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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 nonterminating 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 nonterminating and repeating decimal.
Example:
Q11. What is the difference between "pure recurring decimals" and "mixedrecurring 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 "nonterminating and nonrecurring."
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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