JEE Advanced 2015 · Paper 1 · Mathematics Q57

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MathematicsMultiple correct+4 / -2Question 57 of 60

Let $f(x)=\sin\left(\dfrac\pi6\sin\left(\dfrac\pi2\sin x\right)\right)$ for all $x\in\mathbb R$ and $g(x)=\dfrac\pi2\sin x$ for all $x\in\mathbb R$ . Let $(f\circ g)(x)$ denote $f(g(x))$ and $(g\circ f)(x)$ denote $g(f(x))$ . Then which of the following is (are) true?

Range of

f

\left[-\dfrac12,\dfrac12\right]

Range of

f\circ g

\left[-\dfrac12,\dfrac12\right]

\displaystyle\lim_{x\to0}\frac{f(x)}{g(x)}=\frac\pi6

There is an

x\in\mathbb R

such that

(g\circ f)(x)=1

Answer and worked solution

Official answer: A, B, C

(A) $\frac\pi2\sin x\in[-\frac\pi2,\frac\pi2]$ , so its sine is in $[-1,1]$ . Then $\frac\pi6\cdot$ that is in $[-\frac\pi6,\frac\pi6]$ , and its sine fills $[-\frac12,\frac12]$ . True.
(B) $g$ takes values in $[-\frac\pi2,\frac\pi2]$ , where $\sin$ still covers $[-1,1]$ . The same chain gives $[-\frac12,\frac12]$ . True.
(C) For small $x$ : $f(x)\approx\frac\pi6\cdot\frac\pi2\sin x$ and $g(x)=\frac\pi2\sin x$ . The ratio tends to $\frac\pi6$ . True.
(D) $g(f(x))=\frac\pi2\sin f(x)\le\frac\pi2\sin\frac12\approx0.75<1$ . False.

(A), (B), (C)

Topics:Domain and range Limits

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JEE Advanced 2015 · Paper 1, Mathematics question 57