If y=e^( x) and t=x^(), then which of following statements is/are TRUE ? (A) d^2y/dt^2=y/t^2(((x-1)+1)/())x (B) d^2y/dt^2=y(((x-1)+1)/())x (C) d^2y/dt^2=1 at x=1 (D) d^2y/dt^2=(e^())/() at x=1 Choose the correct answer from the options given below: Option 1: (A) only Option 2: (B) and (C) only Option 3: (A) and (D) only Option 4: (A), (C) and (D) only

Answer: (A) and (D) only. Solution: dy/dt=( e^( x))/( x^(-1))=e^( x)x^(1-). Differentiating again with respect to t means dividing by dt/dx= x^(-1), giving d^2y/dt^2=(e^( x)( x-+1))/( x^(2-1)), which is exactly the expression in (A). At x=1 this equals (e^())/(), so (D) is true and (C) is false.

Calculus

Continuity & Differentiability

Hard

common

If $y=e^{\pi x}$ and $t=x^{\pi}$ , then which of following statements is/are TRUE ?

(A) $\frac{d^2y}{dt^2}=\frac{y}{t^2}\left(\frac{\pi(x-1)+1}{\pi}\right)x$

(B) $\frac{d^2y}{dt^2}=y\left(\frac{\pi(x-1)+1}{\pi}\right)x$

(C) $\frac{d^2y}{dt^2}=1$ at $x=1$

(D) $\frac{d^2y}{dt^2}=\frac{e^{\pi}}{\pi}$ at $x=1$

Choose the correct answer from the options given below:

(A) only

(B) and (C) only

(A) and (D) only

(A), (C) and (D) only

✅ Correct Option: 3

$\frac{dy}{dt}=\frac{\pi e^{\pi x}}{\pi x^{\pi-1}}=e^{\pi x}x^{1-\pi}$ .

Differentiating again with respect to $t$ means dividing by $\frac{dt}{dx}=\pi x^{\pi-1}$ , giving $\frac{d^2y}{dt^2}=\frac{e^{\pi x}\left(\pi x-\pi+1\right)}{\pi x^{2\pi-1}}$ , which is exactly the expression in (A).

At $x=1$ this equals $\frac{e^{\pi}}{\pi}$ , so (D) is true and (C) is false.

2026: 26 May Shift 2

Let $f(x) = |3-x| + (3+x)$ be a function, then which of the following statements are TRUE?

A. $f(x)$ is continuous at $x=3$

B. $f(x)$ is continuous at $x=-3$

C. $f(x)$ is differentiable at $x=3$

D. $f(x)$ is differentiable at $x=-3$

Choose the correct answer from the options given below:

2025: 16 May Shift 1

Match List-I with List-II

List-I	List-II
(A) $f(x) = x \sin x$	(I) is not continuous at $x = -3$
(B) $f(x) = \frac{\vert x\vert }{x}, x \neq 0$ and $f(x) = 1 \text{ at } x = 0$	(II) is continuous everywhere
(C) $f(x) = x - [x]$ , $[x]$ denotes greatest integer function	(III) is not differentiable at $x = 1$
(D) $f(x) = e^{\vert x - 1\vert }$	(IV) is not continuous at $x = 0$

Choose the correct answer from the options given below:

2025: 26 May Shift 2

If $e^x + e^y = e^{x+y}$ , then $\frac{dy}{dx}$ equals

2026: 21 May Shift 2

If $y = ax^{n+1} + bx^{-n}$ , then $x^2 \dfrac{d^2y}{dx^2}$ is equal to

2026: 21 May Shift 1

If $x = 3\sin\theta - \sin 3\theta$ , $y = 3\cos\theta - \cos 3\theta$ , then $\frac{d^2y}{dx^2}$ at $\theta = \frac{\pi}{3}$ is

2025: 29 May Shift 2

Differentiation of $\frac{x^3}{1 - x^3}$ with respect to $x^3$ is equal to:

Solution

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