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Questions by Arjun
1
votes
1
answer
601
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
578
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
maxima-minima
calculus
+
–
0
votes
1
answer
602
ISI2014-DCG-40
Let the following two equations represent two curves $A$ and $B$. $A: 16x^2+9y^2=144\:\: \text{and}\:\: B:x^2+y^2-10x=-21$ Further, let $L$ and $M$ be the tangents to these curves $A$ and $B$, respectively, at the point $(3,0)$. Then the angle between these two tangents, $L$ and $M$, is $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$
Let the following two equations represent two curves $A$ and $B$. $$A: 16x^2+9y^2=144\:\: \text{and}\:\: B:x^2+y^2-10x=-21$$ Further, let $L$ and $M$ be the tangents to ...
268
views
asked
Sep 23, 2019
Others
isi2014-dcg
curves
+
–
1
votes
1
answer
603
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$$14$
578
views
asked
Sep 23, 2019
Combinatory
isi2014-dcg
combinatory
+
–
0
votes
0
answers
604
ISI2014-DCG-42
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then $f$ has no local minima $f$ has no local maxima $f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd integers $k$ and local maxima at $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for even integers $k$ None of the above
Let $f(x)=\sin x^2, \: x \in \mathbb{R}$. Then$f$ has no local minima$f$ has no local maxima$f$ has local minima at $x=0$ and $x=\pm\sqrt{(k+\frac{1}{2} ) \pi}$ for odd i...
413
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
maxima-minima
+
–
0
votes
0
answers
605
ISI2014-DCG-43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid -1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
Let $$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}$$ Then $\underset{x \to a}{\lim}...
363
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
functions
limits
+
–
0
votes
1
answer
606
ISI2014-DCG-44
The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$ has a maximum at $x= \pi /3$ has a maximum at $x= \pi$ has a minimum at $x= \pi /3$ has neither a maximum nor a minimum at $x=\pi/3$
The function $f(x)=\sin x(1+ \cos x)$ which is defined for all real values of $x$has a maximum at $x= \pi /3$has a maximum at $x= \pi$has a minimum at $x= \pi /3$has neit...
433
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
maxima-minima
+
–
0
votes
0
answers
607
ISI2014-DCG-45
Which of the following is true? $\log(1+x) < x- \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x- \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$ $\log(1+x) > x- \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$ $\log(1+x) < x- \frac{x^2}{2} + \frac{x^3}{3} \text{ for some } x>0$
Which of the following is true?$\log(1+x) < x- \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$$\log(1+x) x- \frac{x^2}{2} + \frac{x^3}{3} \text{ for all } x>0$$\log(...
370
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
functions
logarithms
+
–
0
votes
1
answer
608
ISI2014-DCG-46
The maximum value of the real valued function $f(x)=\cos x + \sin x$ is $2$ $1$ $0$ $\sqrt{2}$
The maximum value of the real valued function $f(x)=\cos x + \sin x$ is$2$$1$$0$$\sqrt{2}$
430
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
maxima-minima
+
–
1
votes
1
answer
609
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$$\frac{1}{2}$
518
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
integration
definite-integral
+
–
0
votes
1
answer
610
ISI2014-DCG-48
If $x$ is real, the set of real values of $a$ for which the function $y=x^2-ax+1-2a^2$ is always greater than zero is $- \frac{2}{3} < a \leq \frac{2}{3}$ $- \frac{2}{3} \leq a < \frac{2}{3}$ $- \frac{2}{3} < a < \frac{2}{3}$ None of these
If $x$ is real, the set of real values of $a$ for which the function $$y=x^2-ax+1-2a^2$$ is always greater than zero is$- \frac{2}{3} < a \leq \frac{2}{3}$$- \frac{2}{3} ...
428
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
functions
quadratic-equations
+
–
0
votes
1
answer
611
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...
