Limits, Continuity & Differentiability · JEE Advanced previous year questions

2023 3 questions

1. Q12023 · 4 Jun 2023 · Paper 1· One or more correct

   Let $S=(0,1)\cup (1,2)\cup (3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?

   1. A.There are infinitely many functions from $S$ to $T$
   2. B.There are infinitely many strictly increasing functions from $S$ to $T$
   3. C.The number of continuous functions from $S$ to $T$ is at most 120
   4. D.Every continuous function from $S$ to $T$ is differentiable
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   Answer: A, C, D

2. Q42023 · 4 Jun 2023 · Paper 1

   Let $f:(0,1)\to \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x\in \left[\frac{1}{n+1},\frac{1}{n}\right)$ where $n\in \mathbb{N}$. Let $g:(0,1)\to \mathbb{R}$ be a function such that ${\int }_{x^2}^x\sqrt{\frac{1-t}{t}}dt<g(x)<2\sqrt{x}$ for all $x\in (0,1)$. Then $$\underset{x\to 0}{\lim }f(x)g(x)$$

   1. A.does NOT exist
   2. B.is equal to 1
   3. C.is equal to 2
   4. D.is equal to 3
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   Answer: C

3. Q62023 · 4 Jun 2023 · Paper 2· One or more correct

   Let $f:(0,1)\to \mathbb{R}$ be the function defined as $f(x)=[4x]{\left(x-\frac{1}{4}\right)}^2\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?

   1. A.The function $f$ is discontinuous exactly at one point in $(0,1)$
   2. B.There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable
   3. C.The function $f$ is NOT differentiable at more than three points in $(0,1)$
   4. D.The minimum value of the function $f$ is $-\frac{1}{512}$
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   Answer: A, B