sample paper · 3 marks

Given that √(5) is irrational, prove that 2 + 3√(5) is irrational.

Show answer
Let us assume, to the contrary, that 2 + 3√(5) is rational. Then 2 + 3√(5) = (p)/(q), where p and q are integers and q ≠ 0. ⇒ √(5) = (p - 2q)/(3q). Since p and q are integers, (p - 2q)/(3q) is a rational number, so √(5) would be rational. But √(5) is irrational. This contradiction shows our assumption is incorrect. Hence 2 + 3√(5) is irrational.

CBSE 2027 sample paper · 3 marks

Given that √(3) is irrational number, prove that (2 + 3√(3)) is an irrational number.

Show answer
Assume, to the contrary, that 2 + 3√(3) is rational. Then 2 + 3√(3) = (a)/(b), where a and b are integers and b ≠ 0. ⇒ √(3) = (a - 2b)/(3b). Since a and b are integers, (a - 2b)/(3b) is rational, so √(3) would be rational. This contradicts the given fact that √(3) is irrational. Hence our assumption is wrong, and 2 + 3√(3) is irrational.

CBSE 2027 sample paper · 2 marks

Show that (25)n will never end with digit zero for any natural number n.

Show answer
(25)n = (52)n = 52n, so the prime factorization of (25)n contains only the prime 5, and by the Fundamental Theorem of Arithmetic this factorization is unique. For (25)n to end with the digit 0, its prime factorization must contain both 2 and 5. But 2 is not a factor of (25)n. Hence (25)n can never end with the digit zero for any natural number n.

CBSE 2027 sample paper · 2 marks

There are three bells placed at different swings in a park, which toll at intervals of 5, 6 and 8 minutes, respectively. They all toll together when the park opens for the visitors at 10:00 a.m.. How many more times do they all toll together till the park is closed at 7:00 p.m.?

Show answer
LCM (5, 6, 8) = 120, so the bells toll together every 120 minutes = 2 hours. After 10:00 a.m. they toll together at 12:00 noon, 2:00 p.m., 4:00 p.m. and 6:00 p.m. (the next, 8:00 p.m., is after closing). They toll together 4 more times.

CBSE 2026 board paper · 1 mark

If HCF (850, 325) is 25, then LCM (850, 325) is :

  1. 442
  2. 11050
  3. 8450
  4. 2210
Show answer
11050