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UPSC CDS 1 2020 Math Paper

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Question : 50 of 100

Marks: +1, -0

If

\left( {{x^8} + \frac{1}{{{x^8}}}} \right) = 47

, what is the value of

\left( {{x^6} + \frac{1}{{{x^6}}}} \right)?

36
27
18
9

Solution:

Concept:

Use algebraic identities relating

x^n + \frac{1}{x^n}

to reduce the given power step by step.

Explanation:

Given:

x^8 + \frac{1}{x^8} = 47

.

Apply identity:

(x^4 + \frac{1}{x^4})^2 = x^8 + 2 + \frac{1}{x^8} = 47 + 2 = 49

.

So

x^4 + \frac{1}{x^4} = 7

(positive value assumed).

Next:

(x^2 + \frac{1}{x^2})^2 = x^4 + 2 + \frac{1}{x^4} = 7 + 2 = 9

.

Thus

x^2 + \frac{1}{x^2} = 3

.

Now use the cube identity:

(a + \frac{1}{a})^3 = a^3 + 3a + \frac{3}{a} + \frac{1}{a^3}

.

Rearranging:

a^3 + \frac{1}{a^3} = (a + \frac{1}{a})^3 - 3(a + \frac{1}{a})

.

Put

a = x^2

: then

x^6 + \frac{1}{x^6} = (x^2 + \frac{1}{x^2})^3 - 3(x^2 + \frac{1}{x^2})

.

Substitute

x^2 + \frac{1}{x^2} = 3

:

x^6 + \frac{1}{x^6} = 3^3 - 3 \times 3 = 27 - 9 = 18

.

Answer:

18 (option C).

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