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RRB NTPC Mains or Stage 2 Model Paper 4 with solutions

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Question : 1 of 120

Marks: +1, -0

What is the value of

2 + \sqrt{2} + \frac{1}{2 + \sqrt{2}} - \frac{1}{2 - \sqrt{2}} ?

2
$2 - \sqrt{2}$
$4 + \sqrt{2}$
$2 \sqrt{2}$

Solution:

Concept:

Simplify reciprocal surd terms by rationalizing denominators.

Explanation:

Let

E = 2 + \sqrt{2} + \frac{1}{2+\sqrt{2}} - \frac{1}{2-\sqrt{2}}

.

Rationalize:

\frac{1}{2+\sqrt{2}} = \frac{2-\sqrt{2}}{(2+\sqrt{2})(2-\sqrt{2})} = \frac{2-\sqrt{2}}{4-2} = \frac{2-\sqrt{2}}{2}

.

Also,

\frac{1}{2-\sqrt{2}} = \frac{2+\sqrt{2}}{(2-\sqrt{2})(2+\sqrt{2})} = \frac{2+\sqrt{2}}{4-2} = \frac{2+\sqrt{2}}{2}

.

Subtract:

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

.

Substitute back:

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

.

Answer:

The value is

2

, so option A is correct.

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