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UPTET Paper 1 Model Test 5

Section: Mathematics

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Question : 110 of 150

Marks: +1, -0

If x =

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

, then the value of (x - 7)^{2} is

48
$2\sqrt{3}$
$4\sqrt{3}$
24

Solution:

Concept:

Rationalize the denominator to simplify

x

. Then substitute into the expression.

Explanation:

Given

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

.

Multiply numerator and denominator by

2+\sqrt{3}

:

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

.

Denominator:

2^2 - (\sqrt{3})^2 = 4 - 3 = 1

.

So

x = (2+\sqrt{3})^2 = 4 + 3 + 4\sqrt{3} = 7 + 4\sqrt{3}

.

Now compute

(x - 7)^2 = (7 + 4\sqrt{3} - 7)^2 = (4\sqrt{3})^2 = 16 \times 3 = 48

.

Answer:

Option A: 48

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