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Most viewed questions in Calculus
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41
TIFR ECE 2021 | Question: 3
Consider the following statements: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$. $\lim _{x \rightarrow 0} \frac{1-\cos x}{x}=1$. Which of the following is $\text{TRUE?}$ Only Statement $1$ ... $1$ and $3$ are correct. All of Statements $1, 2,$ and $3$ are correct. None of the three Statements $1,2,$ and $3$ are correct.
Consider the following statements:$\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=1$.$\lim _{x \rightarrow 0} \frac{1-\cos x}...
admin
46.4k
points
98
views
admin
asked
Nov 30, 2022
Calculus
tifrece2021
calculus
limits
+
–
0
votes
0
answers
42
GATE ECE 2014 Set 3 | Question: 1
The maximum value of the function $f(x) = \text{ln } (1+x) – x $ (where $x >-1$) occurs at $x=$_______.
The maximum value of the function $f(x) = \text{ln } (1+x) – x $ (where $x >-1$) occurs at $x=$_______.
Milicevic3306
16.0k
points
97
views
Milicevic3306
asked
Mar 26, 2018
Calculus
gate2014-ec-3
calculus
maxima-minima
numerical-answers
+
–
0
votes
0
answers
43
GATE ECE 2014 Set 2 | Question: 4
The value of $\lim_{x\rightarrow \infty }(1 +\tfrac{1}{x})^{x}$ is $\text{ln }2$ $1.0$ $e$ $\infty$
The value of $$\lim_{x\rightarrow \infty }(1 +\tfrac{1}{x})^{x}$$ is$\text{ln }2$$1.0$$e$$\infty$
Milicevic3306
16.0k
points
97
views
Milicevic3306
asked
Mar 26, 2018
Calculus
gate2014-ec-2
calculus
limits
+
–
1
votes
0
answers
44
TIFR ECE 2014 | Question: 13
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y)$. Then which of the following is $\text{TRUE?}$ $f(x) \leq f(y)$ whenever ... $f$ and $g$ are both convex, then $\min \{f, g\}$ is also convex. For a random variable $X, E(f(X)) \geq f(E(X))$.
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y...
admin
46.4k
points
96
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
functions
+
–
1
votes
0
answers
45
TIFR ECE 2010 | Question: 2
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is $\alpha>1 /(2 \pi)$ $\alpha \geq 1 / 6$ $\alpha \leq 1 /(2 \pi)$ $\alpha=1 / 4$ None of the above
For $x \in[0, \pi / 2], \alpha$ for which $\sin (x) \geq x-\alpha x^{3}$ is$\alpha>1 /(2 \pi)$$\alpha \geq 1 / 6$$\alpha \leq 1 /(2 \pi)$$\alpha=1 / 4$None of the above
admin
46.4k
points
96
views
admin
asked
Nov 30, 2022
Calculus
tifr2010
calculus
maxima-minima
+
–
1
votes
0
answers
46
TIFR ECE 2021 | Question: 7
Consider the function \[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\] where $\log _{e}(x)$ denotes the natural logarithm of $x$. Which of the following is true: The function $f(y)$ ... $y \geq 1$. The derivative of function $f(y)$ does not exist at $y=1$.
Consider the function\[f(y)=\int_{1}^{y} \frac{1}{1+x^{2}} d x-\log _{e}(1+y)\]where $\log _{e}(x)$ denotes the natural logarithm of $x$.Which of the following is true:Th...
admin
46.4k
points
96
views
admin
asked
Nov 30, 2022
Calculus
tifrece2021
calculus
definite-integrals
+
–
1
votes
0
answers
47
TIFR ECE 2013 | Question: 20
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is convex if for $x, y \in \mathbb{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq \alpha f(x)+(1-\alpha) f(y)$. Which of the following is not convex: $x^{2}$ $x^{3}$ $x$ $x^{4}$ $\mathrm{e}^{x}$
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is convex if for $x, y \in \mathbb{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq \alpha f(x)+(1-\alpha) f(y)$.Which...
admin
46.4k
points
92
views
admin
asked
Dec 12, 2022
Calculus
tifr2013
calculus
functions
+
–
1
votes
0
answers
48
TIFR ECE 2011 | Question: 8
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true $\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$. $\max _{x} \min _{y} f(x, y) \geq \min _{y} \max _{x} f(x, y)$ ... $\max _{x} \min _{y} f(x, y)=\min _{y} \max _{x} f(x, y)+\min _{y} \min _{x} f(x, y)$. None of the above.
