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Syllabus: Limits, Continuity, and Differentiability, Maxima and minima, Mean value theorem, Integration.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{2020}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} & 1 &1&1&1&1&1&0&1&1&1&0&0.9&1
\\\hline\textbf{2 Marks Count} & 0 &0&0&0&0&0&1&0&0&0&0&0.1&1
\\\hline\textbf{Total Marks} & 1 &1&1&1&1&1&2&1&1&1&\bf{1}&\bf{1.1}&\bf{2}\\\hline
\end{array}}}$$

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Most answered questions in Calculus

3 votes
4 answers
21
ISI2014-DCG-4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to$\infty$$0$$\log_e 2$$1$
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28 votes
4 answers
23
GATE CSE 2018 | Question: 16
The value of $\int^{\pi/4} _0 x \cos(x^2) dx$ correct to three decimal places (assuming that $\pi = 3.14$) is ____
The value of $\int^{\pi/4} _0 x \cos(x^2) dx$ correct to three decimal places (assuming that $\pi = 3.14$) is ____
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22 votes
4 answers
24
GATE CSE 2017 Set 1 | Question: 28
The value of $\displaystyle \lim_{x\rightarrow 1} \frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}$ is $0$ is $-1$ is $1$ does not exist
The value of $\displaystyle \lim_{x\rightarrow 1} \frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}$is $0$is $-1$is $1$does not exist
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26 votes
4 answers
26
GATE CSE 2016 Set 1 | Question: 3
$\lim _{x\rightarrow 4}\frac{\sin(x-4)}{x-4}=\_\_\_\_\_\_\_\_\_\_\_\_$
$$\lim _{x\rightarrow 4}\frac{\sin(x-4)}{x-4}=\_\_\_\_\_\_\_\_\_\_\_\_$$
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12 votes
4 answers
27
TIFR CSE 2011 | Part A | Question: 14
The limit $\lim_{x \to 0} \frac{d}{dx}\,\frac{\sin^2 x}{x}$ is $0$ $2$ $1$ $\frac{1}{2}$ None of the above
The limit $$\lim_{x \to 0} \frac{d}{dx}\,\frac{\sin^2 x}{x}$$ is$0$$2$$1$$\frac{1}{2}$None of the above
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4 votes
4 answers
28
TIFR2010-Maths-A-8
Let $f(x)= |x|^{3/2}, x \in \mathbb{R}$. Then $f$ is uniformly continuous. $f$ is continuous, but not differentiable at $x=0$. $f$ is differentiable and $f ' $ is continuous. $f$ is differentiable, but $f ' $ is discontinuous at $x=0$.
Let $f(x)= |x|^{3/2}, x \in \mathbb{R}$. Then$f$ is uniformly continuous.$f$ is continuous, but not differentiable at $x=0$.$f$ is differentiable and $f ' $ is continuous...
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0 votes
4 answers
29
Limits
What is $\lim_{ x \to 0} (1-x)^{\frac{1}{x}}$ ? Also please explain the result?
What is $\lim_{ x \to 0} (1-x)^{\frac{1}{x}}$ ? Also please explain the result?
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32 votes
4 answers
30
GATE CSE 2015 Set 2 | Question: GA-3
Consider a function $f(x) = 1- |x| \text{ on } -1 \leq x \leq 1$. The value of $x$ at which the function attains a maximum, and the maximum value of the function are: $0, -1$ $-1, 0$ $0, 1$ $-1, 2$
Consider a function $f(x) = 1- |x| \text{ on } -1 \leq x \leq 1$. The value of $x$ at which the function attains a maximum, and the maximum value of the function are:$0, ...
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18 votes
4 answers
31
GATE IT 2005 | Question: 35
What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$ $-1$ $0$ $1$ $\pi$
What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$$-1$$0$$1$$\pi$
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10 votes
4 answers
32
GATE CSE 1997 | Question: 4.1
What is the maximum value of the function $f(x) = 2x^2 - 2x + 6$ in the interval $\left[0,2 \right]$? 6 10 12 5.5
What is the maximum value of the function $f(x) = 2x^2 - 2x + 6$ in the interval $\left[0,2 \right]$?610125.5
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31 votes
4 answers
33
GATE CSE 2014 Set 3 | Question: 47
The value of the integral given below is $\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$ $-2\pi$ $\pi$ $-\pi$ $2\pi$
The value of the integral given below is$$\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$$$-2\pi$$\pi$$-\pi$$2\pi$
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19 votes
4 answers
34
GATE CSE 2014 Set 1 | Question: 46
The function $f(x) =x \sin x$ satisfies the following equation: $f''(x) + f(x) +t \cos x = 0$The value of $t$ is______.
The function $f(x) =x \sin x$ satisfies the following equation: $$f''(x) + f(x) +t \cos x = 0$$The value of $t$ is______.
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43 votes
4 answers
35
GATE CSE 2014 Set 1 | Question: 6
Let the function ... $\theta \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta)\neq 0$ I only II only Both I and II Neither I nor II
Let the function$$f(\theta) = \begin{vmatrix} \sin\theta & \cos\theta & \tan\theta \\ \sin(\frac{\pi}{6}) & \cos(\frac{\pi}{6}) & \tan(\frac{\pi}{6}) & \\ \sin(\frac{\pi...
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17 votes
4 answers
36
GATE CSE 2013 | Question: 22
Which one of the following functions is continuous at $x = 3?$ $f(x) = \begin{cases} 2,&\text{if $x = 3$ } \\ x-1& \text{if $x > 3$}\\ \frac{x+3}{3}&\text{if $x < 3$ } \end{cases}$ ... $} \end{cases}$ $f(x) = \begin{cases} \frac{1}{x^3-27}&\text{if $x \neq 3$ } \end{cases}$
Which one of the following functions is continuous at $x = 3?$$f(x) = \begin{cases} 2,&\text{if $x = 3$ } \\ x-1& \text{if $x 3$}\\ \frac{x+3}{3}&\text{if $x < 3$ } \end...
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3 votes
3 answers
38
GO Classes Test Series 2023 | Calculus | Test 1 | Question: 3
The function $f(x)=x^{4}-6 x^{2}$ is increasing on the intervals $(0, \sqrt{3})$ only $(\sqrt{3}, \infty)$ only $(-\infty,-\sqrt{3})$ and $(0, \sqrt{3})$ only $(-\sqrt{3}, 0)$ and $(\sqrt{3}, \infty)$ only
The function $f(x)=x^{4}-6 x^{2}$ is increasing on the intervals$(0, \sqrt{3})$ only$(\sqrt{3}, \infty)$ only$(-\infty,-\sqrt{3})$ and $(0, \sqrt{3})$ only$(-\sqrt{3}, 0)...
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0 votes
3 answers
40
Applied Mathematics Question
Let $f(x) = e^{x^2},$ then find $f''(x).$
Let $f(x) = e^{x^2},$ then find $f''(x).$
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