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

$$\small{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count}&1&1&0&1&1&1&0&0.8&1
\\\hline\textbf{2 Marks Count}&0&0&1&0&0&0&0&0.2&1
\\\hline\textbf{Total Marks}&1&1&2&1&1&1&\bf{1}&\bf{1.2}&\bf{2}\\\hline
\end{array}}}$$

Most answered questions in Calculus

35 votes
7 answers
1
Let $f(x)=x^{-\left(\frac{1}{3}\right)}$ and $A$ denote the area of region bounded by $f(x)$ and the X-axis, when $x$ varies from $-1$ to $1$. Which of the following statements is/are TRUE? $f$ is continuous in $[-1, 1]$ $f$ is not bounded in $[-1, 1]$ $A$ is nonzero and finite II only III only II and III only I, II and III
asked Feb 12, 2015 in Calculus jothee 7.7k views
53 votes
7 answers
2
A function $f(x)$ is continuous in the interval $[0,2]$. It is known that $f(0) = f(2) = -1$ and $f(1) = 1$. Which one of the following statements must be true? There exists a $y$ in the interval $(0,1)$ such that $f(y) = f(y+1)$ For every $y$ ... maximum value of the function in the interval $(0,2)$ is $1$ There exists a $y$ in the interval $(0,1)$ such that $f(y)$ = $-f(2-y)$
asked Sep 28, 2014 in Calculus jothee 9.9k views
8 votes
6 answers
3
Compute $\displaystyle \lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}$ $1$ $53/12$ $108/7$ Limit does not exist
asked Feb 7, 2019 in Calculus Arjun 3.1k views
19 votes
6 answers
4
The value of $\int^{\pi/4} _0 x \cos(x^2) dx$ correct to three decimal places (assuming that $\pi = 3.14$) is ____
asked Feb 14, 2018 in Calculus gatecse 7.4k views
9 votes
6 answers
5
Solve min $x^{2}+y^{2}$ subject to $\begin {align*} x + y &\geq 10,\\ 2x + 3y &\geq 20,\\ x &\geq 4,\\ y &\geq 4. \end{align*}$ $32$ $50$ $52$ $100$ None of the above
asked Nov 9, 2015 in Calculus makhdoom ghaya 831 views
13 votes
6 answers
6
The limit of $\dfrac{10^{n}}{n!}$ as $n \to \infty$ is. $0$ $1$ $e$ $10$ $\infty$
asked Oct 2, 2015 in Calculus makhdoom ghaya 1.5k views
16 votes
6 answers
7
$\lim_{x \to \infty}\frac{x-\sin x}{x+\cos x}$ equals $1$ $-1$ $\infty$ $-\infty$
asked Sep 11, 2014 in Calculus Kathleen 4.4k views
5 votes
5 answers
8
Consider the matrix $A = \begin{bmatrix} \frac{1}{2} &\frac{1}{2} & 0\\ 0& \frac{3}{4} & \frac{1}{4}\\ 0& \frac{1}{4} & \frac{3}{4} \end{bmatrix}$ What is $\lim_{n→\infty}$A^n$ ? $\begin{bmatrix} \ 0 & 0 & 0\\ 0& 0 & 0\\ 0 & 0 & 0 \end{bmatrix}$ $\begin{bmatrix} ... $\text{The limit exists, but it is none of the above}$
asked Dec 18, 2018 in Calculus Arjun 997 views
4 votes
5 answers
9
19 votes
5 answers
10
The value of $\lim_{x\rightarrow 1} \frac{x^{7}-2x^{5}+1}{x^{3}-3x^{2}+2}$ is $0$ is $-1$ is $1$ does not exist
asked Feb 14, 2017 in Calculus Arjun 3.3k views
24 votes
5 answers
11
Let $f(x)$ be a polynomial and $g(x)=f'(x)$ be its derivative. If the degree of $(f(x)+f(-x))$ is $10$, then the degree of $(g(x) - g(-x))$ is __________.
asked Feb 12, 2016 in Calculus Akash Kanase 5.2k views
13 votes
5 answers
12
Suppose that $f(x)$ is a continuous function such that $0.4 \leq f(x) \leq 0.6$ for $0 \leq x \leq 1$. Which of the following is always true? $f(0.5) = 0.5$. There exists $x$ between $0$ and $1$ such that $f(x) = 0.8x$. There exists $x$ between $0$ and $0.5$ such that $f(x) = x$. $f(0.5) > 0.5$. None of the above statements are always true.
asked Dec 5, 2015 in Calculus makhdoom ghaya 1.1k views
23 votes
5 answers
13
The value of $\lim_{x \rightarrow \infty} (1+x^2)^{e^{-x}}$ is $0$ $\frac{1}{2}$ $1$ $\infty$
asked Feb 14, 2015 in Calculus jothee 6.1k views
16 votes
5 answers
14
$\lim_{x\rightarrow \infty } x^{ \tfrac{1}{x}}$ is $\infty $ 0 1 Not defined
asked Feb 11, 2015 in Calculus makhdoom ghaya 4k views
18 votes
5 answers
15
Given $i = \sqrt{-1}$, what will be the evaluation of the definite integral $\int \limits_0^{\pi/2} \dfrac{\cos x +i \sin x} {\cos x - i \sin x} dx$ ? $0$ $2$ $-i$ $i$
asked Sep 29, 2014 in Calculus jothee 4.3k views
23 votes
5 answers
16
If $\int \limits_0^{2 \pi} |x \: \sin x| dx=k\pi$, then the value of $k$ is equal to ______.
asked Sep 28, 2014 in Calculus jothee 4.9k views
20 votes
5 answers
17
What is the value of $\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{2n}$ ? 0 $e^{-2}$ $e^{-1/2}$ 1
asked Sep 21, 2014 in Calculus gatecse 3.6k views
4 votes
4 answers
18
$\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$
asked Sep 23, 2019 in Calculus Arjun 423 views
18 votes
4 answers
19
The function $y=|2 - 3x|$​ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{3}{2}$ is continuous $∀ x ∈ R$ and differentiable $∀ x ∈ R$ except at $x=\frac{2}{3}$ is continuous $∀ x ∈ R$ except $x=3$ and differentiable $∀ x ∈ R$
asked Mar 20, 2016 in Calculus Anuraag Nayak 2.5k views
20 votes
4 answers
20
$\lim _{x\rightarrow 4}\frac{\sin(x-4)}{x-4}$=____.
asked Feb 12, 2016 in Calculus Sandeep Singh 3.6k views
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