Let $f(x, y)$ be a function in two variables $x, y$. Then which of the following is true$\max _{x} \min _{y} f(x, y) \leq \min _{y} \max _{x} f(x, y)$.$\max _{x} \min _{y...
admin
46.4k
points
92
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
maxima-minima
+
–
0
votes
0
answers
49
GATE ECE 2014 Set 2 | Question: 3
For $0 \leq t < \infty ,$ the maximum value of the function $f(t)= e^{-t}-2e^{-2t}$ occurs at $t= log_{e}4$ $t= log_{e}2$ $t= 0$ $t= log_{e}8$
For $0 \leq t < \infty ,$ the maximum value of the function $f(t)= e^{-t}-2e^{-2t}$ occurs at$t= log_{e}4$$t= log_{e}2$$t= 0$$t= log_{e}8$
Milicevic3306
16.0k
points
92
views
Milicevic3306
asked
Mar 26, 2018
Calculus
gate2014-ec-2
calculus
maxima-minima
+
–
0
votes
0
answers
50
GATE ECE 2014 Set 1 | Question: 26
The Taylor series expansion of $3\sin x + 2 \cos x$ is $2 + 3x-x^{2} – \frac{x^{3}}{2} + \dots$ $2 – 3x + x^{2} – \frac{x^{3}}{2} + \dots$ $2 + 3x + x^{2} + \frac{x^{3}}{2} + \dots$ $2 – 3x – x^{2} + \frac{x^{3}}{2} + \dots$
The Taylor series expansion of $3\sin x + 2 \cos x$ is$2 + 3x-x^{2} – \frac{x^{3}}{2} + \dots$$2 – 3x + x^{2} – \frac{x^{3}}{2} + \dots$$2 + 3x + x^{2} + \frac...
Milicevic3306
16.0k
points
92
views
Milicevic3306
asked
Mar 25, 2018
Calculus
gate2014-ec-1
calculus
taylor-series
+
–
1
votes
0
answers
51
TIFR ECE 2012 | Question: 2
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then \[ \lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n} \] is $0$ $\infty$ $\alpha_{k}$ $\alpha_{1}$ $\max _{j}|\alpha_{j}|$
Let $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{k}$ be complex numbers. Then\[\lim _{n \rightarrow \infty}\left|\sum_{i=1}^{k} \alpha_{i}^{n}\right|^{1 / n}\]is$0$$\infty$$\...
admin
46.4k
points
90
views
admin
asked
Dec 8, 2022
Calculus
tifr2012
calculus
limits
+
–
0
votes
0
answers
52
GATE ECE 2017 Set 2 | Question: 26
The values of the integrals $\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dy \right )dx$ and $\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dx \right )dy$ are same and equal to $0.5$ same and equal to $-0.5$ $0.5$ and $-0.5$, respectively $-0.5$ and $0.5$, respectively
The values of the integrals $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dy \right )dx$$ and $$\int_{0}^{1}\left ( \int_{0}^{1}\frac{x-y}{(x+y)^3}dx \right )dy$$ ...
admin
46.4k
points
90
views
admin
asked
Nov 23, 2017
Calculus
gate2017-ec-2
calculus
definite-integrals
+
–
1
votes
0
answers
53
TIFR ECE 2010 | Question: 17
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit \[ \lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n} \] is $0$ $\infty$ $a_{k}$ $a_{1}$ $\left(\sum_{i=1}^{k} a_{k}\right) / k$
Let $a_{1} \geq a_{2} \geq \cdots \geq a_{k} \geq 0$. Then the limit\[\lim _{n \rightarrow \infty}\left(\sum_{i=1}^{k} a_{i}^{n}\right)^{1 / n}\]is$0$$\infty$$a_{k}$$a_{1...
admin
46.4k
points
89
views
admin
asked
Nov 30, 2022
Calculus
tifr2010
calculus
limits
+
–
1
votes
0
answers
54
TIFR ECE 2017 | Question: 14
Consider the positive integer sequence \[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\] Which of the following statements is $\text{TRUE?}$ For every $M>0$, there exists an $n$ such that $x_{n}>M$ ... and then increases with $n \geq 1$ Sequence $\left\{x_{n}\right\}$ eventually converges to zero as $n \rightarrow \infty$ None of the above
Consider the positive integer sequence\[x_{n}=n^{50} e^{-(\log (n))^{3 / 2}}, \quad n=1,2,3, \ldots\]Which of the following statements is $\text{TRUE?}$For every $M>0$, t...
