Recent questions in Calculus

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

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\\\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}}}$$

0 votes

1 answer

ISI2014-DCG-12
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

asked Sep 23 in Calculus by Arjun Veteran (421k points) | 44 views

+1 vote

1 answer

ISI2014-DCG-39
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

asked Sep 23 in Calculus by Arjun Veteran (421k points) | 36 views

+1 vote

1 answer

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?

asked Sep 22 in Calculus by Nirmal Gaur Active (2k points) | 35 views

0 votes

1 answer

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

asked Sep 18 in Calculus by gatecse Boss (16.3k points) | 19 views

0 votes

1 answer

Maths: Limits
$\LARGE \lim_{n \rightarrow \infty} \frac{n^{\frac{3}{4}}}{log^9 n}$

asked May 26 in Calculus by Mk Utkarsh Boss (35.2k points) | 136 views

0 votes

1 answer

ISI2018-MMA-28
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$.

asked May 11 in Calculus by akash.dinkar12 Boss (41.6k points) | 59 views

0 votes

0 answers

ISI2018-MMA-30
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 by akash.dinkar12 Boss (41.6k points) | 42 views

0 votes

0 answers

ISI2018-MMA-29
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 by akash.dinkar12 Boss (41.6k points) | 50 views

0 votes

1 answer

ISI2018-MMA-19
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 $i = 1, 2, . . . , n.$ ... $a_n \rightarrow p, b_n \rightarrow p,c_n \rightarrow 1-2p$ $a_n \rightarrow1/2, b_n \rightarrow1/2,c_n \rightarrow0$ $a_n \rightarrow0, b_n \rightarrow0,c_n \rightarrow1$

asked May 11 in Calculus by akash.dinkar12 Boss (41.6k points) | 29 views

+1 vote

1 answer

ISI2019-MMA-30
Consider the function $h$ defined on $\{0,1,…….10\}$ with $h(0)=0, \: h(10)=10 $ and $2[h(i)-h(i-1)] = h(i+1) – h(i) \: \text{ for } i = 1,2, \dots ,9.$ Then the value of $h(1)$ is $\frac{1}{2^9-1}\\$ $\frac{10}{2^9+1}\\$ $\frac{10}{2^{10}-1}\\$ $\frac{1}{2^{10}+1}$

asked May 7 in Calculus by Sayan Bose Loyal (7.1k points) | 354 views

+1 vote

1 answer

ISI2019-MMA-29
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 $f(0)$ $f’(0)$ $f’’(0)$ $f(100)$

asked May 7 in Calculus by Sayan Bose Loyal (7.1k points) | 385 views

0 votes

1 answer

ISI2019-MMA-28
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 $1/4$ $1/6$ $1/8$ $1/24$

asked May 7 in Calculus by Sayan Bose Loyal (7.1k points) | 642 views

0 votes

1 answer

ISI2019-MMA-25
Let $a,b,c$ be non-zero 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 $(0,2)$

asked May 7 in Calculus by Sayan Bose Loyal (7.1k points) | 229 views

+1 vote

1 answer

ISI2019-MMA-24
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 $f''(x) = f \circ f^{n-1}(x)$ for $n \geq 2$ ... $S \subset T$ $T \subset S$ $S = T$ None of the above

asked May 7 in Calculus by Sayan Bose Loyal (7.1k points) | 236 views

0 votes

1 answer

ISI2019-MMA-21
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 $x_2$

asked May 7 in Calculus by Sayan Bose Loyal (7.1k points) | 393 views

0 votes

1 answer

ISI2019-MMA-5
If $f(a)=2, \: f’(a) = 1, \: g(a) =-1$ and $g’(a) =2$, then the value of $\lim _{x\rightarrow a}\frac{g(x) f(a) – f(x) g(a)}{x-a}$ is $-5$ $-3$ $3$ $5$

asked May 6 in Calculus by Sayan Bose Loyal (7.1k points) | 163 views

0 votes

1 answer

Gate 2002 - ME
Which of the following functions is not differentiable in the domain $[-1,1]$ ? (a) $f(x) = x^2$ (b) $f(x) = x-1$ (c) $f(x) = 2$ (d) $f(x) = Maximum (x,-x)$

asked May 4 in Calculus by balchandar reddy san Active (3.2k points) | 76 views

+1 vote

1 answer

ISI-MMA 2019 Sample Questions-23
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.

asked Mar 17 in Calculus by ankitgupta.1729 Boss (15.4k points) | 118 views

0 votes

1 answer

ISI MMA-2015
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 $0$

asked Feb 21 in Calculus by ankitgupta.1729 Boss (15.4k points) | 140 views

+1 vote

1 answer

ISI MMA-2015
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.

asked Feb 20 in Calculus by ankitgupta.1729 Boss (15.4k points) | 133 views

+3 votes

6 answers

GATE2019-13
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 in Calculus by Arjun Veteran (421k points) | 1.7k views

0 votes

0 answers

How to solve such question.
$\frac{d}{dx}\int_{1}^{x^4} sect\space dt$

asked Jan 20 in Calculus by `JEET Loyal (8k points) | 96 views

0 votes

0 answers

MadeEasy Workbook: Calculus - Maxima Minima

asked Jan 19 in Calculus by chanchala3993 (5 points) | 67 views

0 votes

0 answers

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 points) | 37 views

0 votes

0 answers

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 by jhaanuj2108 (191 points) | 76 views

0 votes

1 answer

How to solve such question with different variable in the limit.
$\int_{a}^{x} \frac{sint}{t} dt$ (x > 0), then f(x) has _________________. (A) a maximum at x = $n\pi$ where n is even (B) a minimum at x = $n\pi$ where n is odd (C) a maximum at x = $n\pi$ where n is odd (D) a minimum at x = $\frac{n\pi}{2}$ where n is odd.

asked Jan 11 in Calculus by `JEET Loyal (8k points) | 67 views

0 votes

1 answer

ME TEST SERIES

asked Jan 10 in Calculus by Shankar Kakde (195 points) | 53 views

0 votes

1 answer

Integration
How to solve it?

asked Jan 8 in Calculus by ghostman23111 (105 points) | 61 views

+1 vote

1 answer

UPPCL AE 2018:85

asked Jan 5 in Calculus by Lakshman Patel RJIT Veteran (51.3k points) | 109 views

0 votes

1 answer

UPPCL AE 2018:81

asked Jan 5 in Calculus by Lakshman Patel RJIT Veteran (51.3k points) | 63 views

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