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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

4 votes
1 answer
91
GATE Overflow Test Series | Calculus | Test 1 | Question: 11
The value of $\lim_{x \to 0} x\sin{\frac{1}{x}}$ $1$ $0$ $\frac{1}{2}$ Does not exist
The value of $$\lim_{x \to 0} x\sin{\frac{1}{x}}$$$1$$0$$\frac{1}{2}$Does not exist
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2 votes
1 answer
96
GATE Overflow Test Series | Calculus | Test 1 | Question: 16
If $\displaystyle \lim_{x \to 0}\sin{(mx)}\cot{\left (\frac{x}{\sqrt 3}\right)} = 2$ then $m = $ _________ (rounded off up to $2$ decimal places)
If $\displaystyle \lim_{x \to 0}\sin{(mx)}\cot{\left (\frac{x}{\sqrt 3}\right)} = 2$ then $m = $ _________ (rounded off up to $2$ decimal places)
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2 votes
1 answer
97
GATE Overflow Test Series | Calculus | Test 1 | Question: 17
If $f(x) = \dfrac{1}{4x^2+2x+1}$ then its maximum value is $0.75$ $1.33$ $0.25$ It does not have a maximum value
If $f(x) = \dfrac{1}{4x^2+2x+1}$ then its maximum value is$0.75$$1.33$$0.25$It does not have a maximum value
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1 votes
1 answer
98
GATE Overflow Test Series | Calculus | Test 1 | Question: 18
The minimum value of the function $f(x) = \frac{x^3}{3} - x$ occurs at $x = 1$ $x =-1$ $x = 0$ $ x = \frac{1}{\sqrt{3}}$
The minimum value of the function $$f(x) = \frac{x^3}{3} - x$$ occurs at$x = 1$$x =-1$$x = 0$$ x = \frac{1}{\sqrt{3}}$
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1 votes
2 answers
99
GATE Overflow Test Series | Calculus | Test 1 | Question: 19
Let $f(x)$ be the function defined as $f(x)= \left\{ \begin{array}{ll} \mid x\mid & \text{for } 0 < x \leq 2 \\ 1 & \text{for } x= 0 \end{array} \right. $ At $x = 0, f$ has: has local maximum has local minimum no local maximum no extremum
Let $f(x)$ be the function defined as $$f(x)= \left\{\begin{array}{ll}\mid x\mid & \text{for } 0 < x \leq 2 \\1 & \text{for } x= 0\end{array}\right. $$At $x = 0, f$ has:...
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1 votes
1 answer
100
GATE Overflow Test Series | Calculus | Test 1 | Question: 20
The maximum value of $f(x) = 2x^3 -9x^2 +12x -3$ in the interval $0 \leq x \leq 3$ is _________
The maximum value of $f(x) = 2x^3 -9x^2 +12x -3$ in the interval $0 \leq x \leq 3$ is _________
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2 votes
1 answer
102
GATE Overflow Test Series | Calculus | Test 1 | Question: 22
The minimum value of the function $f(x) = \frac{1}{3}x(x^2-3)$ in the interval $-100 \leq \ x \leq 100$ occurs at $x = $_________
The minimum value of the function $f(x) = \frac{1}{3}x(x^2-3)$ in the interval $-100 \leq \ x \leq 100$ occurs at $x = $_________
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1 votes
1 answer
103
GATE Overflow Test Series | Calculus | Test 1 | Question: 23
Which of the following function is strictly decreasing? $f(x) = ax+b$ for $x \in \mathbb{R}, a,b > 0$ $f(x) = e^{2x}$ for $x \in \mathbb{R}$ $f(x) = \frac{2}{x} + 5$ for $x \in \mathbb{R}$ $f(x) = x^3 -6x^2+15x-18$ for $x \in \mathbb{R}$
Which of the following function is strictly decreasing?$f(x) = ax+b$ for $x \in \mathbb{R}, a,b 0$$f(x) = e^{2x}$ for $x \in \mathbb{R}$$f(x) = \frac{2}{x} + 5$ for $x \...
