Register

Recent questions tagged isi2014-dcg

3 votes
1 answer
31
ISI2014-DCG-31
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is $\alpha$ $[\alpha]$ $1$ $\dfrac{[\alpha] + [\alpha +1]}{2}$
For real $\alpha$, the value of $\int_{\alpha}^{\alpha+1} [x]dx$, where $[x]$ denotes the largest integer less than or equal to $x$, is$\alpha$$[\alpha]$$1$$\dfrac{[\alph...
User Avatar
2 votes
1 answer
32
ISI2014-DCG-32
Consider $30$ multiple-choice questions, each with four options of which exactly one is correct. Then the number of ways one can get only the alternate questions correctly answered is $3^{15}$ $2^{31}$ $2 \times \begin{pmatrix} 30 \\ 15 \end{pmatrix}$ $2 \times 3^{15}$
Consider $30$ multiple-choice questions, each with four options of which exactly one is correct. Then the number of ways one can get only the alternate questions correctl...
User Avatar
1 votes
0 answers
33
ISI2014-DCG-33
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties. $f(0)=0$, $f(1)=1$, and $f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_2<1$. Then the number of such functions is $0$ $1$ $2$ $\infty$
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties.$f(0)=0$,$f(1)=1$, and$f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_...
User Avatar
2 votes
1 answer
36
ISI2014-DCG-36
Consider any integer $I=m^2+n^2$, where $m$ and $n$ are odd integers. Then $I$ is never divisible by $2$ $I$ is never divisible by $4$ $I$ is never divisible by $6$ None of the above
Consider any integer $I=m^2+n^2$, where $m$ and $n$ are odd integers. Then$I$ is never divisible by $2$$I$ is never divisible by $4$$I$ is never divisible by $6$None of t...
User Avatar
3 votes
1 answer
38
ISI2014-DCG-38
Suppose that $A$ is a $3 \times 3$ real matrix such that for each $u=(u_1, u_2, u_3)’ \in \mathbb{R}^3, \: u’Au=0$ where $u’$ stands for the transpose of $u$. Then which one of the following is true? $A’=-A$ $A’=A$ $AA’=I$ None of these
Suppose that $A$ is a $3 \times 3$ real matrix such that for each $u=(u_1, u_2, u_3)’ \in \mathbb{R}^3, \: u’Au=0$ where $u’$ stands for the transpose of $u$. Then ...
User Avatar
1 votes
1 answer
39
ISI2014-DCG-39
The function $f(x) = x^{1/x}, \: x \neq 0$ has a minimum at $x=e$; a maximum at $x=e$; neither a maximum nor a minimum at $x=e$; None of the above
The function $f(x) = x^{1/x}, \: x \neq 0$ hasa minimum at $x=e$;a maximum at $x=e$;neither a maximum nor a minimum at $x=e$;None of the above
User Avatar
1 votes
1 answer
41
ISI2014-DCG-41
The number of permutations of the letters $a, b, c$ and $d$ such that $b$ does not follow $a,c$ does not follow $b$, and $c$ does not follow $d$, is $11$ $12$ $13$ $14$
The number of permutations of the letters $a, b, c$ and $d$ such that $b$ does not follow $a,c$ does not follow $b$, and $c$ does not follow $d$, is$11$$12$$13$$14$
User Avatar
0 votes
0 answers
42
ISI2014-DCG-42
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then $f$ has no local minima $f$ has no local maxima $f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd integers $k$ and local maxima at $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for even integers $k$ None of the above
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then$f$ has no local minima$f$ has no local maxima$f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd i...
User Avatar
0 votes
0 answers
43
ISI2014-DCG-43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid -1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
Let $$f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid -1, & \text{ if } x>0. \end{cases}$$ Then $\underset{x \to a}{\lim}...
User Avatar
0 votes
1 answer
44
ISI2014-DCG-44
The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$ has a maximum at $x= \pi /3$ has a maximum at $x= \pi$ has a minimum at $x= \pi /3$ has neither a maximum nor a minimum at $x=\pi/3$
The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$has a maximum at $x= \pi /3$has a maximum at $x= \pi$has a minimum at $x= \pi /3$has neit...
User Avatar
0 votes
1 answer
46
ISI2014-DCG-46
The maximum value of the real valued function $f(x)=\cos x + \sin x$ is $2$ $1$ $0$ $\sqrt{2}$
The maximum value of the real valued function $f(x)=\cos x + \sin x$ is$2$$1$$0$$\sqrt{2}$
User Avatar
1 votes
1 answer
47
ISI2014-DCG-47
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is $\frac{\pi}{6} + \sqrt{3}$ $\frac{\pi}{6} - \sqrt{3}$ $0$ $\frac{1}{2}$
The value of the definite integral $\int_0^{\pi} \mid \frac{1}{2} + \cos x \mid dx$ is$\frac{\pi}{6} + \sqrt{3}$$\frac{\pi}{6} - \sqrt{3}$$0$$\frac{1}{2}$
User Avatar
0 votes
1 answer
48
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
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}$$- \frac{2}{3} ...