683
views
asked
Sep 23, 2019
Others
isi2014-dcg
calculus
differentiation
functions
+
–
0
votes
0
answers
612
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$$2$
519
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
limits
+
–
1
votes
1
answer
613
ISI2014-DCG-51
The function $f(x)$ defined as $f(x)=x^3-6x^2+24x$, where $x$ is real, is strictly increasing strictly decreasing increasing in $(- \infty, 0)$ and decreasing in $(0, \infty)$ decreasing in $(- \infty, 0)$ and increasing in $(0, \infty)$
The function $f(x)$ defined as $f(x)=x^3-6x^2+24x$, where $x$ is real, isstrictly increasingstrictly decreasingincreasing in $(- \infty, 0)$ and decreasing in $(0, \infty...
565
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
maxima-minima
+
–
0
votes
1
answer
614
ISI2014-DCG-52
The area under the curve $x^2+3x-4$ in the positive quadrant and bounded by the line $x=5$ is equal to $59 \frac{1}{6}$ $61 \frac{1}{3}$ $40 \frac{2}{3}$ $72$
The area under the curve $x^2+3x-4$ in the positive quadrant and bounded by the line $x=5$ is equal to$59 \frac{1}{6}$$61 \frac{1}{3}$$40 \frac{2}{3}$$72$
291
views
asked
Sep 23, 2019
Geometry
isi2014-dcg
curves
area
+
–
0
votes
1
answer
615
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}$$1$
608
views
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
integration
definite-integral
+
–
1
votes
1
answer
616
ISI2014-DCG-54
The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to $0$ $1$ $3$ $4$
The number of real roots of the equation $1+\cos ^2x+\cos ^3 x – \cos^4x=5$ is equal to$0$$1$$3$$4$
562
views
asked
Sep 23, 2019
Quantitative Aptitude
isi2014-dcg
quantitative-aptitude
trigonometry
roots
+
–
0
votes
1
answer
617
ISI2014-DCG-55
If $a,b,c$ are sides of a triangle $ABC$ such that $x^2-2(a+b+c)x+3 \lambda (ab+bc+ca)=0$ has real roots then $\lambda < \frac{4}{3}$ $\lambda > \frac{5}{3}$ $\lambda \in \big( \frac{4}{3}, \frac{5}{3}\big)$ $\lambda \in \big( \frac{1}{3}, \frac{5}{3}\big)$
If $a,b,c$ are sides of a triangle $ABC$ such that $x^2-2(a+b+c)x+3 \lambda (ab+bc+ca)=0$ has real roots then$\lambda < \frac{4}{3}$$\lambda \frac{5}{3}$$\lambda \in \bi...
384
views
asked
Sep 23, 2019
Quantitative Aptitude
isi2014-dcg
quantitative-aptitude
geometry
quadratic-equations
+
–
1
votes
1
answer
618
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$$-3$
435
views
asked
Sep 23, 2019
Quantitative Aptitude
isi2014-dcg
quantitative-aptitude
geometry
rectangles
lines
+
–
1
votes
1
answer
619
ISI2014-DCG-57
If a focal chord of the parabola $y^2=4ax$ cuts it at two distinct points $(x_1,y_1)$ and $(x_2,y_2)$, then $x_1x_2=a^2$ $y_1y_2=a^2$ $x_1x_2^2=a^2$ $x_1^2x_2=a^2$
If a focal chord of the parabola $y^2=4ax$ cuts it at two distinct points $(x_1,y_1)$ and $(x_2,y_2)$, then$x_1x_2=a^2$$y_1y_2=a^2$$x_1x_2^2=a^2$$x_1^2x_2=a^2$
317
views
asked
Sep 23, 2019
Others
isi2014-dcg
parabola
non-gate
+
–
2
votes
1
answer
620
ISI2014-DCG-58
Consider a circle with centre at origin and radius $2\sqrt{2}$. A square is inscribed in the circle whose sides are parallel to the $X$ an $Y$ axes. The coordinates of one of the vertices of this square are $(2, -2)$ $(2\sqrt{2},-2)$ $(-2, 2\sqrt{2})$ $(2\sqrt{2}, -2\sqrt{2})$
Consider a circle with centre at origin and radius $2\sqrt{2}$. A square is inscribed in the circle whose sides are parallel to the $X$ an $Y$ axes. The coordinates of on...
519
views
asked
Sep 23, 2019
Quantitative Aptitude
isi2014-dcg
quantitative-aptitude
geometry
circle
squares
+
–
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