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Search results for isi2015-mma
0
votes
2
answers
1
ISI2015-MMA-69
Consider the function $f(x) = \begin{cases} \int_0^x \{5+ \mid 1-y \mid \} dy & \text{ if } x>2 \\ 5x+2 & \text{ if } x \leq 2 \end{cases}$ Then $f$ is not continuous at $x=2$ $f$ is continuous and differentiable everywhere $f$ is continuous everywhere but not differentiable at $x=1$ $f$ is continuous everywhere but not differentiable at $x=2$
Consider the function $$f(x) = \begin{cases} \int_0^x \{5+ \mid 1-y \mid \} dy & \text{ if } x>2 \\ 5x+2 & \text{ if } x \leq 2 \end{cases}$$ Then$f$ is not continuous at...
Arjun
834
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
continuity
differentiation
definite-integral
non-gate
+
–
1
votes
2
answers
2
ISI2015-MMA-84
For positive real numbers $a_1, a_2, \cdots, a_{100}$, let $p=\sum_{i=1}^{100} a_i \text{ and } q=\sum_{1 \leq i < j \leq 100} a_ia_j.$ Then $q=\frac{p^2}{2}$ $q^2 \geq \frac{p^2}{2}$ $q< \frac{p^2}{2}$ none of the above
For positive real numbers $a_1, a_2, \cdots, a_{100}$, let $$p=\sum_{i=1}^{100} a_i \text{ and } q=\sum_{1 \leq i < j \leq 100} a_ia_j.$$ Then $q=\frac{p^2}{2}$$q^2 \geq ...
Arjun
435
views
Arjun
asked
Sep 23, 2019
Others
isi2015-mma
summation
non-gate
+
–
0
votes
2
answers
3
ISI2015-MMA-70
Let $w=\log(u^2 +v^2)$ where $u=e^{(x^2+y)}$ and $v=e^{(x+y^2)}$. Then $\frac{\partial w }{\partial x} \mid _{x=0, y=0}$ is $0$ $1$ $2$ $4$
Let $w=\log(u^2 +v^2)$ where $u=e^{(x^2+y)}$ and $v=e^{(x+y^2)}$. Then $\frac{\partial w }{\partial x} \mid _{x=0, y=0}$ is$0$$1$$2$$4$
Arjun
528
views
Arjun
asked
Sep 23, 2019
Others
isi2015-mma
partial-derivatives
non-gate
+
–
1
votes
1
answer
4
ISI2015-MMA-81
If $f$ is continuous in $[0,1]$ then $\displaystyle \lim_ {n \to \infty} \sum_{j=0}^{[n/2]} \frac{1}{n} f \left(\frac{j}{n} \right)$ (where $[y]$ is the largest integer less than or equal to $y$) does not exist exists and is equal to $\frac{1}{2} \int_0^1 f(x) dx$ exists and is equal to $ \int_0^1 f(x) dx$ exists and is equal to $\int_0^{1/2} f(x) dx$
If $f$ is continuous in $[0,1]$ then $$\displaystyle \lim_ {n \to \infty} \sum_{j=0}^{[n/2]} \frac{1}{n} f \left(\frac{j}{n} \right)$$ (where $[y]$ is the largest integ...
Arjun
421
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
limits
definite-integral
non-gate
+
–
0
votes
1
answer
5
ISI2015-MMA-73
$f(x)$ is a differentiable function on the real line such that $\underset{x \to \infty=}{\lim} f(x) =1$ and $\underset{x \to \infty=}{\lim} f’(x) =\alpha$. Then $\alpha$ must be $0$ $\alpha$ need not be $0$, but $\mid \alpha \mid <1$ $\alpha >1$ $\alpha < -1$
$f(x)$ is a differentiable function on the real line such that $\underset{x \to \infty=}{\lim} f(x) =1$ and $\underset{x \to \infty=}{\lim} f’(x) =\alpha$. Then$\alpha$...
