Match List-I with List-II (each function is taken where it is defined)
| List-I | List-II | ||
|---|---|---|---|
| Function y | \(\displaystyle\frac{d^{2}y}{dx^{2}}\) | ||
| (A) | \(\tan x\) | (I) | \(\displaystyle\frac{1}{x}\) |
| (B) | \(\log_e(\sin x)\) | (II) | \((4x^{2} - 2)e^{-x^{2}}\) |
| (C) | \(e^{-x^{2}}\) | (III) | \(2\sec^{2}x\tan x\) |
| (D) | \(x\log_e x\) | (IV) | \(-\mathrm{cosec}^{2}x\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
(A) \(\tan x\)
This matches (III) \(2\sec^{2}x\tan x\). Since \(\frac{d}{dx}\sec x = \sec x\tan x\):
\( \displaystyle \frac{dy}{dx} = \sec^{2}x \)
\( \displaystyle \frac{d^{2}y}{dx^{2}} = 2\sec x \cdot \sec x\tan x = 2\sec^{2}x\tan x \)
(B) \(\log_e(\sin x)\)
This matches (IV) \(-\mathrm{cosec}^{2}x\). By the chain rule:
\( \displaystyle \frac{dy}{dx} = \frac{\cos x}{\sin x} = \cot x \)
\( \displaystyle \frac{d^{2}y}{dx^{2}} = -\mathrm{cosec}^{2}x \)
(C) \(e^{-x^{2}}\)
This matches (II) \((4x^{2} - 2)e^{-x^{2}}\):
\( \displaystyle \frac{dy}{dx} = -2x\,e^{-x^{2}} \)
By the product rule:
\( \displaystyle \frac{d^{2}y}{dx^{2}} = -2e^{-x^{2}} + (-2x)(-2x)e^{-x^{2}} \)
\( = (4x^{2} - 2)e^{-x^{2}} \)
(D) \(x\log_e x\)
This matches (I) \(\frac{1}{x}\):
\( \displaystyle \frac{dy}{dx} = \log_e x + x \cdot \frac{1}{x} = \log_e x + 1 \)
\( \displaystyle \frac{d^{2}y}{dx^{2}} = \frac{1}{x} \)