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Recent questions tagged isi2015-mma
2
votes
1
answer
61
ISI2015-MMA-61
Let $ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{pmatrix} \text{ and } B=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{pmatrix}.$ Then there exists a matrix $C$ ... no matrix $C$ such that $A=BC$ there exists a matrix $C$ such that $A=BC$, but $A \neq CB$ there is no matrix $C$ such that $A=CB$
Arjun
asked
in
Linear Algebra
Sep 23, 2019
by
Arjun
572
views
isi2015-mma
linear-algebra
matrix
2
votes
1
answer
62
ISI2015-MMA-62
If the matrix $A = \begin{bmatrix} a & 1 \\ 2 & 3 \end{bmatrix}$ has $1$ as an eigenvalue, then $\textit{trace}(A)$ is $4$ $5$ $6$ $7$
Arjun
asked
in
Linear Algebra
Sep 23, 2019
by
Arjun
848
views
isi2015-mma
linear-algebra
matrix
eigen-value
1
vote
2
answers
63
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 $A^{2010}$ ... $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
Arjun
asked
in
Linear Algebra
Sep 23, 2019
by
Arjun
651
views
isi2015-mma
linear-algebra
matrix
0
votes
0
answers
64
ISI2015-MMA-64
Let the position of a particle in three dimensional space at time $t$ be $(t, \cos t, \sin t)$. Then the length of the path traversed by the particle between the times $t=0$ and $t=2 \pi$ is $2 \pi$ $2 \sqrt{2 \pi}$ $\sqrt{2 \pi}$ none of the above
Arjun
asked
in
Geometry
Sep 23, 2019
by
Arjun
438
views
isi2015-mma
trigonometry
curves
non-gate
0
votes
1
answer
65
ISI2015-MMA-65
Let $n$ be a positive real number and $p$ be a positive integer. Which of the following inequalities is true? $n^p > \frac{(n+1)^{p+1} – n^{p+1}}{p+1}$ $n^p < \frac{(n+1)^{p+1} – n^{p+1}}{p+1}$ $(n+1)^p < \frac{(n+1)^{p+1} – n^{p+1}}{p+1}$ none of the above
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
392
views
isi2015-mma
inequality
non-gate
0
votes
0
answers
66
ISI2015-MMA-66
The smallest positive number $K$ for which the inequality $\mid \sin ^2 x – \sin ^2 y \mid \leq K \mid x-y \mid$ holds for all $x$ and $y$ is $2$ $1$ $\frac{\pi}{2}$ there is no smallest positive value of $K$; any $K>0$ will make the inequality hold.
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
464
views
isi2015-mma
inequality
trigonometry
non-gate
0
votes
0
answers
67
ISI2015-MMA-67
Given two real numbers $a<b$, let $d(x,[a,b]) = \text{min} \{ \mid x-y \mid : a \leq y \leq b \} \text{ for } - \infty < x < \infty$. Then the function $f(x) = \frac{d(x,[0,1])}{d(x,[0,1]) + d(x,[2,3])}$ satisfies $0 \leq f(x) < \frac{1}{2}$ for every $x$ ... $f(x)=1$ if $ 0 \leq x \leq 1$ $f(x)=0$ if $0 \leq x \leq 1$ and $f(x)=1$ if $ 2 \leq x \leq 3$
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
345
views
isi2015-mma
functions
non-gate
1
vote
0
answers
68
ISI2015-MMA-68
Let $f(x,y) = \begin{cases} e^{-1/(x^2+y^2)} & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0). \end{cases}$Then $f(x,y)$ is not continuous at $(0,0)$ continuous at $(0,0)$ but does not have first order partial derivatives continuous at $(0,0)$ and has first order partial derivatives, but not differentiable at $(0,0)$ differentiable at $(0,0)$
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
283
views
isi2015-mma
partial-derivatives
non-gate
0
votes
2
answers
69
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 $x=2$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
818
views
isi2015-mma
calculus
continuity
differentiation
definite-integral
non-gate
0
votes
2
answers
70
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 $0$ $1$ $2$ $4$
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
499
views
isi2015-mma
partial-derivatives
non-gate
1
vote
1
answer
71
ISI2015-MMA-71
Let $f(x,y) = \begin{cases} 1, & \text{ if } xy=0, \\ xy, & \text{ if } xy \neq 0. \end{cases}$ Then $f$ is continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ exists $f$ is not continuous at $(0,0)$ ... $f$ is not continuous at $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ does not exist
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
396
views
isi2015-mma
continuity
partial-derivatives
non-gate
0
votes
1
answer
72
ISI2015-MMA-72
The map $f(x) = a_0 \cos \mid x \mid +a_1 \sin \mid x \mid +a_2 \mid x \mid ^3$ is differentiable at $x=0$ if and only if $a_1=0$ and $a_2=0$ $a_0=0$ and $a_1=0$ $a_1=0$ $a_0, a_1, a_2$ can take any real value
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
477
views
isi2015-mma
calculus
differentiation
0
votes
1
answer
73
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 $\mid \alpha \mid <1$ $\alpha >1$ $\alpha < -1$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
471
views
isi2015-mma
calculus
limits
differentiation
0
votes
1
answer
74
ISI2015-MMA-74
Let $f$ and $g$ be two differentiable functions such that $f’(x)\leq g’(x)$for all $x<1$ and $f’(x) \geq g’(x)$ for all $x>1$. Then if $f(1) \geq g(1)$, then $f(x) \geq g(x)$ for all $x$ if $f(1) \leq g(1)$, then $f(x) \leq g(x)$ for all $x$ $f(1) \leq g(1)$ $f(1) \geq g(1)$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
