Recent questions and answers in Calculus

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1

ISI2016-DCG-69
Consider the differential equation $(x^{2}-y^{2})\frac{\mathrm{d} y}{\mathrm{d} x}=2xy.$ Assuming $y=10$ for $x=0,$ its solution is $x^{2}+(y-5)^{2}=25$ $x^{2}+y^{2}=100$ $(x-5)^{2}+y^{2}=125$ $(x-5)^{2}+(y-5)^{2}=50$

answered Mar 14 in Calculus by haralk10 Active (1.2k points) | 26 views

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1 answer

2

ISI2016-DCG-68
The general solution of the differential equation $x+y-x{y}'=0$ is (assuming $C$ as an arbitrary constant of integration) $y=x(\log x+C)$ $x=y(\log y+C)$ $y=x(\log y+C)$ $y=y(\log x+C)$

answered Mar 14 in Calculus by haralk10 Active (1.2k points) | 25 views

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1 answer

3

ISI2016-DCG-67
The general solution of the differential equation $2y{y}'-x=0$ is (assuming $C$ as an arbitrary constant of integration) $x^{2}-y^{2}=C$ $2x^{2}-y^{2}=C$ $2y^{2}-x^{2}=C$ $x^{2}+y^{2}=C$

answered Mar 14 in Calculus by haralk10 Active (1.2k points) | 27 views

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1 answer

4

ISI2015-DCG-50
The piecewise linear function for the following graph is $f(x) = \begin{cases} = x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$ $f(x) = \begin{cases} = x-2, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x-1, \: x \geq 3 \end{cases}$ ... $f(x) = \begin{cases} = 2-x, \: x \leq -2 \\ =4, \: -2<x<3 \\ =x+1, \: x \geq 3 \end{cases}$

answered Mar 14 in Calculus by haralk10 Active (1.2k points) | 28 views

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5

ISI2014-DCG-48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2-ax+1-2a^2$ is always greater than zero is $- \frac{2}{3} < a \leq \frac{2}{3}$ $- \frac{2}{3} \leq a < \frac{2}{3}$ $- \frac{2}{3} < a < \frac{2}{3}$ None of these

answered Mar 12 in Calculus by haralk10 Active (1.2k points) | 22 views

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1 answer

6

TIFR2020-A-8
Consider a function $f:[0,1]\rightarrow [0,1]$ which is twice differentiable in $(0,1).$ Suppose it has exactly one global maximum and exactly global minimum inside $(0,1).$ What can you say about the behaviour of the first derivative $f'$ and and second derivative $f''$ on ... is zero at at-least one point $f'$ is zero at at-least two points, $f''$ is zero at at-least two points

answered Mar 1 in Calculus by ankitgupta.1729 Boss (18.1k points) | 46 views

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1 answer

7

TIFR2020-A-13
What is the area of the largest rectangle that can be inscribed in a circle of radius $R$? $R^{2}/2$ $\pi \times R^{2}/2$ $R^{2}$ $2R^{2}$ None of the above

answered Feb 19 in Calculus by ankitgupta.1729 Boss (18.1k points) | 32 views

+2 votes

2 answers

8

ISI2014-DCG-6
If $f(x)$ is a real valued function such that $2f(x)+3f(-x)=15-4x$, for every $x \in \mathbb{R}$, then $f(2)$ is $-15$ $22$ $11$ $0$

answered Jan 29 in Calculus by Akumar20 (35 points) | 114 views

+1 vote

2 answers

9

MadeEasy Test Series: Calculus - Limits
How it solve this ?

answered Jan 9 in Calculus by Mayank Harbola (45 points) | 253 views

+1 vote

2 answers

10

GATE1993-02.6
The value of the double integral $\int^{1}_{0} \int_{0}^{\frac{1}{x}} \frac {x}{1+y^2} dxdy$ is_________.

answered Dec 31, 2019 in Calculus by Verma Ashish Boss (13.6k points) | 751 views

+3 votes

5 answers

11

TIFR2019-A-15
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 ... $\text{The limit exists, but it is none of the above}$

answered Dec 5, 2019 in Calculus by severustux (131 points) | 519 views

+22 votes

5 answers

12

GATE2014-3-6
If $\int \limits_0^{2 \pi} |x \: \sin x| dx=k\pi$, then the value of $k$ is equal to ______.

answered Nov 30, 2019 in Calculus by Lakshman Patel RJIT Veteran (61.3k points) | 3.4k views

