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

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

31 votes
4 answers
151
The value of the integral given below is$$\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$$$-2\pi$$\pi$$-\pi$$2\pi$
0 votes
0 answers
152
A solution for the differential equation $x’(t) + 2x(t) = \delta(t)$ with initial condition $x(\overline{0}) = 0$$e^{-2t}u(t)$$e^{2t}u(t)$$e^{-t}u(t)$$e^{t}u(t)$
0 votes
1 answer
153
The value of the integral $\displaystyle{}\int_{-1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$$\frac{1}{2}$$ – \frac{1}{2}$$1$
0 votes
0 answers
154
True/False Question :The function $f\left ( x \right )=cos\left ( e^{x} \right )$ is not uniformly continuous on $\mathbb{R}$.
0 votes
1 answer
156
The map $f(x) = a_0 \cos \mid x \mid +a_1 \sin \mid x \mid +a_2 \mid x \mid ^3$ is differentiable at $x=0$ if and only if$a_1=0$ and $a_2=0$$a_0=0$ and $a_1=0$$a_1=0$$a_0...
3 votes
2 answers
157
The function $f(x)=x^{5}-5x^{4}+5x^{3}-1$ hasone minima and two maximatwo minima and one maximatwo minima and two maximaone minima and one maxima
3 votes
1 answer
158
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is$\alpha$$[\alpha]$$1$$\dfrac{[\alph...
2 votes
2 answers
159
The area enclosed by the curve $\mid\: x \mid + \mid y \mid =1$ is$1$$2$$\sqrt{2}$$4$
4 votes
4 answers
161
1 votes
2 answers
163
1 votes
1 answer
165
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is$\frac{\pi}{6} + \sqrt{3}$$\frac{\pi}{6} - \sqrt{3}$$0$$\frac{1}{2}$
2 votes
2 answers
167
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
0 votes
1 answer
169
$\underset{x \to 0}{\lim} x \sin \left( \frac{1}{x} \right)$ equals$-1$$0$$1$Does not exist
0 votes
1 answer
170
$\underset{x \to 1}{\lim} \dfrac{x^{16}-1}{\mid x-1 \mid}$ equals$-1$$0$$1$Does not exist