User Avatar
0 votes
1 answer
49
ISI2014-DCG-49
Let $f(x) = \dfrac{x}{(x-1)(2x+3)}$, where $x>1$. Then the $4^{th}$ derivative of $f, \: f^{(4)} (x)$ is equal to $- \frac{24}{5} \bigg[ \frac{1}{(x-1)^5} - \frac{48}{(2x+3)^5} \bigg]$ ... $\frac{64}{5} \bigg[ \frac{1}{(x-1)^5} + \frac{48}{(2x+3)^5} \bigg]$
Let $f(x) = \dfrac{x}{(x-1)(2x+3)}$, where $x>1$. Then the $4^{th}$ derivative of $f, \: f^{(4)} (x)$ is equal to$- \frac{24}{5} \bigg[ \frac{1}{(x-1)^5} – \frac{48}{(2...
User Avatar
0 votes
0 answers
50
ISI2014-DCG-50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1- \cos tx}$ is equal to $0$ $1$ $\infty$ $2$
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1- \cos tx}$ is equal to$0$$1$$\infty$$2$
User Avatar
1 votes
1 answer
51
ISI2014-DCG-51
The function $f(x)$ defined as $f(x)=x^3-6x^2+24x$, where $x$ is real, is strictly increasing strictly decreasing increasing in $(- \infty, 0)$ and decreasing in $(0, \infty)$ decreasing in $(- \infty, 0)$ and increasing in $(0, \infty)$
The function $f(x)$ defined as $f(x)=x^3-6x^2+24x$, where $x$ is real, isstrictly increasingstrictly decreasingincreasing in $(- \infty, 0)$ and decreasing in $(0, \infty...
User Avatar
0 votes
1 answer
52
ISI2014-DCG-52
The area under the curve $x^2+3x-4$ in the positive quadrant and bounded by the line $x=5$ is equal to $59 \frac{1}{6}$ $61 \frac{1}{3}$ $40 \frac{2}{3}$ $72$
The area under the curve $x^2+3x-4$ 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}$$72$
User Avatar
0 votes
1 answer
53
ISI2014-DCG-53
The value of the integral $\displaystyle{}\int_{-1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$ $\frac{1}{2}$ $ – \frac{1}{2}$ $1$
The value of the integral $\displaystyle{}\int_{-1}^1 \dfrac{x^2}{1+x^2} \sin x \sin 3x \sin 5x dx$ is $0$$\frac{1}{2}$$ – \frac{1}{2}$$1$
User Avatar
1 votes
1 answer
54
ISI2014-DCG-54
The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to $0$ $1$ $3$ $4$
The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to$0$$1$$3$$4$
User Avatar
1 votes
1 answer
56
ISI2014-DCG-56
Two opposite vertices of a rectangle are $(1,3)$ and $(5,1)$ while the other two vertices lie on the straight line $y=2x+c$. Then the value of $c$ is $4$ $3$ $-4$ $-3$
Two opposite vertices of a rectangle are $(1,3)$ and $(5,1)$ while the other two vertices lie on the straight line $y=2x+c$. Then the value of $c$ is$4$$3$$-4$$-3$
User Avatar
1 votes
1 answer
57
ISI2014-DCG-57
If a focal chord of the parabola $y^2=4ax$ cuts it at two distinct points $(x_1,y_1)$ and $(x_2,y_2)$, then $x_1x_2=a^2$ $y_1y_2=a^2$ $x_1x_2^2=a^2$ $x_1^2x_2=a^2$
If a focal chord of the parabola $y^2=4ax$ cuts it at two distinct points $(x_1,y_1)$ and $(x_2,y_2)$, then$x_1x_2=a^2$$y_1y_2=a^2$$x_1x_2^2=a^2$$x_1^2x_2=a^2$
User Avatar
0 votes
1 answer
59
ISI2014-DCG-59
The equation $5x^2+9y^2+10x-36y-4=0$ 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 $(\pm2,0)$
The equation $5x^2+9y^2+10x-36y-4=0$ representsan ellipse with the coordinates of foci being $(\pm3,0)$a hyperbola with the coordinates of foci being $(\pm3,0)$an ellipse...
User Avatar
Page: