JEE Advanced 2014 · Paper 1, Mathematics question 46

The inner function. Let $u(x)=\log(\sec x+\tan x)$ on $(-\frac\pi2,\frac\pi2)$.

• $u'(x)=\sec x>0$, so $u$ is strictly increasing.
• $\sec(-x)+\tan(-x)=\sec x-\tan x=\dfrac1{\sec x+\tan x}$, so $u(-x)=-u(x)$: $u$ is odd.
• As $x\to\pm\frac\pi2$, $u\to\pm\infty$, so $u$ maps onto $\mathbb R$.

Cubing keeps all three properties.

• (A) Odd. True.
• (B) Strictly increasing, so one-one. True.
• (C) Range is $\mathbb R$, so onto. True.
• (D) Not even (it is odd and not identically zero). False.

(A), (B), (C)