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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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Hot questions in Calculus

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271
ISI2015-MMA-58
Let $\{a_n\}, n \geq 1$, be a sequence of real numbers satisfying $\mid a_n \mid \leq 1$ for all $n$. Define $A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$, for $n \geq 1$. Then $\underset{n \to \infty}{\lim} \sqrt{n}(A_{n+1}-A_n)$ is equal to $0$ $-1$ $1$ none of these
Let $\{a_n\}, n \geq 1$, be a sequence of real numbers satisfying $\mid a_n \mid \leq 1$ for all $n$. Define $A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$, for $n \geq 1$. Then...
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272
ISI2014-DCG-50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1- \cos tx}$ is equal to $0$ $1$ $\infty$ $2$
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1- \cos tx}$ is equal to$0$$1$$\infty$$2$
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1 votes
1 answer
273
ISI2015-DCG-46
Let $I=\int (\sin x – \cos x)(\sin x + \cos x)^3 dx$ and $K$ be a constant of integration. Then the value of $I$ is $(\sin x + \cos x)^4+K$ $(\sin x + \cos x)^2+K$ $- \frac{1}{4} (\sin x + \cos x)^4+K$ None of these
Let $I=\int (\sin x – \cos x)(\sin x + \cos x)^3 dx$ and $K$ be a constant of integration. Then the value of $I$ is$(\sin x + \cos x)^4+K$$(\sin x + \cos x)^2+K$$- \fra...
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1 votes
1 answer
274
ISI2015-DCG-48
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals$\frac{4}{3}$$\frac{3}{4}$$1$None of these
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0 votes
1 answer
276
ISI2018-DCG-24
Let $[x]$ denote the largest integer less than or equal to $x.$ The number of points in the open interval $(1,3)$ in which the function $f(x)=a^{[x^2]},a\gt1$ is not differentiable, is $0$ $3$ $5$ $7$
Let $[x]$ denote the largest integer less than or equal to $x.$ The number of points in the open interval $(1,3)$ in which the function $f(x)=a^{[x^2]},a\gt1$ is not diff...
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1 votes
1 answer
277
ISI2019-MMA-28
Consider the functions $f,g:[0,1] \rightarrow [0,1]$ given by $f(x)=\frac{1}{2}x(x+1) \text{ and } g(x)=\frac{1}{2}x^2(x+1).$ Then the area enclosed between the graphs of $f^{-1}$ and $g^{-1}$ is $1/4$ $1/6$ $1/8$ $1/24$
Consider the functions $f,g:[0,1] \rightarrow [0,1]$ given by$$f(x)=\frac{1}{2}x(x+1) \text{ and } g(x)=\frac{1}{2}x^2(x+1).$$Then the area enclosed between the graphs of...
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278
ISI2014-DCG-42
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then $f$ has no local minima $f$ has no local maxima $f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd integers $k$ and local maxima at $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for even integers $k$ None of the above
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then$f$ has no local minima$f$ has no local maxima$f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd i...
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0 votes
1 answer
279
ISI2016-DCG-49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals$\frac{4}{3}$$\frac{3}{4}$$1$None of these
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0 votes
1 answer
280
ISI2016-DCG-53
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals$\frac{3}{5}$$\frac{5}{3}$$1$$\infty$
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0 answers
281
ISI2014-DCG-43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid -1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
Let $$f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid -1, & \text{ if } x>0. \end{cases}$$ Then $\underset{x \to a}{\lim}...
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1 votes
0 answers
282
ISI2015-DCG-55
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals$-1$$0$$1$Does not exist
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0 answers
283
ISI2018-DCG-29
Let $f(x)=(x-1)(x-2)(x-3)g(x); \: x\in \mathbb{R}$ where $g$ is twice differentiable function. Then there exists $y\in(1,3)$ such that $f’’(y)=0.$ there exists $y\in(1,2)$ such that $f’’(y)=0.$ there exists $y\in(2,3)$ such that $f’’(y)=0.$ none of the above is true.
Let $f(x)=(x-1)(x-2)(x-3)g(x); \: x\in \mathbb{R}$ where $g$ is twice differentiable function. Thenthere exists $y\in(1,3)$ such that $f’’(y)=0.$there exists $y\in(1,...
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284
ISI2016-DCG-57
$\underset{x\rightarrow 1}{\lim} \dfrac{x^{16}-1}{\mid x-1\mid}$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow 1}{\lim} \dfrac{x^{16}-1}{\mid x-1\mid}$ equals$-1$$0$$1$Does not exist
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0 votes
1 answer
286
ISI2017-DCG-26
The value of $\underset{n \to \infty}{\lim} \bigg( \dfrac{1}{1-n^2} + \dfrac{2}{1-n^2} + \dots + \dfrac{n}{1-n^2} \bigg)$ is $0$ $ – \frac{1}{2}$ $\frac{1}{2}$ none of these
The value of $\underset{n \to \infty}{\lim} \bigg( \dfrac{1}{1-n^2} + \dfrac{2}{1-n^2} + \dots + \dfrac{n}{1-n^2} \bigg)$ is$0$$ – \frac{1}{2}$$\frac{1}{2}$none of the...
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0 answers
288
ISI2017-DCG-27
The limit of the sequence $\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \dots$ is $1$ $2$ $2\sqrt{2}$ $\infty$
The limit of the sequence $\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \dots$ is$1$$2$$2\sqrt{2}$$\infty$
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290
ISI2016-DCG-58
Let $y=\left \lfloor x \right \rfloor$ where $\left \lfloor x \right \rfloor$ is greatest integer less than or equal to $x$. Then $y$ is continuous and many-one. $y$ is not differentiable and many-one. $y$ is not differentiable. $y$ is differentiable and many-one.
Let $y=\left \lfloor x \right \rfloor$ where $\left \lfloor x \right \rfloor$ is greatest integer less than or equal to $x$. Then$y$ is continuous and many-one.$y$ is not...
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