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1
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ISI2014DCG13
Let the function $f(x)$ be defined as $f(x)=\mid x1 \mid + \mid x2 \:\mid$. Then which of the following statements is true? $f(x)$ is differentiable at $x=1$ $f(x)$ is differentiable at $x=2$ $f(x)$ is differentiable at $x=1$ but not at $x=2$ none of the above
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isi2014dcg
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3
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2
ISI2014DCG7
If $f(x) = \dfrac{\sqrt{3} \sin x}{2+\cos x}$, then the range of $f(x)$ is the interval $[1 , \sqrt{3}{/2}]$ the interval $[\sqrt{3}{/2}, 1]$ the interval $[1, 1]$ none of these
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Abhishek Kumar 40
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isi2014dcg
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1
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3
ISI2014DCG29
If $f(x) = \sin \bigg( \dfrac{1}{x^2+1} \bigg),$ then $f(x)$ is continuous at $x=0$, but not differentiable at $x=0$ $f(x)$ is differentiable at $x=0$, and $f’(0) \neq 0$ $f(x)$ is differentiable at $x=0$, and $f’(0) = 0$ None of the above
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isi2014dcg
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1
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4
ISI2014DCG60
The equation of any circle passing through the origin and with its centre on the $X$axis is given by $x^2+y^22ax=0$ where $a$ must be positive $x^2+y^22ax=0$ for any given $a \in \mathbb{R}$ $x^2+y^22by=0$ where $b$ must be positive $x^2+y^22by=0$ for any given $b \in \mathbb{R}$
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isi2014dcg
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1
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5
ISI2014DCG57
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 $x_1x_2=a^2$ $y_1y_2=a^2$ $x_1x_2^2=a^2$ $x_1^2x_2=a^2$
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isi2014dcg
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1
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6
ISI2014DCG72
The sum $\sum_{k=1}^n (1)^k \:\: {}^nC_k \sum_{j=0}^k (1)^j \: \: {}^kC_j$ is equal to $1$ $0$ $1$ $2^n$
answered
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21
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isi2014dcg
0
votes
1
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7
ISI2014DCG40
Let the following two equations represent two curves $A$ and $B$. $A: 16x^2+9y^2=144\:\: \text{and}\:\: B:x^2+y^210x=21$ 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 $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$
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isi2014dcg
0
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1
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8
ISI2017DCG15
The number of solutions of $\tan^{1}(x1) + \tan^{1}(x) + \tan^{1}(x+1) = \tan^{1}(3x)$ is $1$ $2$ $3$ $4$
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Oct 9
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ISI2017DCG23
A determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability of choosing a nonzero determinant is $\frac{3}{16}$ $\frac{3}{8}$ $\frac{1}{4}$ none of these
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2
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10
ISI2015DCG1
The sequence $\dfrac{1}{\log_3 2}, \: \dfrac{1}{\log_6 2}, \: \dfrac{1}{\log_{12} 2}, \: \dfrac{1}{\log_{24} 2} \dots $ is in Arithmetic progression (AP) Geometric progression ( GP) Harmonic progression (HP) None of these
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Oct 9
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1
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11
ISI2017DCG14
If $a,b,c$ are the sides of a triangle such that $a:b:c=1: \sqrt{3}:2$, then $A:B:C$ (where $A,B,C$ are the angles opposite to the sides of $a,b,c$ respectively) is $3:2:1$ $3:1:2$ $1:2:3$ $1:3:2$
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Oct 9
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isi2017dcg
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1
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12
ISI2015DCG59
If in a $\Delta ABC$, $\angle B = \frac{2 \pi}{3}$, then $\cos A + \cos C$ lies in $[ \sqrt{3}, \sqrt{3}]$ $( – \sqrt{3}, \sqrt{3}]$ $(\frac{3}{2}, \sqrt{3})$ $(\frac{3}{2}, \sqrt{3}]$
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Oct 9
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isi2015dcg
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13
ISI2014DCG15
Let $\mathbb{N}=\{1,2,3, \dots\}$ be the set of natural numbers. For each $n \in \mathbb{N}$, define $A_n=\{(n+1)k, \: k \in \mathbb{N} \}$. Then $A_1 \cap A_2$ equals $A_3$ $A_4$ $A_5$ $A_6$
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Oct 8
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techbd123
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isi2014dcg
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1
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14
ISI2014DCG37
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^}{2}$ and $f(x) \to – \infty$ as $x \to \dfrac{\pi^+}{2}$. Which one of the following functions satisfies the above properties of $f(x)$? $\cos x$ $\tan x$ $\tan^{1} x$ $\sin x$
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Oct 8
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techbd123
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isi2014dcg
