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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

5 votes
3 answers
61
ISRO-2013-49
What is the least value of the function $f(x) = 2x^{2}-8x-3$ in the interval $[0, 5]$? $-15$ $7$ $-11$ $-3$
What is the least value of the function $f(x) = 2x^{2}-8x-3$ in the interval $[0, 5]$?$-15$$7$$-11$$-3$
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3 votes
3 answers
62
solve calculate the limit
$\lim_{n \to \infty} \left [ \frac{1}{(1+n)} + \frac{1}{(2+n)} + - - - - - + \frac{1}{(n+n)} \right ]$ a) $log 2$ b) $2$ c) $\frac{1}{2}$ d) $\Pi /4$
$\lim_{n \to \infty} \left [ \frac{1}{(1+n)} + \frac{1}{(2+n)} + - - - - - + \frac{1}{(n+n)} \right ]$a) $log 2$ b) $2$c) $\frac{1}{2}$ ...
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7 votes
3 answers
63
TIFR CSE 2012 | Part A | Question: 14
The limit $\displaystyle \lim_{n \rightarrow \infty} \left(\sqrt{n^{2}+n}-n\right)$ equals. $\infty$ $1$ $1 / 2$ $0$ None of the above
The limit $\displaystyle \lim_{n \rightarrow \infty} \left(\sqrt{n^{2}+n}-n\right)$ equals.$\infty$$1$$1 / 2$$0$None of the above
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10 votes
3 answers
64
TIFR CSE 2011 | Part A | Question: 11
$\int_{0}^{1} \log_e(x) dx=$ $1$ $-1$ $\infty $ $-\infty $ None of the above
$$\int_{0}^{1} \log_e(x) dx=$$$1$$-1$$\infty $$-\infty $None of the above
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3 votes
3 answers
65
TIFR2010-Maths-A-13
What is the value of $\lim_{x\to 0} \sin{\left (\frac1 x \right )}$ $1$ $0$ $\frac{1}{2}$ Does Not Exist
What is the value of $$\lim_{x\to 0} \sin{\left (\frac1 x \right )}$$$1$$0$$\frac{1}{2}$Does Not Exist
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7 votes
3 answers
66
TIFR CSE 2010 | Part A | Question: 3
The function $f (x) = 2.5 \log_e \left( 2 + \exp \left( x^2 - 4x + 5 \right)\right)$ attains a minimum at $x = $? $0$ $1$ $2$ $3$ $4$
The function $f (x) = 2.5 \log_e \left( 2 + \exp \left( x^2 - 4x + 5 \right)\right)$ attains a minimum at $x = $?$0$$1$$2$$3$$4$
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1 votes
3 answers
67
Please find the value of this limit
I have applied L'Hospital here. And result came up like this : $1 - \frac{1}{2} (y^{2} + y)^{-\frac{1}{2}} (2y+1)$ Then after applying limit value , result came as $-\infty$. Am I wrong ? Please correct me.
I have applied L'Hospital here. And result came up like this :$$1 - \frac{1}{2} (y^{2} + y)^{-\frac{1}{2}} (2y+1)$$Then after applying limit value , result came as $-\inf...
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1 votes
3 answers
68
Calculate the limit
$\lim_{x \to 0} \frac{\cos(x)-\log(1+x)-1+x}{\sin^2x} = ? $ Please explain the steps also
$$\lim_{x \to 0} \frac{\cos(x)-\log(1+x)-1+x}{\sin^2x} = ? $$Please explain the steps also
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2 votes
3 answers
69
limit
limx->0 (cot x)1/logx ? ans: 0(m getting 1/e)
limx->0 (cot x)1/logx ?ans: 0(m getting 1/e)
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0 votes
3 answers
70
select correct option for limit
$\lim_{x\rightarrow 0}\sin \left( \frac{1}{x}\right)$ (a) 1 (b) 0 (c) does not exist (d) none of these
$\lim_{x\rightarrow 0}\sin \left( \frac{1}{x}\right)$(a) 1(b) 0(c) does not exist(d) none of these
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1 votes
3 answers
71
calculate limit
$\lim_{x\rightarrow \infty }\left(4^{x}+5^{x}\right)^{1/x}$
$\lim_{x\rightarrow \infty }\left(4^{x}+5^{x}\right)^{1/x}$
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0 votes
3 answers
72
calculate limit
$\lim_{x\rightarrow 0}\left ( \frac{a^{x}+b^{x}}{2} \right )^{^{\frac{1}{x}}}$
$\lim_{x\rightarrow 0}\left ( \frac{a^{x}+b^{x}}{2} \right )^{^{\frac{1}{x}}}$
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1 votes
3 answers
73
calculate limit
$\lim_{x\rightarrow \frac{\pi}{2}}{( \sin x)}^{\tan x}$
$\lim_{x\rightarrow \frac{\pi}{2}}{( \sin x)}^{\tan x}$
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19 votes
3 answers
75
GATE CSE 1995 | Question: 2.8
If the cube roots of unity are $1, \omega$ and $\omega^2$, then the roots of the following equation are $(x-1)^3 +8 =0$ $-1, 1 + 2\omega, 1 + 2\omega^2$ $1, 1 - 2\omega, 1 - 2\omega^2$ $-1, 1 - 2\omega, 1 - 2\omega^2$ $-1, 1 + 2\omega, -1 + 2\omega^2$
If the cube roots of unity are $1, \omega$ and $\omega^2$, then the roots of the following equation are $$(x-1)^3 +8 =0$$$-1, 1 + 2\omega, 1 + 2\omega^2$ $1, 1 - 2\omeg...
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26 votes
3 answers
76
GATE CSE 1995 | Question: 1.21
In the interval $[0, \pi]$ the equation $x=\cos x$ has No solution Exactly one solution Exactly two solutions An infinite number of solutions
In the interval $[0, \pi]$ the equation $x=\cos x$ has No solutionExactly one solutionExactly two solutionsAn infinite number of solutions
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24 votes
3 answers
77
GATE CSE 2009 | Question: 25
$\int^{\pi/4}_0 (1-\tan x)/(1+\tan x)\,dx $ $0$ $1$ $\ln 2$ $1/2 \ln 2$
$\int^{\pi/4}_0 (1-\tan x)/(1+\tan x)\,dx $$0$$1$ $\ln 2$$1/2 \ln 2$
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35 votes
3 answers
78
GATE CSE 2000 | Question: 2.3
Let $S = \sum_{i=3}^{100} i \log_{2} i$, and $T = \int_{2}^{100} x \log_{2}x dx$. Which of the following statements is true? $S > T$ $S = T$ $S < T$ and $2S > T$ $2S ≤ T$
Let $S = \sum_{i=3}^{100} i \log_{2} i$, and $T = \int_{2}^{100} x \log_{2}x dx$.Which of the following statements is true?$S T$$S = T$$S < T$ and $2S T$$2S ≤ T$
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15 votes
3 answers
79
GATE CSE 1993 | Question: 02.1
$\displaystyle \lim_{x \to 0} \frac{x(e^x - 1) + 2(\cos x -1)}{x(1 - \cos x)}$ is __________
$\displaystyle \lim_{x \to 0} \frac{x(e^x - 1) + 2(\cos x -1)}{x(1 - \cos x)}$ is __________
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