JEE 2022 — Mathematics PYQ

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Question

JEE 2022 — Mathematics PYQ

JEE | Mathematics | 2022

Let $f(x)$ and $g(x)$ be two real polynomials of degree 2 and 1 respectively. If $f(g(x)) = 8x^2 - 2x$ , and $g(f(x)) = 4x^2 + 6x + 1$ , then the value of $f(2) + g(2)$ is ________.

Choose the correct answer:

Explanation

Solution

Let $g(x) = ax + b$ .

From $g(f(x)) = a f(x) + b = 4x^2 + 6x + 1$ , we have:

f(x) = \frac{4x^2 + 6x + 1 - b}{a}

Substitute $f(x)$ into $f(g(x)) = 8x^2 - 2x$ :

\frac{4(ax+b)^2 + 6(ax+b) + 1 - b}{a} = 8x^2 - 2x

Comparing the coefficient of $x^2$ :

\frac{4a^2}{a} = 8 \implies 4a = 8 \implies a = 2

Now compare the coefficient of $x$ :

\frac{8ab + 6a}{a} = -2 \implies 8b + 6 = -2 \implies 8b = -8 \implies b = -1

So, $g(x) = 2x - 1$ .

Using $a=2, b=-1$ in the $f(x)$ formula:

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

Final Calculation:

$f(2) = 2(2^2) + 3(2) + 1 = 8 + 6 + 1 = 15$
$g(2) = 2(2) - 1 = 3$
$f(2) + g(2) = 15 + 3 = 18$

Choose the correct answer:

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