+22 votes

4 answers

13

GATE2014-3-47
The value of the integral given below is $\int \limits_0^{\pi} \: x^2 \: \cos x\:dx$ $-2\pi$ $\pi$ $-\pi$ $2\pi$

answered Nov 30, 2019 in Calculus by Lakshman Patel RJIT Veteran (61.3k points) | 2.3k views

+6 votes

3 answers

14

ISRO-2013-49
What is the least value of the function $f(x) = 2x^{2}-8x-3$ in the interval $[0, 5]$? $-15$ $7$ $-11$ $-3$

answered Nov 28, 2019 in Calculus by Lakshman Patel RJIT Veteran (61.3k points) | 2.1k views

+1 vote

1 answer

15

TIFR-2011-Maths-A-19
The derivative of the function $\int_{0}^{\sqrt{x}} e^{-t^{2}}dt$ at $x = 1$ is $e^{-1}$ .

answered Nov 25, 2019 in Calculus by seetal samal (37 points) | 188 views

+1 vote

1 answer

16

TIFR-2011-Maths-A-21
Any continuous function from the open unit interval $(0, 1)$ to itself has a fixed point.

answered Nov 25, 2019 in Calculus by seetal samal (37 points) | 120 views

+2 votes

2 answers

17

Integration
Solve the following $\int_{0}^{\infty}e^{-x^2}x^4dx$

answered Nov 22, 2019 in Calculus by Lakshman Patel RJIT Veteran (61.3k points) | 217 views

+2 votes

1 answer

18

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 $64/5$ $32/5$ $37/5$ $67/5$

answered Nov 19, 2019 in Calculus by `JEET Boss (19.7k points) | 61 views

+1 vote

3 answers

19

ISI2017-DCG-23
A determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability of choosing a non-zero determinant is $\frac{3}{16}$ $\frac{3}{8}$ $\frac{1}{4}$ none of these

answered Nov 19, 2019 in Calculus by chirudeepnamini Loyal (5.3k points) | 69 views

+39 votes

4 answers

20

GATE2012-9
Consider the function $f(x) = \sin(x)$ in the interval $x =\left[\frac{\pi}{4},\frac{7\pi}{4}\right]$. The number and location(s) of the local minima of this function are One, at $\dfrac{\pi}{2}$ One, at $\dfrac{3\pi}{2}$ Two, at $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2}$ Two, at $\dfrac{\pi}{4}$ and $\dfrac{3\pi}{2}$

answered Nov 13, 2019 in Calculus by Obafgkme (67 points) | 4.8k views

0 votes

1 answer

21

MadeEasy Workbook: Calculus - Maxima Minima

answered Oct 31, 2019 in Calculus by Kushagra गुप्ता Loyal (5.9k points) | 119 views

+2 votes

2 answers

22

ISI2015-MMA-10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^3-1}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above

answered Oct 25, 2019 in Calculus by techbd123 Active (3.7k points) | 83 views

0 votes

1 answer

23

ISI2017-DCG-6
Let $f(x) = \dfrac{x-1}{x+1}, \: f^{k+1}(x)=f\left(f^k(x)\right)$ for all $k=1, 2, 3, \dots , 99$. Then $f^{100}(10)$ is $1$ $10$ $100$ $101$

answered Oct 21, 2019 in Calculus by `JEET Boss (19.7k points) | 56 views

+1 vote

1 answer

24

ISI2015-MMA-36
For non-negative integers $m$, $n$ define a function as follows $f(m,n) = \begin{cases} n+1 & \text{ if } m=0 \\ f(m-1, 1) & \text{ if } m \neq 0, n=0 \\ f(m-1, f(m,n-1)) & \text{ if } m \neq 0, n \neq 0 \end{cases}$ Then the value of $f(1,1)$ is $4$ $3$ $2$ $1$

answered Oct 19, 2019 in Calculus by chirudeepnamini Loyal (5.3k points) | 30 views

