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**Q1. Find a rational number between 1 and 2.**

Let x = 1 and y = 2, then

Thus, 3/2 is a rational number between 1 and 2.

**Q2. Write a rational number equivalent to 5/9 such that its numerator is 25.**

∵ 25/5 = 5

Thus, 25/45 is the required rational number whose numerator is 25.

**Q3. Find two rational numbers between 0.1 and 0.3.**

Let x = 0.1, y = 0.3 and n = 2

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

**Q4. Express **** in the form of a decimal.**

We have,

Now, dividing 25 by 8,

Since, the remainder is 0.

∴ The process of division terminates.

**Q5. Express as a rational number.**

Multiplying (1) by 100, we have 100x = 100 x 0.3333…

⇒ 100x = 33.3333 …(2)

Subtracting (2) from (1), we have

100x – x = 33.3333… – 0.3333…

⇒ 99x = 33

**Q6. Simplify: (4+ ****√3****) (4 −****√****3****)**

∵ (a + b)(a – b) = a

^{2}– b^{2}

(4 +√3) (4 −√3) = (4)^{2}– ( √3)^{2}= 16 – 3 = 13

Thus, (4 +√3) (4 −√3) = 13

**Q7. Simplify: (**√3** +**√2)^{2}

∵ (a + b)

^{2}= a^{2 }+ 2ab + b^{2}

(√3 +√2)^{2}= (√3)^{2}+ √2 ( √3 ×2) + (√2)^{2}= 3 + 2 √6 + 2 = 5 + 2 √6

Thus, (√3 +(√2)^{2}= 5 + 2√6

**Q8. Rationalise the denominator of **

Since RF of (√x - √y ) is (√x +√y )

∴ RF of (√3 - √2 ) is (√3 +√2 )

Now, we have

Thus,

**Q9. Find **

As, 64 = 4 x 4 x 4 = 4

^{3}

**Q10. Define a non-terminating decimal and repeating decimals.**

The decimal expansion of some rational numbers do not have a 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:

**Q11. What is the difference between "pure recurring decimals" and "mixed-recurring decimals"?**

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.

**Q12. What type of decimal expansion does an irrational number have?**

The decimal expansion of an irrational number is "non-terminating and non-recurring."

**Q13. Find a rational number lying between **

Obviously x < y

A rational number lying between x and yHence, 7/20 is a rational number lying between

∴

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

NowThus, three rational numbers between 0 and 0.1 are: 0.025, 0.050 and 0.075.

**Q14. Express **** as a fraction in the simplest form.**

Let X = = 0.24545 ... (1)

Then, multiplying (1) by 10,

We have 10X = 10 x 0.24545...

⇒ 10x = 2.4545 ... (2)

Again multiplying (1) by 1000,

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

⇒ 1000X = 245.4545 ... (3)

Subtracting (2) from (3),

1000X – 10X = 245.4545... – 2.4545...Thus

**Q15. If x = (2 +**√5**) , find the value of**

We have x = 2 + √5

∴

∴

Now

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