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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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Recent questions in Calculus

2 votes
1 answer
621
TIFR2010-Maths-A-14
The solution of the ordinary differential equation. $\frac{dy}{dx}=y, y(0)=0$ Is unbounded Is positive Is negative Is zero
The solution of the ordinary differential equation.$$\frac{dy}{dx}=y, y(0)=0$$Is unboundedIs positiveIs negativeIs zero
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3 votes
3 answers
622
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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4 votes
4 answers
623
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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3 votes
1 answer
624
TIFR2010-Maths-A-6
The maximum value of $f(x)=x^{n}(1 - x)^{n}$ for a natural numbers $n\geq 1$ and $0\leq x\leq 1$ is $\frac{1}{2^{n}}$ $\frac{1}{3^{n}}$ $\frac{1}{5^{n}}$ $\frac{1}{4^{n}}$
The maximum value of $f(x)=x^{n}(1 - x)^{n}$ for a natural numbers $n\geq 1$ and $0\leq x\leq 1$ is$\frac{1}{2^{n}}$$\frac{1}{3^{n}}$$\frac{1}{5^{n}}$$\frac{1}{4^{n}}$
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15 votes
6 answers
625
TIFR CSE 2010 | Part A | Question: 7
The limit of $\dfrac{10^{n}}{n!}$ as $n \to \infty$ is. $0$ $1$ $e$ $10$ $\infty$
The limit of $\dfrac{10^{n}}{n!}$ as $n \to \infty$ is.$0$$1$$e$$10$$\infty$
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7 votes
3 answers
626
TIFR CSE 2010 | Part A | Question: 3
The function $f (x) = 2.5 \log_e \left( 2 + \exp \left( x^2 - 4x + 5 \right)\right)$ attains a minimum at $x = $? $0$ $1$ $2$ $3$ $4$
The function $f (x) = 2.5 \log_e \left( 2 + \exp \left( x^2 - 4x + 5 \right)\right)$ attains a minimum at $x = $?$0$$1$$2$$3$$4$
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1 votes
3 answers
627
Please find the value of this limit
I have applied L'Hospital here. And result came up like this : $1 - \frac{1}{2} (y^{2} + y)^{-\frac{1}{2}} (2y+1)$ Then after applying limit value , result came as $-\infty$. Am I wrong ? Please correct me.
I have applied L'Hospital here. And result came up like this :$$1 - \frac{1}{2} (y^{2} + y)^{-\frac{1}{2}} (2y+1)$$Then after applying limit value , result came as $-\inf...
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1 votes
3 answers
628
Calculate the limit
$\lim_{x \to 0} \frac{\cos(x)-\log(1+x)-1+x}{\sin^2x} = ? $ Please explain the steps also
$$\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
629
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
630
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
1 answer
631
calculate limit
$\lim_{x\rightarrow 0}\frac{ \sin x^{\circ}}{x}$
$\lim_{x\rightarrow 0}\frac{ \sin x^{\circ}}{x}$
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0 votes
1 answer
632
calculate limit
$\lim_{x\rightarrow 0} \left( \frac{1}{x^{2}} -\frac{1}{\sin^{2}x} \right)$
$\lim_{x\rightarrow 0} \left( \frac{1}{x^{2}} -\frac{1}{\sin^{2}x} \right)$
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0 votes
2 answers
633
calculate limit
$\lim_{\theta \rightarrow 0} \frac{ 1-\cos \theta }{ \theta \sin\theta }$
$\lim_{\theta \rightarrow 0} \frac{ 1-\cos \theta }{ \theta \sin\theta }$
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1 votes
2 answers
634
calculate limit
$\lim_{x\rightarrow 0} \frac{e^{x}-e^{\sin x}} {x-\sin x}$
$\lim_{x\rightarrow 0} \frac{e^{x}-e^{\sin x}} {x-\sin x}$
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0 votes
2 answers
635
calculate limit
$\lim_{x\rightarrow \infty }\frac{\sin x}{x}$
$\lim_{x\rightarrow \infty }\frac{\sin x}{x}$
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0 votes
2 answers
636
calculate limit
$\lim_{x\rightarrow 0} \frac{x^{x}-1}{x}$
$\lim_{x\rightarrow 0} \frac{x^{x}-1}{x}$
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0 votes
2 answers
637
calculate limit
$\lim_{x\rightarrow 0} \frac{6^{x}-2^{x}-3^{x}+1}{x^{2}}$
$\lim_{x\rightarrow 0} \frac{6^{x}-2^{x}-3^{x}+1}{x^{2}}$
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0 votes
1 answer
638
calculate limit
$\lim_{x\rightarrow 0}\frac{1^{x}+2^{x}+3^{x}+ \dots +n^{x}-n}{x}$
$\lim_{x\rightarrow 0}\frac{1^{x}+2^{x}+3^{x}+ \dots +n^{x}-n}{x}$
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1 votes
3 answers
639
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
1 answer
640
calculate limit
$\lim_{x\rightarrow \infty }x^{m}e^{-x}$ where m is +ve integer
$\lim_{x\rightarrow \infty }x^{m}e^{-x}$ where m is +ve integer
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