+14 votes

3 answers

25

GATE1996-1.6
The formula used to compute an approximation for the second derivative of a function $f$ at a point $X_0$ is $\dfrac{f(x_0 +h) + f(x_0 – h)}{2}$ $\dfrac{f(x_0 +h) - f(x_0 – h)}{2h}$ $\dfrac{f(x_0 +h) + 2f(x_0) + f(x_0 – h)}{h^2}$ $\dfrac{f(x_0 +h) - 2f(x_0) + f(x_0 – h)}{h^2}$

answered Oct 18, 2019 in Calculus by neeraj_bhatt (407 points) | 1.7k views

+1 vote

1 answer

26

ISI2014-DCG-13
Let the function $f(x)$ be defined as $f(x)=\mid x-1 \mid + \mid x-2 \:\mid$. Then which of the following statements is true? $f(x)$ is differentiable at $x=1$ $f(x)$ is differentiable at $x=2$ $f(x)$ is differentiable at $x=1$ but not at $x=2$ none of the above

answered Oct 14, 2019 in Calculus by joshi_nitish Boss (31.7k points) | 83 views

+2 votes

3 answers

27

ISI2014-DCG-7
If $f(x) = \dfrac{\sqrt{3} \sin x}{2+\cos x}$, then the range of $f(x)$ is the interval $[-1 , \sqrt{3}{/2}]$ the interval $[-\sqrt{3}{/2}, 1]$ the interval $[-1, 1]$ none of these

answered Oct 11, 2019 in Calculus by Abhishek Kumar 40 (253 points) | 74 views

0 votes

1 answer

28

ISI2014-DCG-29
If $f(x) = \sin \bigg( \dfrac{1}{x^2+1} \bigg),$ then $f(x)$ is continuous at $x=0$, but not differentiable at $x=0$ $f(x)$ is differentiable at $x=0$, and $f’(0) \neq 0$ $f(x)$ is differentiable at $x=0$, and $f’(0) = 0$ None of the above

answered Oct 11, 2019 in Calculus by techbd123 Active (3.7k points) | 58 views

+1 vote

1 answer

29

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, 2019 in Calculus by Nirmal Gaur Active (2.3k points) | 68 views

+1 vote

1 answer

30

ISI2014-DCG-37
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^-}{2}$ and $f(x) \to – \infty$ as $x \to -\dfrac{\pi^+}{2}$. Which one of the following functions satisfies the above properties of $f(x)$? $\cos x$ $\tan x$ $\tan^{-1} x$ $\sin x$

answered Oct 8, 2019 in Calculus by techbd123 Active (3.7k points) | 43 views

0 votes

1 answer

31

ISI2015-MMA-78
The value of $\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$ is $0$ $\ln 2$ $\ln 3$ $\infty$

answered Oct 4, 2019 in Calculus by `JEET Boss (19.7k points) | 30 views

0 votes

1 answer

32

ISI2015-MMA-22
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1- \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$

answered Oct 4, 2019 in Calculus by `JEET Boss (19.7k points) | 29 views

0 votes

1 answer

33

ISI2015-MMA-19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k-1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$

answered Oct 4, 2019 in Calculus by `JEET Boss (19.7k points) | 47 views

+2 votes

1 answer

34

ISI2016-DCG-45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:x-x\:\tan\:x}{\sin\:x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these

answered Oct 2, 2019 in Calculus by `JEET Boss (19.7k points) | 55 views

+1 vote

1 answer

35

ISI2015-MMA-25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$

answered Oct 1, 2019 in Calculus by `JEET Boss (19.7k points) | 56 views

+1 vote

1 answer

36

ISI2015-MMA-20
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals $e^{-1}$ $e^{-1/2}$ $e^{-2}$ $1$

answered Oct 1, 2019 in Calculus by `JEET Boss (19.7k points) | 36 views

+1 vote

1 answer

37

ISI2018-DCG-9
Let $f(x)=1+x+\dfrac{x^2}{2}+\dfrac{x^3}{3}...+\dfrac{x^{2018}}{2018}.$ Then $f’(1)$ is equal to $0$ $2017$ $2018$ $2019$

answered Sep 30, 2019 in Calculus by `JEET Boss (19.7k points) | 80 views

0 votes

1 answer

38

ISI2018-DCG-10
Let $f’(x)=4x^3-3x^2+2x+k,$ $f(0)=1$ and $f(1)=4.$ Then $f(x)$ is equal to $4x^4-3x^3+2x^2+x+1$ $x^4-x^3+x^2+2x+1$ $x^4-x^3+x^2+2(x+1)$ none of these

answered Sep 30, 2019 in Calculus by `JEET Boss (19.7k points) | 35 views

+3 votes

2 answers

39

ISI2014-DCG-4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$

answered Sep 30, 2019 in Calculus by techbd123 Active (3.7k points) | 140 views

+4 votes

4 answers

40

ISI2014-DCG-3
$\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$

answered Sep 29, 2019 in Calculus by techbd123 Active (3.7k points) | 175 views

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