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UPTET Paper 2 Math and Science 2019 Solved Paper

Question : 97 of 120

Marks: +1, -0

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

x^{3}-2x^{2}-7x+5

Solution:

Concept:

Rationalize the denominator to simplify

x

, then use the resulting quadratic relation to reduce the given expression.

Explanation:

First, rationalize

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

Multiply numerator and denominator by

2+\sqrt{3}

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

x-2 = \sqrt{3}

. Square both sides:

(x-2)^2 = 3 \Rightarrow x^2 - 4x + 4 = 3 \Rightarrow x^2 - 4x + 1 = 0

Hence

x^2 = 4x - 1

and

x^2 - 4x = -1

Now multiply

x^2 = 4x - 1

x

x^3 = 4x^2 - x

Substitute into the given expression:

x^3 - 2x^2 - 7x + 5 = (4x^2 - x) - 2x^2 - 7x + 5 = 2x^2 - 8x + 5

Factor:

2(x^2 - 4x) + 5

Use

x^2 - 4x = -1

2(-1) + 5 = -2 + 5 = 3

Answer:

3 (Option D)

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