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**Q.1. Find a rational number between 1 and 2.****Solution.**

Let x = 1 and y = 2, then

Thus, 3/2 is a rational number between 1 and 2.**Q.2. Write a rational number equivalent to 5/9 such that its numerator is 25.****Solution.**

Thus, 25/45 is the required rational number whose numerator is 25.**Q.3. Find two rational numbers between 0.1 and 0.3.****Solution.**** **Let x = 0.1, y = 0.3 and n = 2

âˆ´ Two rational numbers between 0.1 and 0.3 are: x + d and x + 2d

**Q.4. Express **** in the form of decimal.**

**Solution.**

We have ,

Now, dividing 25 by 8, we have:

Since, the remainder is 0.

âˆ´ The process of division terminates.

**Q.5. Express as a rational number.Solution.**

Multiplying by 100, we have 100x = 100 x 0.3333â€¦

â‡’ 100x = 33.3333 â€¦ ...(2)

Subtracting (2) from (1), we have

100x â€“ x = 33.3333â€¦ â€“ 0.3333â€¦

â‡’ 99x = 33**Q.6. Simplify: (4+ **âˆš3**) (4 âˆ’**âˆš3**)****Solution. **(4 +âˆš3) (4 âˆ’âˆš3) = (4)^{2} â€“ ( âˆš3)^{2} [âˆµ (a + b)(a â€“ b) = a^{2} â€“ b^{2}]

= 16 â€“ 3 = 13

Thus, (4 +âˆš3) (4 âˆ’âˆš3) = 13**Q.7. Simplify: (**âˆš3** +**âˆš2)^{2}**Solution.**

(âˆš3 +âˆš2)^{2} = (âˆš3)^{2} + âˆš2 ( âˆš3 Ã—2) + (âˆš2)^{2 } [âˆµ (a + b)^{2} = a^{2 }+ 2ab + b^{2}]

= 3 + 2 âˆš6 + 2 = 5 + 2 âˆš6

Thus, (âˆš3 +**(**âˆš2)^{2} = 5 + 2âˆš6**Q.8. Rationalise the denominator of ****Solution.**

Since RF of (âˆšx - âˆšy ) is (âˆšx +âˆšy )

âˆ´ RF of (âˆš3 - âˆš2 ) is (âˆš3 +âˆš2 )

Now, we have

Thus, **Q.9. Find ****Solution.**

Since

64 = 4 x 4 x 4 = 4^{3}

**Q.10. Define a non-terminating decimal and repeating decimals ?****Solution.** The decimal expansion of some rational numbers do not have finite number of decimal places in their decimal parts, rather they have a repeating block of digits in decimal parts. Such decimal expansion is called non-terminating and repeating decimal.**Example: ****Q.11. What is the difference between "pure recurring decimals" and "mixed-recurring decimals."?****Solution. **A decimal in which all the digits after the decimal point are repeated, is called a pure recurring decimal. A decimal in which at least one of the digits after the decimal point is not repeated and then some digit(s) repeated is called a mixed recurring decimal.**Example:** are pure recurring decimals.

are mixed recurring decimals.**Q.12. What type of decimal expansion does an irrational number have?Solution.** The decimal expansion of an irrational number is "non-terminating non-recurring."

Obviously x < y

A rational number lying between x and y

Hence, 7/20 is a rational number lying between

âˆ´

âˆ´

âˆ´ Three required numbers between 0 and 0.1 are: (x + d), (x + 2d) and (x + 3d)

Now

Thus, three rational numbers between 0 and 0.1 are: 0.025, 0.050 and 0.075.**Q.14. Express **** as a fraction in the simplest form.****Solution.** Let x = = 0.24545 ... ...(1)

Then, multiplying (1) by 10,

we have 10x = 10 x 0.24545...

â‡’ 10x = 2.4545 ... ...(2)

Again multiplying (1) by 1000,

we get 1000 x x = 0.24545... x 1000

â‡’ 1000x = 245.4545 ... ...(3)

Subtracting (2) from (3),

we have 1000x â€“ 10x = 245.4545... â€“ 2.4545...

Thus **Q.15. If x = (2 +**âˆš5**) , find the value of** **Solution. **We have x = 2 + âˆš5

âˆ´

âˆ´

Now

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