If x = 3 - 3, y = 3 - 3, then d^2y/dx^2 at = ()/3 is Option 1: -16/27 Option 2: 16/27 Option 3: 9/17 Option 4: -9/17

Answer: -16/27. Solution: dy/dx=(3 3-3)/(3-3 3)=(2 2)/(2 2)= 2. Then d^2y/dx^2=(-2^2 2)/(dx/d). At =()/3, ^2(2)/3=43 and dx/(d)=92, giving (-8/3)/(9/2)=-16/27.

Calculus

Continuity & Differentiability

Hard

core

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

$-\frac{16}{27}$

$\frac{16}{27}$

$\frac{9}{17}$

$-\frac{9}{17}$

✅ Correct Option: 1

$\frac{dy}{dx}=\frac{3\sin 3\theta-3\sin\theta}{3\cos\theta-3\cos 3\theta}=\frac{2\cos 2\theta\sin\theta}{2\sin 2\theta\sin\theta}=\cot 2\theta$ . Then $\frac{d^2y}{dx^2}=\frac{-2\csc^2 2\theta}{dx/d\theta}$ . At $\theta=\frac{\pi}{3}$ , $\csc^2\frac{2\pi}{3}=\frac43$ and $\frac{dx}{d\theta}=\frac92$ , giving $\frac{-8/3}{9/2}=-\frac{16}{27}$ .

2026: 30 May Shift 2

Match List-I with List-II

List-I	List-II
(A) $\frac{d}{dx}(\sqrt{\sin x})$	(I) $\frac{-\sin x}{2\sqrt{\cos x}}$
(B) $\frac{d}{dx}(\sqrt{\cos x})$	(II) $-\sin 2x$
(C) $\frac{d}{dx}(\sin^2 x)$	(III) $\frac{\cos x}{2\sqrt{\sin x}}$
(D) $\frac{d}{dx}(\cos^2 x)$	(IV) $\sin 2x$

Choose the correct answer from the options given below:

2025: 14 May Shift 1

If the function $f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x} & : x \neq \frac{\pi}{2} \\ 3 & : x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$ , then $k$ is equal to

2025: 22 May Shift 1

If $y = \sin^{-1} x + \sin^{-1} \sqrt{1-x^2}, x \in (-1, 0)$ , then $\frac{dy}{dx}$ is equal to

2025: 14 May Shift 1

Match List-I with List-II

$\begin{array}{|l|l|} \hline \rule{0pt}{2.8ex}\text{List-I} & \text{List-II} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(A) } f(x) = |x| & \text{(I) Not differentiable at } x=-2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(B) } f(x) = |x+2| & \text{(II) Not differentiable at } x=0 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(C) } f(x) = |x^2-4| & \text{(III) Not differentiable at } x=2 \text{ only} \\[1.2ex] \hline \rule{0pt}{2.8ex}\text{(D) } f(x) = |x-2| & \text{(IV) Not differentiable at } x=2,-2 \text{ only} \\[1.2ex] \hline \end{array}$

Choose the correct answer from the options given below:

2023: 15 June Shift 2

If $y = \log\left[\frac{x^2}{e^2}\right]$ then value of $\frac{d^2y}{dx^2}$ is :

2023: 15 June Shift 2

If $\cos y = x\cos(a+y)$ , then $\frac{dy}{dx} =$

If x=3sin⁡θ−sin⁡3θx = 3\sin\theta - \sin 3\thetax=3sinθ−sin3θ, y=3cos⁡θ−cos⁡3θy = 3\cos\theta - \cos 3\thetay=3cosθ−cos3θ, then d2ydx2\frac{d^2y}{dx^2}dx2d2y​ at θ=π3\theta = \frac{\pi}{3}θ=3π​ is

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CUET Mathematics Past Year Questions (Topic-Wise)

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