CG Board Class 9 Mathematics Chapter 2 Solutions: Real Numbers (वास्तविक संख्याएँ) – 2026-27

(i)** Any positive integer like 1 belongs to N, W, and Z.

• (ii) 0 is a whole number (W) but not a natural number (N).
• (iii) 0 (or -2, 1/2) is a rational number (Q) but not a natural number (N).
• (i) 126/5:
• (ii) 335/16:
• (iii) 22/7:
• (iv) -118/3:
• (i) 0.53: 53 / 100.
• (ii) 16.8: 168 / 10 = 84 / 5.
• (iii) 105.25: 10525 / 100 = 421 / 4.
• (iv) 7.36: 736 / 100 = 184 / 25.
• (i) 0.70 bar (0.707070…..):

Let x = 0.7070..... -> 100x = 70.7070.....

99x = 70 -> x = 70/99.

• (ii) 8.39 bar (8.393939…..):

Let x = 8.3939..... -> 100x = 839.3939.....

99x = 831 -> x = 831/99 = 277/33.

• (iii) 5.63180 bar:

Expressing repeating block yields fraction = 563180 / 99900 = 56318 / 9990.

• (iv) 5.125 bar (5.125125125…..):

Let x = 5.125125..... -> 1000x = 5125.125125.....

999x = 5120 -> x = 5120/999.

(i)* 3sqrt(3) + 15sqrt(3) = (3 + 15)sqrt(3) = 18*sqrt(3).

• (ii) 5sqrt(7) - 2sqrt(7) = (5 - 2)sqrt(7) = 3sqrt(7).
• (i) Using (a + b)(a - b) = a2 - b2

(sqrt(3))2 - (sqrt(2))2 = 3 - 2 = 1.

• (ii) Using (a + b)(a - b) = a2 - b2

(5)2 - (sqrt(5))2 = 25 - 5 = 20.

• (iii) Expanding terms

sqrt(2)sqrt(3) + sqrt(2)sqrt(8) + sqrt(3)sqrt(3) + sqrt(3)sqrt(8)

= sqrt(6) + sqrt(16) + 3 + sqrt(24)

= sqrt(6) + 4 + 3 + 2sqrt(6) = 7 + 3sqrt(6).

• (iv) Using (a - b)2 = a2 + b2 - 2ab

(sqrt(7))2 + (sqrt(3))2 - 2sqrt(7)sqrt(3) = 7 + 3 - 2sqrt(21) = 10 - 2sqrt(21).

• (i) Multiply numerator and denominator by sqrt(5)

[1 sqrt(5)] / [sqrt(5) sqrt(5)] = sqrt(5) / 5.

• (ii) Multiply numerator and denominator by sqrt(6)

[2 sqrt(6)] / [sqrt(6) sqrt(6)] = 2*sqrt(6) / 6 = sqrt(6) / 3.

• (iii) Simplify radical fraction

sqrt(21 / 3) = sqrt(7) = sqrt(7) / 1.

• (i) Rationalising LHS

[(6 + sqrt(3))(6 + sqrt(3))] / [(6 - sqrt(3))(6 + sqrt(3))]

= [(6)2 + (sqrt(3))2 + 2(6)(sqrt(3))] / [(6)2 - (sqrt(3))2]

= (36 + 3 + 12*sqrt(3)) / (36 - 3)

= (39 + 12sqrt(3)) / 33 = 39/33 + (12/33)sqrt(3) = 13/11 + (4/11)*sqrt(3).

Comparing with a + b*sqrt(3) gives

a = 13/11, b = 4/11.

• (ii) Rationalising LHS

[(sqrt(5) + sqrt(3))(sqrt(5) + sqrt(3))] / [(sqrt(5) - sqrt(3))(sqrt(5) + sqrt(3))]

= [(sqrt(5))2 + (sqrt(3))2 + 2sqrt(5)sqrt(3)] / [(sqrt(5))2 - (sqrt(3))2]

= (5 + 3 + 2sqrt(15)) / (5 - 3) = (8 + 2sqrt(15)) / 2 = 4 + 1*sqrt(15).

Comparing with a + b*sqrt(15) gives

a = 4, b = 1.

• (i) Rationalise each term
• First term: [(5 + sqrt(3))(2 + sqrt(3))] / (4 - 3) = 10 + 5sqrt(3) + 2sqrt(3) + 3 = 13 + 7*sqrt(3)
• Second term: [(2 - sqrt(3))(2 - sqrt(3))] / (4 - 3) = 4 + 3 - 4sqrt(3) = 7 - 4sqrt(3)
• Sum = (13 + 7sqrt(3)) + (7 - 4sqrt(3)) = 20 + 3*sqrt(3).
• (ii) Rationalise each term
• First term: [(2 + sqrt(3))2] / (4 - 3) = 4 + 3 + 4sqrt(3) = 7 + 4sqrt(3)
• Second term: [(2 - sqrt(3))2] / (4 - 3) = 4 + 3 - 4sqrt(3) = 7 - 4sqrt(3)
• Third term: [(sqrt(3) - 1)2] / (3 - 1) = (3 + 1 - 2sqrt(3)) / 2 = (4 - 2sqrt(3)) / 2 = 2 - sqrt(3)
• Total Sum = (7 + 4sqrt(3)) + (7 - 4sqrt(3)) + (2 - sqrt(3)) = 16 - sqrt(3) (or 16 - 3*sqrt(8) for print representation variant).