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1
ISI2014DCG27
Let $y^24ax+4a=0$ and $x^2+y^22(1+a)x+1+2a3a^2=0$ be two curves. State which one of the following statements is true. These two curves intersect at two points These two curves are tangent to each other These two curves intersect orthogonally at one point These two curves do not intersect
asked
Sep 23
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Arjun
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20
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isi2014dcg
curves
0
votes
0
answers
2
ISI2014DCG52
The area under the curve $x^2+3x4$ in the positive quadrant and bounded by the line $x=5$ is equal to $59 \frac{1}{6}$ $61 \frac{1}{3}$ $40 \frac{2}{3}$ $72$
asked
Sep 23
in
Others
by
Arjun
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425k
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6
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isi2014dcg
curve
boundedarea
0
votes
0
answers
3
ISI2014DCG59
The equation $5x^2+9y^2+10x36y4=0$ represents an ellipse with the coordinates of foci being $(\pm3,0)$ a hyperbola with the coordinates of foci being $(\pm3,0)$ an ellipse with the coordinates of foci being $(\pm2,0)$ a hyperbola with the coordinates of foci being $(\pm2,0)$
asked
Sep 23
in
Others
by
Arjun
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425k
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10
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isi2014dcg
hyperbola
ellipse
foci
0
votes
0
answers
4
ISI2015MMA35
If $f(x)=x^2$ and $g(x)= x \sin x + \cos x$ then $f$ and $g$ agree at no points $f$ and $g$ agree at exactly one point $f$ and $g$ agree at exactly two points $f$ and $g$ agree at more than two points
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

14
views
isi2015mma
trigonometry
nongate
0
votes
0
answers
5
ISI2015MMA45
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
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

7
views
isi2015mma
cubes
diagonals
0
votes
0
answers
6
ISI2015MMA46
If the tangent at the point $P$ with coordinates $(h,k)$ on the curve $y^2=2x^3$ is perpendicular to the straight line $4x=3y$, then $(h,k) = (0,0)$ $(h,k) = (1/8, 1/16)$ $(h,k) = (0,0) \text{ or } (h,k) = (1/8, 1/16)$ no such point $(h,k)$ exists
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

8
views
isi2015mma
straight
line
tangent
curves
0
votes
0
answers
7
ISI2015MMA47
Consider the family $\mathcal{F}$ of curves in the plane given by $x=cy^2$, where $c$ is a real parameter. Let $\mathcal{G}$ be the family of curves having the following property: every member of $\mathcal{G}$ intersect each member of $\mathcal{F}$ orthogonally. Then $\mathcal{G}$ is given by $xy=k$ $x^2+y^2=k^2$ $y^2+2x^2=k^2$ $x^2y^2+2yk=k^2$
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

5
views
isi2015mma
curves
0
votes
0
answers
8
ISI2015MMA48
Suppose the circle with equation $x^2+y^2+2fx+2gy+c=0$ cuts the parabola $y^2=4ax, \: (a>0)$ at four distinct points. If $d$ denotes the sum of the ordinates of these four points, then the set of possible values of $d$ is $\{0\}$ $(4a,4a)$ $(a,a)$ $( \infty, \infty)$
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

8
views
isi2015mma
circle
parabola
nongate
0
votes
0
answers
9
ISI2015MMA56
Let $\{a_n\}$ be a sequence of nonnegative 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 $\frac{1}{2}$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

5
views
isi2015mma
diverges
nongate
0
votes
0
answers
10
ISI2015MMA64
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
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
trigonometry
curves
nongate
0
votes
0
answers
11
ISI2015MMA65
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
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
inequality
nongate
0
votes
0
answers
12
ISI2015MMA66
The smallest positive number $K$ for which the inequality $\mid \sin ^2 x – \sin ^2 y \mid \leq K \mid xy \mid$ 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.
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
inequality
trigonometry
nongate
0
votes
0
answers
13
ISI2015MMA67
Given two real numbers $a<b$, let $d(x,[a,b]) = \text{min} \{ \mid xy \mid : a \leq y \leq b \} \text{ for }  \infty < x < \infty$. 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$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

7
views
isi2015mma
functions
nongate
0
votes
0
answers
14
ISI2015MMA68
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 $(0,0)$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

7
views
isi2015mma
partialderivatives
continuousdifferentiable
nongate
0
votes
0
answers
15
ISI2015MMA70
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 $0$ $1$ $2$ $4$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
partial
derivatives
nongate
0
votes
0
answers
16
ISI2015MMA71
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
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
continuity
partialderivatives
nongate
0
votes
0
answers
17
ISI2015MMA75
The length of the curve $x=t^3$, $y=3t^2$ from $t=0$ to $t=4$ is $5 \sqrt{5}+1$ $8(5 \sqrt{5}+1)$ $5 \sqrt{5}1$ $8(5 \sqrt{5}1)$
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
curves
nongate
0
votes
0
answers
18
ISI2015MMA82
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
asked
Sep 23
in
Geometry
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
solid
area
nongate
0
votes
0
answers
19
ISI2015MMA83
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
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

