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Recent questions and answers in Calculus
+11
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3
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1
GATE19961.6
The formula used to compute an approximation for the second derivative of a function $f$ at a point $X_0$ is $\dfrac{f(x_0 +h) + f(x_0 – h)}{2}$ $\dfrac{f(x_0 +h)  f(x_0 – h)}{2h}$ $\dfrac{f(x_0 +h) + 2f(x_0) + f(x_0 – h)}{h^2}$ $\dfrac{f(x_0 +h)  2f(x_0) + f(x_0 – h)}{h^2}$
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gate1996
calculus
differentiability
normal
+1
vote
1
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2
Mean Value Theorem
f(x) is a differentiable function that satisfies 5 ≤ f′(x) ≤ 14 for all x. Let a and b be the maximum and minimum values, respectively, that f(11)−f(3) can possibly have, then what is the value of a+b?
answered
Oct 10
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Calculus
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Nirmal Gaur
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0
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3
ISI2016DCG45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:xx\:\tan\:x}{\sin\:x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these
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Oct 2
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18
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isi2016dcg
limits
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1
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4
ISI2014DCG39
The function $f(x) = x^{1/x}, \: x \neq 0$ has a minimum at $x=e$; a maximum at $x=e$; neither a maximum nor a minimum at $x=e$; None of the above
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Sep 24
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Calculus
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isi2014dcg
maximaminima
calculus
0
votes
1
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5
ISI2014DCG12
The integral $\int _0^{\frac{\pi}{2}} \frac{\sin^{50} x}{\sin^{50}x +\cos^{50}x} dx$ equals $\frac{3 \pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{4}$ none of these
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Sep 24
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isi2014dcg
calculus
definiteintegrals
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+2
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3
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6
ISI2017MMA13
An even function $f(x)$ has left derivative $5$ at $x=0$. Then the right derivative of $f(x)$ at $x=0$ need not exist the right derivative of $f(x)$ at $x=0$ exists and is equal to $5$ the right derivative of $f(x)$ at $x=0$ exists and is equal to $5$ none of the above is necessarily true
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Sep 22
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Calculus
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techbd123
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isi2017mma
engineeringmathematics
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+16
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5
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7
GATE2016202
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 __________.
answered
Aug 25
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Calculus
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King Suleiman
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gate20162
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2
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8
Continuity and Differentiability
If the function f(x) =[(x2)3 /a] sin(x2) + acos(x2), [.] denotes greatest integer function, is continuous & differentiable in (4,6) then find ‘a’ range: (A) a ϵ (∞,∞) (B) a ϵ [64, ∞) (C) a ϵ [128, ∞) (D) Not defined
answered
Aug 20
in
Calculus
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moharanaguruprasad
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0
votes
1
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9
Gate 2002  ME
Which of the following functions is not differentiable in the domain $[1,1]$ ? (a) $f(x) = x^2$ (b) $f(x) = x1$ (c) $f(x) = 2$ (d) $f(x) = Maximum (x,x)$
answered
Jul 24
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Calculus
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35
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76
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engineeringmathematics
usergate2002
usermod
calculus
differentiability
0
votes
1
answer
10
ISI2018MMA28
Consider the following functions $f(x)=\left\{\begin{matrix} 1 &, if\ x \leq 1 \\ 0 & ,if\ x>1 \end{matrix}\right.$ ... at $ 1$ $h_2$ is continuous everywhere and $h_1$ has discontinuity at $ 2$ $h_1$ has discontinuity at $ 2$ and $h_2$ has discontinuity at $ 1$.
