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

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

Highest voted questions in Calculus

1 votes
1 answer
271
1 votes
0 answers
272
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties.$f(0)=0$,$f(1)=1$, and$f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_...
1 votes
1 answer
273
The function $f(x) = x^{1/x}, \: x \neq 0$ hasa minimum at $x=e$;a maximum at $x=e$;neither a maximum nor a minimum at $x=e$;None of the above
1 votes
1 answer
274
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is$\frac{\pi}{6} + \sqrt{3}$$\frac{\pi}{6} - \sqrt{3}$$0$$\frac{1}{2}$
1 votes
1 answer
275
The function $f(x)$ defined as $f(x)=x^3-6x^2+24x$, where $x$ is real, isstrictly increasingstrictly decreasingincreasing in $(- \infty, 0)$ and decreasing in $(0, \infty...
1 votes
3 answers
276
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals$e^{-1}$$e^{-1/2}$$e^{-2}$$1$
1 votes
1 answer
277
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals$1$$0$$e^{-8/3}$$e^{4/9}$
1 votes
1 answer
278
If $f(x)$ is a real valued function such that $$2f(x)+3f(-x)=15-4x,$$ for every $x \in \mathbb{R}$, then $f(2)$ is$-15$$22$$11$$0$
1 votes
1 answer
279
1 votes
1 answer
280
Suppose $a>0$. Consider the sequence $a_n = n \{ \sqrt[n]{ea} – \sqrt[n]{a}, \:\:\:\:\: n \geq 1$. Then$\underset{n \to \infty}{\lim} a_n$ does not exist$\underset{n \t...
1 votes
1 answer
281
Let $R$ be the triangle in the $xy$ – plane bounded by the $x$-axis, the line $y=x$, and the line $x=1$. The value of the double integral $$ \int \int_R \frac{\sin x}{x...
1 votes
1 answer
283
f(x) is a differentiable function that satisfies 5 ≤ f′(x) ≤ 14 for all x. Let a and b be the maximum and minimum values, respectively, that f(11)−f(3) can possib...
1 votes
2 answers
284
The value of $\underset{x \to 0}{\lim} \dfrac{\tan ^2 x – x \tan x }{\sin x}$ is$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$0$None of these
1 votes
1 answer
285
Let $I=\int (\sin x – \cos x)(\sin x + \cos x)^3 dx$ and $K$ be a constant of integration. Then the value of $I$ is$(\sin x + \cos x)^4+K$$(\sin x + \cos x)^2+K$$- \fra...
1 votes
1 answer
286
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals$\frac{4}{3}$$\frac{3}{4}$$1$None of these
1 votes
1 answer
287
The area bounded by $y=x^2-4$, $y=0$ and $x=4$ is$\frac{64}{3}$$6$$\frac{16}{3}$$\frac{32}{3}$
1 votes
1 answer
288
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals$\frac{3}{5}$$\frac{5}{3}$$1$$\infty$
1 votes
0 answers
289
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals$-1$$0$$1$Does not exist
1 votes
1 answer
290
$\underset{x \to \infty}{\lim} \left( 1 + \dfrac{1}{x^2} \right) ^x$ equals$-1$$0$$1$Does not exist