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Recent questions tagged non-gate

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ISI2015-MMA-48
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 ... $ is$\{0\}$(-4a,4a)$(-a,a)$(- \infty, \infty)$
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518
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ISI2015-MMA-49
The polar equation $r=a \cos \theta$ representsa spirala parabolaa circlenone of the above
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584
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2 answers
2 votes
ISI2015-MMA-50
Let ... +8+15+25}{4} \right) ^2 . \end{array}$ Then$V_3<V_2<V_1$V_3<V_1<V_2$V_1<V_2<V_3$V_2<V_3<V_1$
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800
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1 answers
1 votes
ISI2015-MMA-54
If $0 <x<1$, then the sum of the infinite series $\frac{1}{2}x^2+\frac{2}{3}x^3+\frac{3}{4}x^4+ \cdots$ is$\log \frac{1+x}{1-x}$\frac{x}{1-x} + \log(1+x)$\frac{1}{1-x} + \log(1-x)$\frac{x}{1-x} + \log(1-x)$
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558
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ISI2015-MMA-56
Let $\{a_n\}$ be a sequence of non-negative real numbers such that the series $\Sigma_{n=1}^{\infty} a_n$ is convergent. If $p$ is a real number such that the ... strictly less than $1$ but can be greater than or equal to $\frac{1}{2}$
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475
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ISI2015-MMA-58
Let $\{a_n\}, n \geq 1$, be a sequence of real numbers satisfying $\mid a_n \mid \leq 1$ for all $n$. Define $A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$ ... $0$-1$1$none of these
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617
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ISI2015-MMA-59
In the Taylor expansion of the function $f(x)=e^{x/2}$ about $x=3$, the coefficient of $(x-3)^5$ is$e^{3/2} \frac{1}{5!}$e^{3/2} \frac{1}{2^5 5!}$e^{-3/2} \frac{1}{2^5 5!}$none of the above
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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 $t=0$ ... $2 \pi$2 \sqrt{2 \pi}$\sqrt{2 \pi}$none of the above
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436
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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+ ... }$(n+1)^p < \frac{(n+1)^{p+1} - n^{p+1}}{p+1}$none of the above
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484
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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$2$ ... $K$; any $K>0$ will make the inequality hold.
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ISI2015-MMA-67
Given two real numbers $a<b$, let $d(x,[a,b]) = \text{min} \{ \mid x-y \mid : a \leq y \leq b \} \text{ for } - \infty < x < \infty$ ... 1$f(x)=0$ if $0 \leq x \leq 1$ and $f(x)=1$ if $ 2 \leq x \leq 3$
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319
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1 votes
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 ... $(0,0)$ and has first order partial derivatives, but not differentiable at $(0,0)$differentiable at $(0,0)$
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893
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ISI2015-MMA-69
Consider the function ... $f$ is continuous everywhere but not differentiable at $x=2$
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560
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2 answers
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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$4$
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437
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1 answers
1 votes
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 $(0,0)$ ... $(0,0)$ and $\frac{\partial f}{\partial x}(0,0)$ does not exist
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529
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1 answers
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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$8(5 \sqrt{5}-1)$
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484
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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}$2 \pi / \sqrt{3}$
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614
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1 answers
1 votes
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$
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540
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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$
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556
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1 answers
1 votes
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$3$
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562
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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$
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452
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1 answers
1 votes
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 ... and is equal to $ \int_0^1 f(x) dx$exists and is equal to $\int_0^{1/2} f(x) dx$
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455
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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
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475
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ISI2015-MMA-83
If $\alpha, \beta$ are complex numbers then the maximum value of $\dfrac{\alpha \overline{\beta}+\overline{\alpha}\beta}{\mid \alpha \beta \mid}$ ... may not always be a real number and hence maximum does not make sensenone of the above
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459
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2 answers
1 votes
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
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340
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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
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494
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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 ... $y=\sin 2x$y=\cos 2x+1$y=\sin ^2 x-1$
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429
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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}}$
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336
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1 answers
1 votes
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 ... $y=\frac{\left[f(x)\right]^2}{x+c}$
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277
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1 answers
1 votes
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 $c<0$, $k>0$ ... exists and is finitenone of the above is true
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