If f(x) = cases (k x)/( - 2x), & x ≠ ()/2 \\ 3, & x = ()/2 cases is continuous at x = ()/2, then k is : Option 1: 6 Option 2: 4 Option 3: 3 Option 4: 2

Answer: 6. Solution: Let h = ()/2 - x. Then x = h and - 2x = 2h. So _(x /2) (k x)/( - 2x) = _(h 0) (k h)/2h = k/2. For continuity: k/2 = 3, so k = 6.

Calculus

Continuity & Differentiability

Medium

core

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

✅ Correct Option: 1

Let $h = \frac{\pi}{2} - x$ . Then $\cos x = \sin h$ and $\pi - 2x = 2h$ . So $\lim_{x \to \pi/2} \frac{k\cos x}{\pi - 2x} = \lim_{h \to 0} \frac{k \sin h}{2h} = \frac{k}{2}$ . For continuity: $\frac{k}{2} = 3$ , so $k = 6$ .

2025: 16 May Shift 1

Match List-I with List-II

List-I	List-II
(Parametric equations)	$\left(\frac{dy}{dx}\right)$
(A) $x = \frac{2}{t}, y = 2t$	(I) $4t^2$
(B) $x = t^3, y = 3t + 2$	(II) $2(t+1)$
(C) $x = \log t, y = 2t^2$	(III) $-t^2$
(D) $x = e^t, y = 2te^t$	(IV) $t^{-2}$

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

2025: 13 May Shift 1

If $f(x) = x^3\log_e x$ , Then $f''(e^2)$ is equal to

2022: 16 July Shift 2

Match List I with List II

List I	List II
A. If $f(x) = 2x$ and $g(x) = \frac{x^2}{2} + 1$ , then $\frac{g(x)}{f(x)}$ is	I. discontinuous at exactly three points.
B. The function $f(x) = \frac{4-x^2}{4x-x^3}$ is	II. continuous everywhere
C. The function $f(x) = \lvert x \rvert + \lvert x-1 \rvert$ is	III. discontinuous at $x = 0$ .
D. The function $f(x) = \lvert \sin x \rvert$ is	IV. continuous at $x = 0$ and $x = 1$

Choose the correct answer from the options given below:

2026: 18 May Shift 2

For the greatest integer function $f(x)=[x]$ , which of the following statements are correct?

A. It is continuous at all real numbers.

B. It is continuous at $x=2$ and 3.

C. It is continuous and differentiable for $x\in(2,3)$

D. It is continuous and differentiable at $x=4.5$

Choose the correct answer from the options given below:

2026: 29 May Shift 2

If $x=a(1-\cos\theta), y=a(\theta+\sin\theta)$ then $\frac{d^2y}{dx^2}$ is equal to

If f(x)={kcos⁡xπ−2x,x≠π23,x=π2f(x) = \begin{cases} \frac{k\cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases}f(x)={π−2xkcosx​,3,​x=2π​x=2π​​ is continuous at x=π2x = \frac{\pi}{2}x=2π​, then k is :

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