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Telangana TET Paper 2 Math and Science Model Paper 1

Section: Math and Science

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Question : 92 of 150

Marks: +1, -0

If x = 5 +

2\sqrt{6}

\frac{(x-1)}{\sqrt{x}}

is equal to

$\sqrt{2}$
$2\sqrt{2}$
$\sqrt{3}$
$2\sqrt{3}$

Solution:

Concept:

Express

x

as a perfect square of a binomial:

( \sqrt{2} + \sqrt{3} )^2 = 2 + 3 + 2\sqrt{6} = 5 + 2\sqrt{6}

.

Explanation:

Given

x = 5 + 2\sqrt{6}

.

Write

x = ( \sqrt{2} )^2 + ( \sqrt{3} )^2 + 2 \cdot \sqrt{2} \cdot \sqrt{3} = ( \sqrt{2} + \sqrt{3} )^2

.

Thus

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

.

Now

(x-1) = 5 + 2\sqrt{6} - 1 = 4 + 2\sqrt{6}

.

Factor

2\sqrt{2}

:

4 + 2\sqrt{6} = 2\sqrt{2} ( \sqrt{2} + \sqrt{3} )

.

Hence

\frac{x-1}{\sqrt{x}} = \frac{2\sqrt{2} ( \sqrt{2} + \sqrt{3} )}{\sqrt{2} + \sqrt{3}} = 2\sqrt{2}

.

Answer:

B.

2\sqrt{2}

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