Arjun
487
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
differentiation
+
–
1
votes
1
answer
6
ISI2015-MMA-89
Let $y(x)$ be a non-trivial solution of the second order linear differential equation $\frac{d^2y}{dx^2}+2c\frac{dy}{dx}+ky=0,$ where $c<0$, $k>0$ and $c^2>k$. Then $\mid y(x) \mid \to \infty$ as $x \to \infty$ $\mid y(x) \mid \to 0$ as $x \to \infty$ $\underset{x \to \pm \infty}{\lim} \mid y(x) \mid$ exists and is finite none of the above is true
Let $y(x)$ be a non-trivial solution of the second order linear differential equation $$\frac{d^2y}{dx^2}+2c\frac{dy}{dx}+ky=0,$$ where $c<0$, $k>0$ and $c^2>k$. Then$\mi...
Arjun
258
views
Arjun
asked
Sep 23, 2019
Others
isi2015-mma
differential-equation
non-gate
+
–
1
votes
1
answer
7
ISI2015-MMA-88
Let $f(x)$ be a given differentiable function. Consider the following differential equation in $y$ $f(x) \frac{dy}{dx} = yf’(x)-y^2.$ The general solution of this equation is given by $y=-\frac{x+c}{f(x)}$ $y^2=\frac{f(x)}{x+c}$ $y=\frac{f(x)}{x+c}$ $y=\frac{\left[f(x)\right]^2}{x+c}$
Let $f(x)$ be a given differentiable function. Consider the following differential equation in $y$ $$f(x) \frac{dy}{dx} = yf’(x)-y^2.$$ The general solution of this equ...
Arjun
306
views
Arjun
asked
Sep 23, 2019
Others
isi2015-mma
differential-equation
general-solution
non-gate
+
–
1
votes
3
answers
8
ISI2015-MMA-20
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals $e^{-1}$ $e^{-1/2}$ $e^{-2}$ $1$
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals$e^{-1}$$e^{-1/2}$$e^{-2}$$1$
Arjun
701
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
1
votes
1
answer
9
ISI2015-MMA-77
Let $R$ be the triangle in the $xy$ – plane bounded by the $x$-axis, the line $y=x$, and the line $x=1$. The value of the double integral $ \int \int_R \frac{\sin x}{x}\: dxdy$ is $1-\cos 1$ $\cos 1$ $\frac{\pi}{2}$ $\pi$
Let $R$ be the triangle in the $xy$ – plane bounded by the $x$-axis, the line $y=x$, and the line $x=1$. The value of the double integral $$ \int \int_R \frac{\sin x}{x...
Arjun
548
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
integration
non-gate
+
–
2
votes
2
answers
10
ISI2015-MMA-11
The number of positive integers which are less than or equal to $1000$ and are divisible by none of $17$, $19$ and $23$ equals $854$ $153$ $160$ none of the above
The number of positive integers which are less than or equal to $1000$ and are divisible by none of $17$, $19$ and $23$ equals$854$$153$$160$none of the above
Arjun
1.1k
views
Arjun
asked
Sep 23, 2019
Quantitative Aptitude
isi2015-mma
quantitative-aptitude
number-system
remainder-theorem
+
–
3
votes
1
answer
11
ISI2015-MMA-93
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true? $x=e, \: y=e$ $x\neq e, \: y=e$ $x=e, \: y \neq e$ $x\neq e, \: y \neq e$
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true?$x=e, \: y=e...
Arjun
907
views
Arjun
asked
Sep 23, 2019
Set Theory & Algebra
isi2015-mma
group-theory
+
–
1
votes
2
answers
12
ISI2015-MMA-63
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ ... $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
Let $\theta=2\pi/67$. Now consider the matrix $A = \begin{pmatrix} \cos \theta & \sin \theta \\ – \sin \theta & \cos \theta \end{pmatrix}$. Then the matrix $A^{2010}$ i...
Arjun
661
views
Arjun
asked
Sep 23, 2019
Linear Algebra
isi2015-mma
linear-algebra
matrix
+
–
0
votes
1
answer
13
ISI2015-MMA-55
Let $\{a_n\}$ be a sequence of real numbers. Then $\underset{n \to \infty}{\lim} a_n$ exists if and only if $\underset{n \to \infty}{\lim} a_{2n}$ and $\underset{n \to \infty}{\lim} a_{2n+2}$ exists $\underset{n \to \infty}{\lim} a_{2n}$ ... $\underset{n \to \infty}{\lim} a_{3n}$ exist none of the above
Let $\{a_n\}$ be a sequence of real numbers. Then $\underset{n \to \infty}{\lim} a_n$ exists if and only if$\underset{n \to \infty}{\lim} a_{2n}$ and $\underset{n \to \in...
