JEE Advanced 2016 · Paper 1 · Mathematics Q47

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MathematicsMultiple correct+4 / -2Question 47 of 54

Let $f:\mathbb R\to\mathbb R$ , $g:\mathbb R\to\mathbb R$ and $h:\mathbb R\to\mathbb R$ be differentiable functions such that $f(x)=x^3+3x+2$ , $g(f(x))=x$ and $h(g(g(x)))=x$ for all $x\in\mathbb R$ . Then

g'(2)=\dfrac1{15}

h'(1)=666

h(0)=16

h(g(3))=36

Answer and worked solution

Official answer: B, C

Identify g and h. $g$ is the inverse of $f$ . From $h(g(g(x)))=x$ , $h=f\circ f$ .

(A) $f(0)=2$ and $f'(0)=3$ , so $g'(2)=\frac13$ . False.
(B) $h'(1)=f'(f(1))f'(1)=f'(6)\cdot6=111\cdot6=666$ . True.
(C) $h(0)=f(f(0))=f(2)=16$ . True.
(D) $h(g(3))=f(f(g(3)))=f(3)=38$ . False.

(B), (C)

Topics:Chain rule and implicit differentiation Composition and inverse functions

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JEE Advanced 2016 · Paper 1, Mathematics question 47