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KCET 2026 Mathematics Solved Paper

Question : 8 of 60

Marks: +1, -0

The value at

x=2

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

Solution:

Concept:

The expression simplifies using binomial expansions of

(x+1)^3

and

(x+1)^4

Explanation:

Identify the numerator as a perfect cube.

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

Identify the denominator as a perfect fourth power.

x^{4}+4x^{3}+6x^{2}+4x+1 = (x+1)^4

So the given expression becomes:

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

Cancel the common factor

(x+1)^3

\frac{(x+1)^3}{(x+1)^4} = \frac{1}{x+1}

Now substitute

x=2

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

Answer:

\frac{1}{3}

Hence, the correct option is C.

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