0
votes
1
answer
15
ISI2014DCG35
Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ elements and also has the property that $S \cap A$ contains $q$ ... $\begin{pmatrix} m \\ pq \end{pmatrix} \times \begin{pmatrix} n \\ q \end{pmatrix}$
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Oct 7
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techbd123
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isi2014dcg
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1
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16
ISI2014DCG26
Let $x_1 > x_2>0$. Then which of the following is true? $\log \big(\frac{x_1+x_2}{2}\big) > \frac{\log x_1+ \log x_2}{2}$ $\log \big(\frac{x_1+x_2}{2}\big) < \frac{\log x_1+ \log x_2}{2}$ There exist $x_1$ and $x_2$ such that $x_1 > x_2 >0$ and $\log \big(\frac{x_1+x_2}{2}\big) = \frac{\log x_1+ \log x_2}{2}$ None of these
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Oct 6
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techbd123
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20
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isi2014dcg
0
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1
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17
ISI2014DCG22
The conditions on $a$, $b$ and $c$ under which the roots of the quadratic equation $ax^2+bx+c=0 \: ,a \neq 0, \: b \neq 0 $ and $c \neq 0$, are unequal magnitude but of the opposite signs, are the following: $a$ and $c$ have the same sign while $b$ has the ... $c$ has the opposite sign. $a$ and $c$ have the same sign. $a$, $b$ and $c$ have the same sign.
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Oct 6
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techbd123
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11
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isi2014dcg
0
votes
1
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18
ISI2014DCG56
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 $4$ $3$ $4$ $3$
answered
Oct 4
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isi2014dcg
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1
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19
ISI2017DCG16
If $\cos x = \dfrac{1}{2}$, the value of the expression $\dfrac{\cos 6x+6 \cos 4x+15 \cos 2x +10}{\cos 5x+5 \cos 3x +10 \cos x}$ is $\frac{1}{2}$ $1$ $\frac{1}{4}$ $0$
answered
Oct 2
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isi2017dcg
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1
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20
ISI2014DCG5
Consider the sets defined by the real solutions of the inequalities $A = \{(x,y):x^2+y^4 \leq 1\} \:\:\:\:\:\:\: B=\{(x,y):x^4+y^6 \leq 1\}$ Then $B \subseteq A$ $A \subseteq B$ Each of the sets $A – B, \: B – A$ and $A \cap B$ is nonempty none of the above
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Oct 1
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Sourajit25
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65
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isi2014dcg
0
votes
1
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21
ISI2014DCG20
If $A(t)$ is the area of the region bounded by the curve $y=e^{\mid x \mid}$ and the portion of the $x$axis between $t$ and $t$, then $\underset{t \to \infty}{\lim} A(t)$ equals $0$ $1$ $2$ $4$
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Oct 1
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techbd123
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14
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isi2014dcg
0
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2
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22
ISI2015DCG4
If $\tan x=p+1$ and $\tan y=p1$, then the value of $2 \cot (xy)$ is $2p$ $p^2$ $(p+1)(p1)$ $\frac{2p}{p^21}$
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Oct 1
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isi2015dcg
0
votes
1
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23
ISI2014DCG68
The number of integer solutions for the equation $x^2+y^2=2011$ is $0$ $1$ $2$ $3$
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Oct 1
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isi2014dcg
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1
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24
ISI2014DCG58
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 $(2, 2)$ $(2\sqrt{2},2)$ $(2, 2\sqrt{2})$ $(2\sqrt{2}, 2\sqrt{2})$
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Oct 1
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isi2014dcg
0
votes
2
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25
ISI2014DCG8
If $M$ is a $3 \times 3$ matrix such that $\begin{bmatrix} 0 & 1 & 2 \end{bmatrix}M=\begin{bmatrix}1 & 0 & 0 \end{bmatrix}$ and $\begin{bmatrix}3 & 4 & 5 \end{bmatrix} M = \begin{bmatrix}0 & 1 & 0 \end{bmatrix}$ ... $\begin{bmatrix} 1 & 2 & 0 \end{bmatrix}$ $\begin{bmatrix} 9 & 10 & 8 \end{bmatrix}$
answered
Oct 1
in
Others
by
amit166
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695
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34
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isi2014dcg
0
votes
2
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26
ISI2017DCG17
If $\cos ^{2}x+ \cos ^{4} x=1$, then $\tan ^{2} x+ \tan ^{4} x$ is equal to $1$ $0$ $2$ none of these
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Oct 1
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isi2017dcg
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2
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27