answered
Jul 24
in
Calculus
by
Arif_Faizan
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35
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58
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isi2018mma
engineeringmathematics
calculus
continuity
+1
vote
2
answers
11
TIFR2014A18
We are given a collection of real numbers where a real number $a_{i}\neq 0$ occurs $n_{i}$ times. Let the collection be enumerated as $\left\{x_{1}, x_{2},...x_{n}\right\}$ so that $x_{1}=x_{2}=...=x_{n_{1}}=a_{1}$ and so on, and $n=\sum _{i}n_{i}$ is finite. What is ... $\min_{i} a_{i}$ $\min_{i} \left(n_{i}a_{i}\right)$ $\max_{i} a_{i}$ None of the above.
answered
Jun 17
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Calculus
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Arjun
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tifr2014
limits
+2
votes
2
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12
TIFR2019A13
Consider the integral $\int^{1}_{0} \frac{x^{300}}{1+x^2+x^3} dx$ What is the value of this integral correct up to two decimal places? $0.00$ $0.02$ $0.10$ $0.33$ $1.00$
answered
Jun 9
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Calculus
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Arjun
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235
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tifr2019
engineeringmathematics
calculus
integration
+19
votes
3
answers
13
GATE201416
Let the function ... There exists $\theta \in (\frac{\pi}{6},\frac{\pi}{3})$ such that $f'(\theta)\neq 0$ I only II only Both I and II Neither I Nor II
answered
Jun 8
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Calculus
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97apoorva singh
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153
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3.7k
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gate20141
calculus
differentiability
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0
votes
1
answer
14
Differentiability
$\varphi \left ( x \right )=x^{2}\cos \frac{1}{x}$ when $x\neq 0$ $=0$ when $x=0$ Is it differentiable at $x=0$? Is it continuous ?
answered
Jun 7
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Calculus
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ankitgupta.1729
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calculus
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0
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1
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15
Maths: Limits
$\LARGE \lim_{n \rightarrow \infty} \frac{n^{\frac{3}{4}}}{log^9 n}$
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May 27
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Calculus
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Mk Utkarsh
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engineeringmathematics
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limits
+12
votes
5
answers
16
GATE201514
$\lim_{x\rightarrow \infty } x^{ \tfrac{1}{x}}$ is $\infty $ 0 1 Not defined
answered
May 26
in
Calculus
by
SALIL KUMAR
(
25
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2k
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gate20151
calculus
limits
normal
+3
votes
1
answer
17
Limits
answered
May 16
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Calculus
by
Satbir
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limits
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+3
votes
1
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18
MadeEasy Test Series 2018: Calculus  Limits
The value of $\lim_{x\rightarrow \infty }\left ( \frac{4^{x+2} + 3^{x}}{4^{x2}} \right )$ is ____________
answered
May 16
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Calculus
by
Satbir
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engineeringmathematics
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madeeasytestseries
+3
votes
6
answers
19
GATE201913
Compute $\displaystyle \lim_{x \rightarrow 3} \frac{x^481}{2x^25x3}$ $1$ $53/12$ $108/7$ Limit does not exist
answered
May 13
in
Calculus
by
Bikki_gupta
(
87
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1.7k
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gate2019
engineeringmathematics
calculus
limits
0
votes
1
answer
20
ISI2019MMA21
A function $f:\mathbb{R^2} \rightarrow \mathbb{R}$ is called degenerate on $x_i$, if $f(x_1,x_2)$ remains constant when $x_i$ varies $(i=1,2)$. Define $f(x_1,x_2) = \mid 2^{\pi _i/x_1} \mid ^{x_2} \text{ for } x_1 \neq 0$, where $i = \sqrt {1}$. ... $x_1$ but not on $x_2$ $f$ is degenerate on $x_2$ but not on $x_1$ $f$ is neither degenerate on $x_1$ nor on $x_2$
answered
May 13
in
Calculus
by
pratekag
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393
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isi2019mma
engineeringmathematics
calculus
0
votes
0
answers
21
ISI2018MMA30
Consider the function $f(x)=\bigg(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^n}{n!}\bigg)e^{x}$, where $n\geq4$ is a positive integer. Which of the following statements is correct? $f$ has no local maximum For every $n$, $f$ has a local maximum at $x = 0$ ... at $x = 0$ when $n$ is even $f$ has no local extremum if $n$ is even and has a local maximum at $x = 0$ when $n$ is odd.
