Question 12If a - 2 -3 - then find the value of a-1-a -

We have a =2 +√(3) Then,1/a=1/2+√(3)×2 √(3)/2 √(3) [Multiplying numerator and denominator by (2 √(3))] 2 √(3)/(2)^2 (√(3))^2=2 √(3)/4 3=2 √(3) [Us ...

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Standard IX

Mathematics

Operations on Irrational Numbers

Question 12If...

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Question 12
If $a = 2 + \sqrt{3}$ , then find the value of $(a - \frac{1}{a})$ .

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Solution

We have $a = 2 + \sqrt{3}$
Then, $\frac{1}{a} = \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ [Multiplying numerator and denominator by $(2 - \sqrt{3})]$
$\frac{2 - \sqrt{3}}{(2)^{2} - (\sqrt{3})^{2}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$ [Using identity $(a + b) (a - b) = a^{2} - b^{2}]$
$\Rightarrow \frac{1}{a} = 2 - \sqrt{3} ∴ a - \frac{1}{a} = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2 + \sqrt{3} - 2 + \sqrt{3} = 2 \sqrt{3}$

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Operations on Irrational Numbers

Standard IX Mathematics

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Question 12 If a=2+√3 , then find the value of (a−1a).

We have a=2+√3 Then,1a=12+√3×2−√32−√3 [Multiplying numerator and denominator by (2−√3)] 2−√3(2)2−(√3)2=2−√34−3=2−√3 [Using identity (a+b)(a−b)=a2−b2] ⇒1a=2−√3∴a−1a=(2+√3)−(2−√3)=2+√3−2+√3=2√3

Question 12
If $a = 2 + \sqrt{3}$ , then find the value of $(a - \frac{1}{a})$ .