ISI2015DCG64
If $\cos 2 \theta = \sqrt{2}(\cos \theta – \sin \theta)$ then $\tan \theta$ equals $1$ $1$ or $1$ $\frac{1}{\sqrt{2}}, – \frac{1}{\sqrt{2}}$ or $1$ None of these
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Oct 1
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isi2015dcg
0
votes
1
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28
ISI2014DCG34
The following sum of $n+1$ terms $2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \end{pmatrix} + 17 \times \begin{pmatrix} n \\ 4 \end{pmatrix} + \cdots$ up to $n+1$ terms is equal to $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$
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Sep 30
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techbd123
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isi2014dcg
0
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1
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29
ISI2014DCG14
$x^43x^2+2x^2y^23y^2+y^4+2=0$ represents A pair of circles having the same radius A circle and an ellipse A pair of circles having different radii none of the above
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Sep 30
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techbd123
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isi2014dcg
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2
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30
ISI2014DCG4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
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Sep 30
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techbd123
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isi2014dcg
0
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1
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31
ISI2014DCG55
If $a,b,c$ are sides of a triangle $ABC$ such that $x^22(a+b+c)x+3 \lambda (ab+bc+ca)=0$ has real roots then $\lambda < \frac{4}{3}$ $\lambda > \frac{5}{3}$ $\lambda \in ( \frac{4}{3}, \frac{5}{3})$ $\lambda \in ( \frac{1}{3}, \frac{5}{3})$
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Sep 30
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isi2014dcg
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4
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32
ISI2014DCG3
$\underset{x \to \infty}{\lim} \bigg( \frac{3x1}{3x+1} \bigg) ^{4x}$ equals $1$ $0$ $e^{8/3}$ $e^{4/9}$
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Sep 29
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isi2014dcg
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1
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33
ISI2014DCG9
The values of $\eta$ for which the following system of equations $\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$ has a solution are $\eta=1, 2$ $\eta=1, 2$ $\eta=3, 3$ $\eta=1, 2$
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Sep 29
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isi2014dcg
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1
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34
ISI2015DCG52
$\underset{x \to 1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
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Sep 28
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isi2015dcg
0
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1
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35
ISI2014DCG69
The number of ways in which the number $1440$ can be expressed as a product of two factors is equal to $18$ $720$ $360$ $36$
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Sep 28
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Ashwani Kumar 2
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isi2014dcg
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36
ISI2017DCG9
The solution of $\log_5(\sqrt{x+5}+\sqrt{x})=1$ is $2$ $4$ $5$ none of these
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Sep 28
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isi2017dcg
0
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1
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37
ISI2014DCG71
Five letters $A, B, C, D$ and $E$ are arranged so that $A$ and $C$ are always adjacent to each other and $B$ and $E$ are never adjacent to each other. The total number of such arrangements is $24$ $16$ $12$ $32$
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Sep 28
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isi2014dcg
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1
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38
ISI2017DCG8
If $x,y,z$ are in $A.P.$ and $a>1$, then $a^x, a^y, a^z$ are in $A.P.$ $G.P$ $H.P$ none of these
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Sep 28
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isi2017dcg
0
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1
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39
ISI2017DCG27
The limit of the sequence $\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \dots$ is $1$ $2$ $2\sqrt{2}$ $\infty$
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Sep 27
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isi2017dcg
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2
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40
ISI2014DCG16
The sum of the series $\dfrac{1}{1.2} + \dfrac{1}{2.3}+ \cdots + \dfrac{1}{n(n+1)} + \cdots $ is $1$ $1/2$ $0$ nonexistent
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Sep 27
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isi2014dcg
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