asked
May 11
in
Calculus
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akash.dinkar12
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42
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isi2018mma
engineeringmathematics
calculus
maximaminima
0
votes
0
answers
22
ISI2018MMA29
Let $f$ be a continuous function with $f(1) = 1$. Define $F(t)=\int_{t}^{t^2}f(x)dx$. The value of $F’(1)$ is $2$ $1$ $1$ $2$
asked
May 11
in
Calculus
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akash.dinkar12
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50
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isi2018mma
engineeringmathematics
calculus
integration
0
votes
1
answer
23
ISI2018MMA19
Let $X_1,X_2, . . . ,X_n$ be independent and identically distributed with $P(X_i = 1) = P(X_i = −1) = p\ $and$ P(X_i = 0) = 1 − 2p$ for all $i = 1, 2, . . . , n.$ ... $a_n \rightarrow p, b_n \rightarrow p,c_n \rightarrow 12p$ $a_n \rightarrow1/2, b_n \rightarrow1/2,c_n \rightarrow0$ $a_n \rightarrow0, b_n \rightarrow0,c_n \rightarrow1$
answered
May 11
in
Calculus
by
srestha
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116k
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29
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isi2018mma
engineeringmathematics
calculus
limits
+1
vote
1
answer
24
ISI2019MMA24
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f''(x)$ exists for every $x \in \mathbb{R}$, where $f''(x) = f \circ f^{n1}(x)$ for $n \geq 2$ ... $S \subset T$ $T \subset S$ $S = T$ None of the above
answered
May 7
in
Calculus
by
pratekag
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236
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isi2019mma
engineeringmathematics
calculus
limits
+1
vote
1
answer
25
ISI2019MMA30
Consider the function $h$ defined on $\{0,1,…….10\}$ with $h(0)=0, \: h(10)=10 $ and $2[h(i)h(i1)] = h(i+1) – h(i) \: \text{ for } i = 1,2, \dots ,9.$ Then the value of $h(1)$ is $\frac{1}{2^91}\\$ $\frac{10}{2^9+1}\\$ $\frac{10}{2^{10}1}\\$ $\frac{1}{2^{10}+1}$
answered
May 7
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Calculus
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pratekag
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354
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isi2019mma
engineeringmathematics
discretemathematics
settheory&algebra
functions
0
votes
1
answer
26
ISI2019MMA28
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 $1/4$ $1/6$ $1/8$ $1/24$
answered
May 7
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Calculus
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pratekag
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calculus
engineeringmathematics
+1
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1
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27
ISI2019MMA29
Let $\psi : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $\psi(y) =0$ for all $y \notin [0,1]$ and $\int_{0}^{1} \psi(y) dy=1$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function. Then the value of $\lim _{n\rightarrow \infty}n \int_{0}^{100} f(x)\psi(nx)dx$ is $f(0)$ $f’(0)$ $f’’(0)$ $f(100)$
answered
May 7
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pratekag
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385
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isi2019mma
engineeringmathematics
calculus
integration
0
votes
1
answer
28
ISI2019MMA25
Let $a,b,c$ be nonzero real numbers such that $\int_{0}^{1} (1 + \cos^8x)(ax^2 + bx +c)dx = \int_{0}^{2}(1+ \cos^8x)(ax^2 + bx + c) dx =0$ Then the quadratic equation $ax^2 + bx +c=0$ has no roots in $(0,2)$ one root in $(0,2)$ and one root outside this interval one repeated root in $(0,2)$ two distinct real roots in $(0,2)$
answered
May 7
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pratekag
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isi2019mma
engineeringmathematics
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integration
0
votes
1
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29
ISI2019MMA5
If $f(a)=2, \: f’(a) = 1, \: g(a) =1$ and $g’(a) =2$, then the value of $\lim _{x\rightarrow a}\frac{g(x) f(a) – f(x) g(a)}{xa}$ is $5$ $3$ $3$ $5$
answered
May 6
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Calculus
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pratekag
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isi2019mma
calculus
limits
+1
vote
1
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30
ISIMMA 2019 Sample Questions23
For $n \geq1$, Let $a_{n} = \frac{1}{2^{2}} + \frac{2}{3^{2}} +.....+ \frac{n}{(n+1)^{2}}$ and $b_{n} = c_{0} + c_{1}r + c_{2}r^{2}+.....+c_{n}r^{n},$ where$c_{k} \leq M$ for all integers $k$ ... not a Cauchy sequence (C) $\{a_n\}$ is not a Cauchy sequence but $\{b_n\}$ is a Cauchy sequence (D) neither $\{a_n\}$ nor $\{b_n\}$ is a Cauchy sequence.