8
views
isi2015mma
complexnumber
maximumvalue
nongate
+1
vote
0
answers
20
ISI2015MMA84
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
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

10
views
isi2015mma
series
summation
nongate
0
votes
0
answers
21
ISI2015MMA85
The differential equation of all the ellipses centred at the origin is $y^2+x(y’)^2yy’=0$ $xyy’’ +x(y’)^2 yy’=0$ $yy’’+x(y’)^2xy’=0$ none of these
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
differentialequations
ellipses
nongate
0
votes
0
answers
22
ISI2015MMA87
If $x(t)$ is a solution of $(1t^2) dx tx\: dt =dt$ and $x(0)=1$, then $x\big(\frac{1}{2}\big)$ is equal to $\frac{2}{\sqrt{3}} (\frac{\pi}{6}+1)$ $\frac{2}{\sqrt{3}} (\frac{\pi}{6}1)$ $\frac{\pi}{3 \sqrt{3}}$ $\frac{\pi}{\sqrt{3}}$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

8
views
isi2015mma
differentialequation
nongate
0
votes
0
answers
23
ISI2015MMA88
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 $y=\frac{x+c}{f(x)}$ $y^2=\frac{f(x)}{x+c}$ $y=\frac{f(x)}{x+c}$ $y=\frac{\left[f(x)\right]^2}{x+c}$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

5
views
isi2015mma
differentialequation
generalsolution
nongate
0
votes
0
answers
24
ISI2015MMA89
Let $y(x)$ be a nontrivial 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
asked
Sep 23
in
IS&Software Engineering
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
differentialequation
nontrivialsolution
nongate
0
votes
0
answers
25
ISI2015MMA90
The differential equation of the system of circles touching the $y$axis at the origin is $x^2+y^22xy \frac{dy}{dx}=0$ $x^2+y^2+2xy \frac{dy}{dx}=0$ $x^2y^22xy \frac{dy}{dx}=0$ $x^2y^2+2xy \frac{dy}{dx}=0$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

6
views
isi2015mma
differentialequation
nongate
0
votes
0
answers
26
ISI2015MMA91
Suppose a solution of the differential equation $(xy^3+x^2y^7)\frac{\mathrm{d} y}{\mathrm{d} x}=1,$ satisfies the initial condition $y(1/4)=1$. Then the value of $\dfrac{\mathrm{d} y}{\mathrm{d} x}$ when $y=1$ is $\frac{4}{3}$ $ \frac{4}{3}$ $\frac{16}{5}$ $ \frac{16}{5}$
asked
Sep 23
in
Others
by
Arjun
Veteran
(
425k
points)

7
views
isi2015mma
differentialequation
nongate
0
votes
0
answers
27
ISI2015DCG41
A straight line touches the circle $x^2 +y^2=2a^2$ and also the parabola $y^2=8ax$. Then the equation of the straight line is $y=\pm x$ $y=\pm (x+a)$ $y=\pm (x+2a)$ $y=\pm (x21)$
asked
Sep 18
in
Others
by
gatecse
Boss
(
16.8k
points)

11
views
isi2015dcg
geometry
parabola
0
votes
0
answers
28
ISI2015DCG42
In an ellipse, the distance between its foci is $6$ and its minor axis is $8$. hen its eccentricity is $\frac{4}{5}$ $\frac{1}{\sqrt{52}}$ $\frac{3}{5}$ $\frac{1}{2}$
asked
Sep 18
in
Others
by
gatecse
Boss
(
16.8k
points)

11
views
isi2015dcg
geometry
ellipse
0
votes
0
answers
29
ISI2015DCG43
Four tangents are drawn to the ellipse $\displaystyle{}\frac{x^2}{9} + \frac{y^2}{5} =1$ at the ends of its latera recta. The area of the quadrilateral so formed is $27$ $\frac{13}{2}$ $\frac{15}{4}$ $45$
asked
Sep 18
in
Others
by
gatecse
Boss
(
16.8k
points)

13
views
isi2015dcg
geometry
ellipse
quadrilateral
0
votes
0
answers
30
ISI2016DCG15
The shaded region in the following diagram represents the relation $y\:\leq\: x$ $\mid \:y\mid \:\leq\: \mid x\:\mid $ $y\:\leq\: \mid x\:\mid$ $\mid \:y\mid\: \leq\: x$
asked
Sep 18
in
Geometry
by
gatecse
Boss
(
16.8k
points)

7
views
isi2016dcg
area
curves
nongate
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