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1 votes
1 answer
104
GATE Overflow Test Series | Calculus | Test 1 | Question: 24
Find the 'c' of Rolle's mean value theorem for $f(x) = x^3 - 4x \in [-2,2].$ $\frac{2}{\sqrt{3}}$ $-\frac{2}{\sqrt{3}}$ $\pm \frac{2}{\sqrt{3}}$ None of the above
Find the 'c' of Rolle's mean value theorem for $f(x) = x^3 - 4x \in [-2,2].$$\frac{2}{\sqrt{3}}$$-\frac{2}{\sqrt{3}}$$\pm \frac{2}{\sqrt{3}}$None of the above
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2 votes
1 answer
105
GATE Overflow Test Series | Calculus | Test 1 | Question: 25
Compute the value of $\int_{0}^{\frac{\pi}{2}}\frac{\tan^7 x}{\cot^7x + \tan^7x}dx$ $\pi$ $\frac{\pi}{4}$ $1$ $\frac{\pi}{2}$
Compute the value of $$\int_{0}^{\frac{\pi}{2}}\frac{\tan^7 x}{\cot^7x + \tan^7x}dx$$$\pi$$\frac{\pi}{4}$$1$$\frac{\pi}{2}$
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1 votes
1 answer
106
GATE Overflow Test Series | Calculus | Test 1 | Question: 26
Which of the following function(s) is/are continuous at $x = 3?$ ... $f(x) = \left\{ \begin{matrix} \frac{1}{\sqrt{x^3- 27}}; & \text{when } x<3 \end{matrix} \right.$
Which of the following function(s) is/are continuous at $x = 3?$ (Mark all the appropriate choices)$f(x) = \left\{\begin{matrix}2; & \text{when } x = 3 \\x-1; & \text{whe...
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2 votes
1 answer
107
GATE Overflow Test Series | Calculus | Test 1 | Question: 27
Consider the following definite integral $I =\int_{0}^{1}\frac{(\sin^{-1} x)^2}{\sqrt{1-x^2}}dx$ The value of the integral is $\frac{\pi^3}{24}$ $\frac{\pi^3}{12}$ $\frac{\pi^3}{48}$ $\frac{\pi^3}{64}$
Consider the following definite integral $$I =\int_{0}^{1}\frac{(\sin^{-1} x)^2}{\sqrt{1-x^2}}dx$$ The value of the integral is$\frac{\pi^3}{24}$$\frac{\pi^3}{12}$$\frac{...
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4 votes
1 answer
108
GATE Overflow Test Series | Calculus | Test 1 | Question: 28
What is the value of the integration $I =\int_{0}^{\frac{\pi}{2}}\log(\sin x)\;dx$ 0 $\frac{-\pi}{2}\log 2$ $-\pi \log 2$ $\log 2$
What is the value of the integration $$I =\int_{0}^{\frac{\pi}{2}}\log(\sin x)\;dx$$0$\frac{-\pi}{2}\log 2$$-\pi \log 2$$\log 2$
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2 votes
1 answer
109
GATE Overflow Test Series | Calculus | Test 1 | Question: 29
The value of the integral $I =\int_{-1}^{2}\left(1 + \mid x \mid \right)dx$ $=$ _________ (rounded up to $1$ decimal place)
The value of the integral $$I =\int_{-1}^{2}\left(1 + \mid x \mid \right)dx$$ $=$ _________ (rounded up to $1$ decimal place)
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3 votes
1 answer
110
GATE Overflow Test Series | Calculus | Test 1 | Question: 30
The value of the integral $I =\int_{-\infty }^{\infty}\frac{dx}{1+x^2}$ $-\pi$ $-\frac{\pi}{2}$ $\frac{\pi}{2}$ $\pi$
The value of the integral $$I =\int_{-\infty }^{\infty}\frac{dx}{1+x^2}$$$-\pi$$-\frac{\pi}{2}$$\frac{\pi}{2}$$\pi$
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