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UPSC CDS 1 2022 Math Paper

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Question : 9 of 100

Marks: +1, -0

If

x = 9999

, then what is the value of the following?

\frac{4x^{3} - x}{(2x + 1)(6x - 3)}

1111
2222
3333
6666

Solution:

Concept:

Factorize and simplify the rational expression, then substitute the given value of

x

.

Explanation:

Start with the expression

\frac{4x^3 - x}{(2x + 1)(6x - 3)}

.

Factor

x

from the numerator:

\frac{x(4x^2 - 1)}{(2x + 1)(6x - 3)}

.

Recognize

4x^2 - 1

as

(2x)^2 - 1^2

:

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

.

Use the identity

a^2 - b^2 = (a-b)(a+b)

:

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

.

Factor

3

from

6x - 3

:

\frac{x(2x - 1)(2x + 1)}{(2x + 1)\cdot 3(2x - 1)}

.

Cancel the common factors

(2x + 1)

and

(2x - 1)

:

\frac{x}{3}

.

Substitute

x = 9999

:

\frac{9999}{3} = 3333

.

Answer:

3333

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