415
views
isi2015-mma
calculus
differentiation
0
votes
1
answer
75
ISI2015-MMA-75
The length of the curve $x=t^3$, $y=3t^2$ from $t=0$ to $t=4$ is $5 \sqrt{5}+1$ $8(5 \sqrt{5}+1)$ $5 \sqrt{5}-1$ $8(5 \sqrt{5}-1)$
Arjun
asked
in
Geometry
Sep 23, 2019
by
Arjun
488
views
isi2015-mma
curves
non-gate
0
votes
1
answer
76
ISI2015-MMA-76
Given that $\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$, the value of $ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+xy+y^2)} dxdy$ is $\sqrt{\pi/3}$ $\pi/\sqrt{3}$ $\sqrt{2 \pi/3}$ $2 \pi / \sqrt{3}$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
442
views
isi2015-mma
calculus
definite-integral
non-gate
1
vote
1
answer
77
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 $1-\cos 1$ $\cos 1$ $\frac{\pi}{2}$ $\pi$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
515
views
isi2015-mma
integration
non-gate
0
votes
2
answers
78
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 $0$ $\ln 2$ $\ln 3$ $\infty$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
499
views
isi2015-mma
calculus
limits
definite-integral
non-gate
1
vote
1
answer
79
ISI2015-MMA-79
Let $g(x,y) = \text{max}\{12-x, 8-y\}$. Then the minimum value of $g(x,y)$ $ $ as $(x,y)$ varies over the line $x+y =10$ is $5$ $7$ $1$ $3$
Arjun
asked
in
Geometry
Sep 23, 2019
by
Arjun
500
views
isi2015-mma
lines
non-gate
0
votes
0
answers
80
ISI2015-MMA-80
Let $0 < \alpha < \beta < 1$. Then $ \Sigma_{k=1}^{\infty} \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x} dx$ is equal to $\log_e \frac{\beta}{\alpha}$ $\log_e \frac{1+ \beta}{1 + \alpha}$ $\log_e \frac{1+\alpha }{1+ \beta}$ $\infty$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
500
views
isi2015-mma
calculus
definite-integral
summation
non-gate
1
vote
1
answer
81
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 $\int_0^{1/2} f(x) dx$
Arjun
asked
in
Calculus
Sep 23, 2019
by
Arjun
380
views
isi2015-mma
limits
definite-integral
non-gate
0
votes
0
answers
82
ISI2015-MMA-82
The volume of the solid, generated by revolving about the horizontal line $y=2$ the region bounded by $y^2 \leq 2x$, $x \leq 8$ and $y \geq 2$, is $2 \sqrt{2\pi}$ $28 \pi/3$ $84 \pi$ none of the above
Arjun
asked
in
Geometry
Sep 23, 2019
by
Arjun
401
views
isi2015-mma
area
non-gate
0
votes
1
answer
83
ISI2015-MMA-83
If $\alpha, \beta$ are complex numbers then the maximum value of $\dfrac{\alpha \overline{\beta}+\overline{\alpha}\beta}{\mid \alpha \beta \mid}$ is $2$ $1$ the expression may not always be a real number and hence maximum does not make sense none of the above
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
409
views
isi2015-mma
complex-number
non-gate
1
vote
2
answers
84
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
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
429
views
isi2015-mma
summation
non-gate
0
votes
1
answer
85
ISI2015-MMA-85
The differential equation of all the ellipses centred at the origin is $y^2+x(y’)^2-yy’=0$ $xyy’’ +x(y’)^2 -yy’=0$ $yy’’+x(y’)^2-xy’=0$ none of these
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
311
views
isi2015-mma
differential-equation
ellipse
non-gate
0
votes
1
answer
86
ISI2015-MMA-86
The coordinates of a moving point $P$ satisfy the equations $\frac{dx}{dt} = \tan x, \:\:\:\: \frac{dy}{dt}=-\sin^2x, \:\:\:\:\: t \geq 0.$ If the curve passes through the point $(\pi/2, 0)$ when $t=0$, then the equation of the curve in rectangular co-ordinates is $y=1/2 \cos ^2 x$ $y=\sin 2x$ $y=\cos 2x+1$ $y=\sin ^2 x-1$
Arjun
asked
in
Geometry
Sep 23, 2019
by
Arjun
451
views
isi2015-mma
trigonometry
curves
non-gate
0
votes
1
answer
87
ISI2015-MMA-87
If $x(t)$ is a solution of $(1-t^2) dx -tx\: dt =dt$ and $x(0)=1$, then $x\big(\frac{1}{2}\big)$ is equal to $\frac{2}{\sqrt{3}} (\frac{\pi}{6}+1)$ $\frac{2}{\sqrt{3}} (\frac{\pi}{6}-1)$ $\frac{\pi}{3 \sqrt{3}}$ $\frac{\pi}{\sqrt{3}}$
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
391
views
isi2015-mma
differential-equation
non-gate
1
vote
1
answer
88
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 $y=-\frac{x+c}{f(x)}$ $y^2=\frac{f(x)}{x+c}$ $y=\frac{f(x)}{x+c}$ $y=\frac{\left[f(x)\right]^2}{x+c}$
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
292
views
isi2015-mma
differential-equation
general-solution
non-gate
1
vote
1
answer
89
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
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
260
views
isi2015-mma
differential-equation
non-gate
0
votes
1
answer
90
ISI2015-MMA-90
The differential equation of the system of circles touching the $y$-axis at the origin is $x^2+y^2-2xy \frac{dy}{dx}=0$ $x^2+y^2+2xy \frac{dy}{dx}=0$ $x^2-y^2-2xy \frac{dy}{dx}=0$ $x^2-y^2+2xy \frac{dy}{dx}=0$
Arjun
asked
in
Others
Sep 23, 2019
by
Arjun
312
views
isi2015-mma
differential-equation
non-gate
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