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Q.

Find the value of $a - \frac{1}{a},$ If $a = 2 + \sqrt{3}$

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a

4

b

5

c

3

d

1

answer is A.

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Detailed Solution

Given,

a = 2 + \sqrt{3}

Now we try to find the value of

\frac{1}{a}

and given in question

a = 2 + \sqrt{3},

So, we will put it to get the value of

\frac{1}{a}

Thus,

\frac{1}{a} = \frac{1}{2 + \sqrt{3}}

Now, multiply both the numerator and denominator with

2 - \sqrt{3},

= \frac{1}{2 + \sqrt{3}} \times \frac{2 - \sqrt{3}}{2 - \sqrt{3}}

= \frac{2 - \sqrt{3}}{(2 - \sqrt{3}) (2 + \sqrt{3})}

= \frac{2 - \sqrt{3}}{2^{2} - (\sqrt{3})^{2}}

= \frac{2 - \sqrt{3}}{4 - 3}

= 2 - \sqrt{3}

Thus,

\frac{1}{a} = 2 - \sqrt{3}

\frac{1}{a} = 2 - \sqrt{3}

Put it in

a - \frac{1}{a}

to get the needed value as,

a - \frac{1}{a}

= 2 + \sqrt{3} + 2 - \sqrt{3}

= 2 + 2

= 4

Thus,

a - \frac{1}{a} = 4

.
Correct option is 1.

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