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** Ques 1: Classify the following numbers as rational or irrational:**

**Ans:** (i) 2 –

Since it is a difference of a rational and irrational number,

∴ 2 – is an irrational number.

We have which is a rational number

is a rational number

(iv)

∵The quotient of rational and irrational is an irrational number.

∴ is an irrational number.

( v )2π

∵ 2π = 2 x π = Product of a rational and an irrational (which is an irrational number)

∴ 2π is an irrational number.

REMEMBER

For real numbers a, b, c and d, we have:

**Ques 2: Simplify each of the following expressions:**

**Ans:** (i) (3+ √3)(2+√2) - 2(3 +√3 ) + √2 (3 +√3)

= (2 x 3 + 2√3 ) + (3√2 +√2 x √3 )

= 6 + 2√3 + 3√2 + √6

Thus, (3 +√3 )(2 +√2 ) = 6 + 2√3 + 3√2 +√6

(ii) (3 +√3 )(3 – √3 ) = (3)^{2} – ( √3)^{2 } [∵ (a + b)(a – b) = a^{2} – b^{2}]

= 3^{2} – 3

= 9 – 3 = 6

∴ (3 + √3 ) (3 – √3 ) = 6.

(iii) ( √5 +√2 )^{2}= (√5 )^{2} + (√2)^{2} + 2( √5)(√2) [∵ (a + b)^{2} = a^{2} + b^{2} + 2ab]

(iv) ( √5 – √2 )( √5 + √2) = (√5 )^{2} – (√2 )^{2} [∵ (a + b)(a – b) = a^{2} – b^{2}]

= 5 – 2 = 3

∴ ( √5 – √2 )( √5 + √2) = 3

**Ques 3: Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π = **** . This seems to contradict the fact that π is irrational. How will you resolve this contradiction?Ans: **When we measure the length of a line with a scale or with any other device, we only get an approximate rational value, i.e. c and d both are irrational.

∴

**Ques 4: Rationalise the denominators of the following:**

**Ans:**

(i)

(ii)

(iv) ∴

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