Test Index

UPSC CDS II 2025 Mathematics Paper

© examsnet.com

Question : 6 of 100

Marks: +1, -0

If

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

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

equal to?

0
0.5
1
1.5

Solution:

Concept:

The numbers

(2+\sqrt{3})

and

(2-\sqrt{3})

are reciprocals because their product is

1

. This property simplifies the given equation.

Explanation:

Let

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

. Since

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

, we have

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

.

The given equation becomes

y + \frac{1}{y} = 2

.

Multiply both sides by

y

:

y^{2} + 1 = 2y

.

Bring all terms to one side:

y^{2} - 2y + 1 = 0

.

Factor:

(y-1)^{2} = 0

, so

y = 1

.

Thus

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

, which implies

x = 0

.

Now the required expression is

(2+\sqrt{3})^{x} - (2-\sqrt{3})^{x} = 1 - 1 = 0

.

Answer:

0

(Option A)

© examsnet.com

Go to Question:

Prev Question

Next Question