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**Q.1. Classify the following numbers as rational or irrational:****(i) 2 - âˆš5**

**(iv) 1/âˆš2****(v) 2Ï€****Ans.** **(i)** 2 â€“âˆš5

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

âˆ´ 2 â€“âˆš5 is an irrational number.

We have

which is a rational number

is a rational number

which is a rational number.

is a rational number.

**(iv)** 1/âˆš2

âˆµThe quotient of rational and irrational is an irrational number.

âˆ´ 1/âˆš2 is an irrational number.

**(v)** 2Ï€

âˆµ 2Ï€ = 2 x Ï€ = Product of a rational and an irrational (which is an irrational number)

âˆ´ 2Ï€ is an irrational number.

REMEMBER

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

**Q.2. Simplify each of the following expressions:****(i) (3 + âˆš3) (2 + âˆš2)****(ii) (3 + âˆš3)(3 - âˆš3)****(iii) (âˆš5 + âˆš2) ^{2} **

= (2 x 3 + 2âˆš3) + (3âˆš2 +âˆš2 x âˆš3)

= 6 + 2âˆš3 + 3âˆš2 + âˆš6

Thus, (3 +âˆš3)(2 +âˆš2) = 6 + 2âˆš3 + 3âˆš2 +âˆš6

= 3

= 9 â€“ 3 = 6

âˆ´ (3 + âˆš3) (3 â€“ âˆš3) = 6.

âˆ´ **(iv)** (âˆš5 â€“ âˆš2)(âˆš5 + âˆš2) = (âˆš5)^{2} â€“ (âˆš2)^{2} [âˆµ (a + b)(a â€“ b) = a^{2} â€“ b^{2}]

= 5 â€“ 2 = 3

âˆ´ (âˆš5 â€“ âˆš2)(âˆš5 + âˆš2) = 3**Q.3. Recall, Ï€ is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, Ï€ = c/d****. This seems to contradict the fact that Ï€ 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.

âˆ´

**(ii)**

Thus,

**(iii)**

[âˆµ RE of (âˆšx + âˆšy) is (âˆšx - âˆšy)]

Thus,

**(iv) **

[âˆµ R.F. of (âˆš7 - 2) is (âˆš7 + 2)]

Thus,

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