admin
46.4k
points
88
views
admin
asked
Nov 29, 2022
Calculus
tifrece2017
calculus
maxima-minima
+
–
0
votes
0
answers
55
GATE ECE 2012 | Question: 46
The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1,6]$ is $21$ $25$ $41$ $46$
The maximum value of $f(x)=x^3-9x^2+24x+5$ in the interval $[1,6]$ is$21$$25$$41$$46$
Milicevic3306
16.0k
points
88
views
Milicevic3306
asked
Mar 25, 2018
Calculus
gate2012-ec
calculus
maxima-minima
+
–
1
votes
0
answers
56
TIFR ECE 2014 | Question: 2
Evaluate the limit \[ \lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} . \] $e$ $1$ $2^{\frac{1}{3}}$ $0$ None of the above
Evaluate the limit\[\lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} .\]$e$$1$$2^{\frac{1}{3}}$$0$None of the above
admin
46.4k
points
87
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
+
–
1
votes
0
answers
57
TIFR ECE 2016 | Question: 1
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible? $\int_{1}^{\infty} x f(x) \mathrm{d} x=\infty$ ... $\int_{1}^{\infty} \frac{f(x)}{1+\ln x} \mathrm{~d} x=\infty$ None of the above
Suppose $f(x)=c x^{-\alpha}$ for some $c>0$ and $\alpha>0$ such that $\int_{1}^{\infty} f(x) \mathrm{d} x=1$. Then, which of the following is possible?$\int_{1}^{\infty} ...
admin
46.4k
points
87
views
admin
asked
Nov 29, 2022
Calculus
tifrece2016
calculus
definite-integrals
+
–
0
votes
0
answers
58
GATE ECE 2012 | Question: 25
If $x=\sqrt{-1}$, then the value of $x^x$ is $e^{\frac{-\pi}{2}}$ $e^{\frac{\pi}{2}}$ $x$ $1$
If $x=\sqrt{-1}$, then the value of $x^x$ is$e^{\frac{-\pi}{2}}$$e^{\frac{\pi}{2}}$$x$$1$
Milicevic3306
16.0k
points
85
views
Milicevic3306
asked
Mar 25, 2018
Calculus
gate2012-ec
calculus
+
–
1
votes
0
answers
59
TIFR ECE 2014 | Question: 20
What is \[ \lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ? \] $0$ $\pi / 2$ $1 / \sqrt{2}$ $2 / \pi$ None of the above.
What is\[\lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ?\]$0$$\pi / 2$$1 / \sqrt{2}$$2 / \pi$None of the above....
admin
46.4k
points
84
views
admin
asked
Dec 14, 2022
Calculus
tifr2014
calculus
limits
+
–
1
votes
0
answers
60
TIFR ECE 2022 | Question: 4
Evaluate the value of \[\max \left(x^{2}+(1-y)^{2}\right),\] where the maximisation above is over $x$ and $y$ such that $0 \leq x \leq y \leq 1$. $0$ $2$ $1 / 2$ $1 / 4$ $1$
Evaluate the value of\[\max \left(x^{2}+(1-y)^{2}\right),\]where the maximisation above is over $x$ and $y$ such that $0 \leq x \leq y \leq 1$.$0$$2$$1 / 2$$1 / 4$$1$
admin
46.4k
points
83
views
admin
asked
Nov 30, 2022
Calculus
tifrece2022
calculus
maxima-minima
+
–
1
votes
0
answers
61
TIFR ECE 2021 | Question: 10
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem \[d_{1}=\min _{a_{1} \in \mathbb{R}}\left\|\vec{u}-a_{1} \vec{v}_{1}\right\|,\] where we denote the length ... $\left\|\vec{u}-\left(\vec{p}_{2}-\vec{p}_{1}\right)\right\|$ $0$
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem\[d_{1}=\min _{a_...
admin
46.4k
points
83
views
admin
asked
Nov 30, 2022
Calculus
tifrece2021
vector-analysis
vector-in-planes
+
–
1
votes
0
answers
62
TIFR ECE 2018 | Question: 4
Consider \[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\] What can we say about $\lim _{x \rightarrow \infty} f(x)$ ? The function $f(x)$ does not have a limit as $x \rightarrow \infty$ ... $\lim _{x \rightarrow \infty} f(x)=e^{1 / 2}$ $\lim _{x \rightarrow \infty} f(x)=0$ $\lim _{x \rightarrow \infty} f(x)=\infty$
Consider\[f(x)=\frac{(x \log x+x)^{5}(1+2 / x)^{x}}{(x+1 / x)^{5}(\log x+1 / \log x)^{6}}\]What can we say about $\lim _{x \rightarrow \infty} f(x)$ ?The function $f(x)$ ...
admin
46.4k
points
82
views
admin
asked
Nov 29, 2022
Calculus
tifrece2018
calculus
limits
+
–
0
votes
0
answers
63
GATE ECE 2014 Set 4 | Question: 29
For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the triangle, the angle between the hypotenuse and the side is $12^{\circ}$ $36^{\circ}$ $60^{\circ}$ $45^{\circ}$
For a right angled triangle, if the sum of the lengths of the hypotenuse and a side is kept constant, in order to have maximum area of the triangle, the triangle, the ang...
