If f(x) = cases 1/(, x, ), & x ≥ 1 \\ ax^2+b, & x < 1 cases is differentiable at x=1 then Option 1: a=1/2, b=-3/2 Option 2: a=-1/2, b=3/2 Option 3: a=-1/2, b=-3/2 Option 4: a=1/2, b=3/2

Answer: a=-1/2, b=3/2. Solution: For x ≥ 1, f(x)=1/x with f'(x)=-1/x^2. Matching derivatives at x=1 gives 2a=-1, so a=-1/2. Continuity requires a+b=1, hence b=3/2.

Calculus

Continuity & Differentiability

Medium

core

If $f(x) = \begin{cases} \dfrac{1}{|x|}, & x \geq 1 \\ ax^2+b, & x < 1 \end{cases}$ is differentiable at $x=1$ then

$a=\dfrac{1}{2}, b=-\dfrac{3}{2}$

$a=-\dfrac{1}{2}, b=\dfrac{3}{2}$

$a=-\dfrac{1}{2}, b=-\dfrac{3}{2}$

$a=\dfrac{1}{2}, b=\dfrac{3}{2}$

✅ Correct Option: 2

For $x \geq 1$ , $f(x)=\frac{1}{x}$ with $f'(x)=-\frac{1}{x^2}$ . Matching derivatives at $x=1$ gives $2a=-1$ , so $a=-\frac{1}{2}$ . Continuity requires $a+b=1$ , hence $b=\frac{3}{2}$ .

2026: 18 May Shift 2

If $x=\log_e t$ and $y=\dfrac{1}{t}$ , then $\dfrac{d^2y}{dx^2}$ at $t=0.1$ is

2026: 30 May Shift 2

If $y = \log_3(\log x)$ , (Consider $\log_e x = \log x$ ) then the value of $\frac{dy}{dx}$ is

2022: 4 Aug Shift 1

If $y = \frac{\log_e x}{x}$ , then $\frac{d^2y}{dx^2} =$

2025: 22 May Shift 2

Match List-I with List-II

Where $\mathbb{R}$ is set of real numbers

List-I	List-II
(A) $\sin x$ is continuous on:	(I) $\mathbb{R} - \{0\}$
(B) $\tan x$ is continuous on:	(II) $\mathbb{R}$
(C) $\cot x$ is continuous on:	(III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$
(D) $x^{-n}, n \in \mathbb{N}$ is continuous on:	(IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$

Choose the correct answer from the options given below:

2026: 30 May Shift 1

Let $f(x)=\begin{cases}\frac{1-\cos4x}{x^2}, & \text{if } x<0\\2\lambda, & \text{if } x=0\\\frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}, & \text{if } x>0\end{cases}$ such that $f$ continuous at $x=0$ . Then, the value of $\lambda$ is equal to:

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:

If f(x)={1∣x∣,x≥1ax2+b,x<1f(x) = \begin{cases} \dfrac{1}{|x|}, & x \geq 1 \\ ax^2+b, & x < 1 \end{cases}f(x)=⎩⎨⎧​∣x∣1​,ax2+b,​x≥1x<1​ is differentiable at x=1x=1x=1 then

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CUET Mathematics Past Year Questions (Topic-Wise)

If $f(x) = \begin{cases} \dfrac{1}{|x|}, & x \geq 1 \\ ax^2+b, & x < 1 \end{cases}$ is differentiable at $x=1$ then