Arjun
880
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
+
–
0
votes
2
answers
14
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 $2$ $2e$ $2 \pi$ $2i$
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 ...
Arjun
940
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
1
votes
2
answers
15
ISI2015-MMA-27
Let $\cos ^6 \theta = a_6 \cos 6 \theta + a_5 \cos 5 \theta + a_4 \cos 4 \theta + a_3 \cos 3 \theta + a_2 \cos 2 \theta + a_1 \cos \theta +a_0$. Then $a_0$ is $0$ $1/32$ $15/32$ $10/32$
Let $\cos ^6 \theta = a_6 \cos 6 \theta + a_5 \cos 5 \theta + a_4 \cos 4 \theta + a_3 \cos 3 \theta + a_2 \cos 2 \theta + a_1 \cos \theta +a_0$. Then $a_0$ is$0$$1/32$$...
Arjun
628
views
Arjun
asked
Sep 23, 2019
Quantitative Aptitude
isi2015-mma
quantitative-aptitude
trigonometry
non-gate
+
–
0
votes
1
answer
16
ISI2015-MMA-23
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by $f(x,A)=\begin{cases} 1 & \text{ if } x \in A \\ 0 & \text{ if } x \notin A \end{cases}$ Then $f(x, A \cup B)$ ... $f(x,A)+f(x,B)\: - f(x,A) \cdot f(x,B)$ $f(x,A)\:+ \mid f(x,A)\: - f(x,B) \mid $
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ by$$f(x,A)=\begin{cases...
Arjun
760
views
Arjun
asked
Sep 23, 2019
Set Theory & Algebra
isi2015-mma
set-theory
functions
non-gate
+
–
0
votes
1
answer
17
ISI2015-MMA-56
Let $\{a_n\}$ be a sequence of non-negative real numbers such that the series $\Sigma_{n=1}^{\infty} a_n$ is convergent. If $p$ is a real number such that the series $\Sigma \frac{\sqrt{a_n}}{n^p}$ diverges, then $p$ must be strictly less than $\frac{1}{2}$ ... but can be greater than$\frac{1}{2}$ $p$ must be strictly less than $1$ but can be greater than or equal to $\frac{1}{2}$
Let $\{a_n\}$ be a sequence of non-negative real numbers such that the series $\Sigma_{n=1}^{\infty} a_n$ is convergent. If $p$ is a real number such that the series $\Si...
Arjun
494
views
Arjun
asked
Sep 23, 2019
Others
isi2015-mma
convergence-divergence
non-gate
+
–
0
votes
3
answers
18
ISI2015-MMA-92
Consider the group $G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$ ... is of finite order $N$ is a normal subgroup and the quotient group is isomorphic to $\mathbb{R}^+$ (the group of positive reals with multiplication).
Consider the group $$G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$$ with usual matrix multiplication. Le...
Arjun
1.4k
views
Arjun
asked
Sep 23, 2019
Set Theory & Algebra
isi2015-mma
group-theory
subgroups
normal
non-gate
+
–
0
votes
1
answer
19
ISI2015-MMA-45
Angles between any pair of $4$ main diagonals of a cube are $\cos^{-1} 1/\sqrt{3}, \pi – \cos ^{-1} 1/\sqrt{3}$ $\cos^{-1} 1/3, \pi – \cos ^{-1} 1/3$ $\pi/2$ none of the above
Angles between any pair of $4$ main diagonals of a cube are$\cos^{-1} 1/\sqrt{3}, \pi – \cos ^{-1} 1/\sqrt{3}$$\cos^{-1} 1/3, \pi – \cos ^{-1} 1/3$$\pi/2$none of the ...
Arjun
551
views
Arjun
asked
Sep 23, 2019
Geometry
isi2015-mma
cubes
non-gate
+
–
0
votes
1
answer
20
ISI2015-MMA-26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to$\infty$$0$$\log_e 2$$1$
Arjun
741
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
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