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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

0 votes
2 answers
271
ISI2016-PCB-A-2
Let $n$ be a fixed positive integer. For any real number $x,$ if for some integer $q,$ $x=qn+r, \: \: \: 0 \leq r < n,$ then we define $x \text{ mod } n=r$. Specify the points of discontinuity of the function $f(x)=x \text{ mod } 3$ with proper reasoning.
Let $n$ be a fixed positive integer. For any real number $x,$ if for some integer $q,$ $$x=qn+r, \: \: \: 0 \leq r < n,$$ then we define $x \text{ mod } n=r$.Specify the...
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272
calculus
what is difference between this and in questions if asked to find derivative or check differentiable or not, which formula should i use? 1) $lim_{h\rightarrow 0} \frac{f(a+h)-f(a)}{h}$ 2) $lim_{x\rightarrow a} \frac{f(x)-f(a)}{x-a}$
what is difference between this and in questions if asked to find derivative or check differentiable or not, which formula should i use?1) $lim_{h\rightarrow 0} \frac{f(a...
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0 votes
2 answers
273
ISI2017-MMA-4
Let $S\subseteq \mathbb{R}$. Consider the statement “There exists a continuous function $f:S\rightarrow S$ such that $f(x) \neq x$ for all $x \in S.$ ” This statement is false if $S$ equals $[2,3]$ $(2,3]$ $[-3,-2] \cup [2,3]$ $(-\infty,\infty)$
Let $S\subseteq \mathbb{R}$. Consider the statement “There exists a continuous function $f:S\rightarrow S$ such that $f(x) \neq x$ for all $x \in S.$ ”This statement ...
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0 votes
1 answer
274
ISI2017-MMA-9
A function $y(x)$ that satisfies $\dfrac{dy}{dx}+4xy=x$ with the boundary condition $y(0)=0$ is $y(x)=(1-e^x)$ $y(x)=\frac{1}{4}(1-e^{-2x^2})$ $y(x)=\frac{1}{4}(1-e^{2x^2})$ $y(x)=\frac{1}{4}(1-\cos x)$
A function $y(x)$ that satisfies $\dfrac{dy}{dx}+4xy=x$ with the boundary condition $y(0)=0$ is$y(x)=(1-e^x)$$y(x)=\frac{1}{4}(1-e^{-2x^2})$$y(x)=\frac{1}{4}(1-e^{2x^2})$...
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0 votes
0 answers
275
ISI2017-MMA-14
Let $(v_n)$ be a sequence defined by $v_1=1$ and $v_{n+1}=\sqrt{v_n^2 +(\frac{1}{5})^n}$ for $n\geq1$. Then $\lim _{n\rightarrow \infty} v_n$ is $\sqrt{5/3}$ $\sqrt{5/4}$ $1$ nonexistent
Let $(v_n)$ be a sequence defined by $v_1=1$ and $v_{n+1}=\sqrt{v_n^2 +(\frac{1}{5})^n}$ for $n\geq1$. Then $\lim _{n\rightarrow \infty} v_n$ is$\sqrt{5/3}$$\sqrt{5/4}$$1...
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0 votes
0 answers
277
ISI2016-MMA-5
Let $ f(x, y) = \begin{cases} \dfrac{x^2y}{x^4+y^2}, & \text{ if } (x, y) \neq (0, 0) \\ 0 & \text{ if } (x, y) = (0, 0) \end{cases}$ Then $\lim_{(x, y) \rightarrow (0,0)}$f(x,y)$ equals $0$ equals $1$ equals $2$ does not exist
Let $ f(x, y) = \begin{cases} \dfrac{x^2y}{x^4+y^2}, & \text{ if } (x, y) \neq (0, 0) \\ 0 & \text{ if } (x, y) = (0, 0) \end{cases}$Then $\lim_{(x, y) \rightarrow (0,0)}...
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0 answers
278
ISI2016-MMA-7
The set of value(s) of $\alpha$ for which $y(t)=t^{\alpha}$ is a solution to the differential equation $t^2 \frac{d^2y}{dx^2}-2t \frac{dy}{dx}+2y =0 \: \text{ for } t>0$ is $\{1\}$ $\{1, -1\}$ $\{1, 2\}$ $\{-1, 2\}$
The set of value(s) of $\alpha$ for which $y(t)=t^{\alpha}$ is a solution to the differential equation $$t^2 \frac{d^2y}{dx^2}-2t \frac{dy}{dx}+2y =0 \: \text{ for } t>0$...
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2 votes
2 answers
279
ISI2016-MMA-8
Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $g'(x^2)=x^3$ for all $x>0$ and $g(1) =1$. Then $g(4)$ equals $64/5$ $32/5$ $37/5$ $67/5$
Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable with $g'(x^2)=x^3$ for all $x>0$ and $g(1) =1$. Then $g(4)$ equals$64/5$$32/5$$37/5$$67/5$
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1 votes
0 answers
281
ISI2016-MMA-21
Let $A=\{1, 2, 3, 4, 5, 6, 7, 8 \}$. How many functions $f: A \rightarrow A$ can be defined such that $f(1)< f(2) < f(3)$? $\begin{pmatrix} 8 \\ 3 \end{pmatrix}$ $\begin{pmatrix} 8 \\ 3 \end{pmatrix} 5^8$ $\begin{pmatrix} 8 \\ 3 \end{pmatrix} 8^5$ $\frac{8!}{3!}$
Let $A=\{1, 2, 3, 4, 5, 6, 7, 8 \}$. How many functions $f: A \rightarrow A$ can be defined such that $f(1)< f(2) < f(3)$?$\begin{pmatrix} 8 \\ 3 \end{pmatrix}$$\begin{pm...
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0 votes
0 answers
282
ISI2016-MMA-23
Given that $\int_{-\infty}^{\infty} e^{-x^2/2} dx = \sqrt{2 \pi}$, what is the value of $\int_{- \infty}^{\infty} \mid x \mid ^{-1/2} e^{- \mid x \mid} dx$? $0$ $\sqrt{\pi}$ $2 \sqrt{\pi}$ $\infty$
Given that $\int_{-\infty}^{\infty} e^{-x^2/2} dx = \sqrt{2 \pi}$, what is the value of $\int_{- \infty}^{\infty} \mid x \mid ^{-1/2} e^{- \mid x \mid} dx$?$0$$\sqrt{\pi}...
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0 answers
285
limit
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286
limit
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287
definite integral when infinity is multiplied by zero
in this question, infinity is multiplied by zero generally infinity multiplied by zero gives undefined result.
in this question, infinity is multiplied by zero generally infinity multiplied by zero gives undefined result.
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288
vectors x1= 5i-2j+k and x2=4i-j+2k are transformed into y1=ax1 and y2=ax2.
vectors x1= 5i-2j+k and x2=4i-j+2k are transformed into y1=ax1 and y2=ax2.
vectors x1= 5i-2j+k and x2=4i-j+2k are transformed into y1=ax1 and y2=ax2.
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0 votes
1 answer
289
integration
now if we use sin^2 x=(1-cos2x)/2 then we are not getting solution like this please check
now if we use sin^2 x=(1-cos2x)/2then we are not getting solution like this please check
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290
limit
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