answered
Mar 18
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sequenceseries
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0
votes
1
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31
UPPCL AE 2018:81
answered
Mar 11
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uppcl2018
0
votes
2
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32
ISRO2012ECE Engineering Mathematics
a) 1x b) (1x)/x c)1/x d)x/(1x)
answered
Mar 10
in
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isro2012ece
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33
ISI MMA2015
Let, $a_{n} \;=\; \left ( 1\frac{1}{\sqrt{2}} \right ) ... \left ( 1 \frac{1}{\sqrt{n+1}} \right )$ , $n \geq 1$. Then $\lim_{n\rightarrow \infty } a_{n}$ (A) equals $1$ (B) does not exist (C) equals $\frac{1}{\sqrt{\pi }}$ (D) equals $0$
answered
Feb 21
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1
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34
ISI MMA2015
If two real polynomials $f(x)$ and $g(x)$ of degrees $m\;(\geq2)$ and $n\;(\geq1)$ respectively, satisfy $f(x^{2}+1) = f(x)g(x)$ $,$ for every $x\in \mathbb{R}$ , then (A) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) \neq 0$ (B) $f$ has exactly one real root $x_{0}$ such that $f'(x_{0}) = 0$ (C) $f$ has $m$ distinct real roots (D) $f$ has no real root.
answered
Feb 21
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usermod
+2
votes
3
answers
35
VALUE OF LIMIT
What is value of $\lim_{x\rightarrow 0, Y\rightarrow 0}\frac{xY}{x^2+Y^2}$ 1 1 0 Does not exist
answered
Jan 28
in
Calculus
by
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189
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292
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limits
+6
votes
3
answers
36
TIFR2011A11
$\int_{0}^{1} \ln x\, \mathrm{d}x=$ $1$ $1$ $\infty $ $\infty $ None of the above.
answered
Jan 26
in
Calculus
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510
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tifr2011
calculus
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0
votes
0
answers
37
How to solve such question.
$\frac{d}{dx}\int_{1}^{x^4} sect\space dt$
asked
Jan 20
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Calculus
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`JEET
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96
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calculus
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0
votes
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38
MadeEasy Workbook: Calculus  Maxima Minima
asked
Jan 19
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by
chanchala3993
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5
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67
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engineeringmathematics
calculus
maximaminima
madeeasybooklet
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39
Calculating average mean
The aggregate monthly expenditure of a family was $ 6240 during first 3 months, $ 6780 during next 4 months, and $7236 during last 5 months of a year. If total saving during the year is $ 7080. Find average monthly income of family?
asked
Jan 13
in
Calculus
by
Alina
(
9
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37
views
0
votes
0
answers
40
Ace Test Series: Calculus  Integration
Can anyone help me with solving this type of problem? I want some resource from where I can learn to solve this type on integration, as according to solution it is a function of α, so I did not understand the solution.
asked
Jan 12
in
Calculus
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191
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76
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acetestseries
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