Milicevic3306
16.0k
points
81
views
Milicevic3306
asked
Mar 26, 2018
Calculus
gate2014-ec-4
calculus
maxima-minima
+
–
1
votes
0
answers
64
TIFR ECE 2022 | Question: 6
Consider a degree-$5$ polynomial function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$. If $f$ exhibits at least four local maxima, which of the following is necessarily true? (Note: A local maximum is a point where the function value is the maximum in a ... derivative of $f(x)$ is negative for some $x \in[0,100]$ $f$ has exactly $4$ local maxima None of the above
Consider a degree-$5$ polynomial function $f:(-\infty, \infty) \rightarrow(-\infty, \infty)$. If $f$ exhibits at least four local maxima, which of the following is necess...
admin
46.4k
points
80
views
admin
asked
Nov 30, 2022
Calculus
tifrece2022
calculus
maxima-minima
+
–
1
votes
0
answers
65
TIFR ECE 2011 | Question: 4
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$ There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$. There exists an $n_{0}$ ... $n>n_{0},\left|\frac{x_{n}}{x}\right| \leq \epsilon$ for any $\epsilon>0$. None of the above.
Let $\lim _{n \rightarrow \infty} x_{n}=x$. Then which of the following is $\text{TRUE.}$There exists an $n_{0}$, such that for all $n>n_{0},\left|x_{n}-x\right|=0$.There...
admin
46.4k
points
73
views
admin
asked
Dec 5, 2022
Calculus
tifr2011
calculus
limits
+
–
1
votes
0
answers
66
TIFR ECE 2020 | Question: 4
Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions that are continuous and differentiable. Consider the following statements: $\min \{f, g\}$ is continuous $\max \{f, g\}$ is continuous $\max \{f, g\}$ is differentiable Which ... is correct Only statement $2$ is correct Only statement $3$ is correct Only statements $1$ and $2$ are correct None of the above
Let $f, g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions that are continuous and differentiable. Consider the following statements:$\min \{f, g\}$ is continuous$\ma...
admin
46.4k
points
47
views
admin
asked
Nov 30, 2022
Calculus
tifrece2020
calculus
continuity-and-differentiability
+
–
1
votes
0
answers
67
GATE ECE 2010 | Question: 26
If $e^{y}=x^{\frac{1}{x}}$, then $y$ has a maximum at $x=e$ minimum at $x=e$ maximum at $x=e^{-1}$ minimum at $x=e^{-1}$
If $e^{y}=x^{\frac{1}{x}}$, then $y$ has amaximum at $x=e$minimum at $x=e$maximum at $x=e^{-1}$minimum at $x=e^{-1}$
admin
46.4k
points
43
views
admin
asked
Sep 15, 2022
Calculus
gate2010-ec
calculus
maxima-minima
+
–
1
votes
0
answers
68
TIFR ECE 2013 | Question: 19
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$. Every odd degree polynomial has a real root. Every odd degree polynomial has at least one complex root. Every even degree polynomial has at least one complex root. Every even degree polynomial has a real root. None of the above
Which of the following is true for polynomials defined over real numbers $\mathbb{R}$.Every odd degree polynomial has a real root.Every odd degree polynomial has at least...
admin
46.4k
points
40
views
admin
asked
Dec 12, 2022
Calculus
tifr2013
calculus
polynomials
+
–
1
votes
0
answers
69
TIFR ECE 2019 | Question: 3
Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which of the following is $\text{TRUE}?$ $\lim _{x \rightarrow \infty} f(x)=1$ ... $1-\epsilon$ $\max _{x \geq 1} f(x)=1$ None of the above
Consider a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f(x)=1$ if $x$ is rational, and $f(x)=1-\epsilon,$ where $0<\epsilon<1$, if $x$ is irrational. Which...
admin
46.4k
points
33
views
admin
asked
Nov 30, 2022
Calculus
tifrece2019
calculus
limits
+
–
1
votes
0
answers
70
TIFR ECE 2019 | Question: 5
Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$ $4$ $2$ $3$ $5$
Consider the function $f(x)=e^{x^{2}}-8 x^{2}$ for all $x$ on the real line. For how many distinct values of $x$ do we have $f(x)=0?$ $1$$4$$2$$3$$5$
admin
46.4k
points
32
views
admin
asked
Nov 30, 2022
Calculus
tifrece2019
calculus
functions
+
–
1
votes
0
answers
71
TIFR ECE 2019 | Question: 4
Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the following is $\text{TRUE}?$ $f(x)$ is ... differentiable $f(x)$ is continuous and differentiable $f(x)$ is continuous but not differentiable $f(x)$ is neither continuous nor differentiable None of the above
Let $f(x)=\sqrt{x^{2}-4 x+4},$ for $x \in(-\infty, \infty)$. Here, $\sqrt{y}$ denotes the non-negative square root of $y$ when $y$ is non-negative. Then, which of the fol...
admin
46.4k
points
29
views
admin
asked
Nov 30, 2022
Calculus
tifrece2019
calculus
continuity-and-differentiability
+
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