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Recent questions and answers in Geometry

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ISRO2014-65
A cube of side $1$ unit is placed in such a way that the origin coincides with one of its top vertices and the three axes along three of its edges. What are the co-ordinates of the vertex which ... (0, 0, 0)$(0, -1, 0)$(0, 1, 0)$(1, 1, 1)$
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Game Theory(SELF DOUBT)
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3 answers
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ISRO2011-76
What is the matrix that represents rotation of an object by $\theta^0$ about the origin in $\text{2D}?$\cos \theta$- \sin \theta$\sin \theta$\cos \theta$\sin \ ... \cos \theta$\sin \theta$\sin \theta$- \cos \theta$\cos \theta$\sin \theta$
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ISI2015-MMA-45
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
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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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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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ISI2015-MMA-47
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 ... xy=k$x^2+y^2=k^2$y^2+2x^2=k^2$x^2-y^2+2yk=k^2$
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ISI2018-DCG-23
Let $A$ be the point of intersection of the lines $3x-y=1$ and $y=1$. Let $B$ be the point of reflection of the point $A$ with respect to the $y$-axis. Then the equation of ... $, is$3x-3y=2$2x+3=0$3x+2=0$3y-2=0$
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ISI2016-DCG-44
If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2},$ then the equation of the hyperbola is$y^{2}-x^{2}=32$x^{2}-y^{2}=16$y^{2}-x^{2}=16$x^{2}-y^{2}=32$
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ISI2015-MMA-46
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$(h,k) = (0,0)$ ... $(h,k)$ exists
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386
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ISI2016-DCG-43
Four tangents are drawn to the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{5}=1$ at the ends of its latera recta. The area of the quadrilateral so formed is$27$\frac{13}{2}$\frac{15}{4}$45$
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396
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ISI2016-DCG-52
The area bounded by $y=x^{2}-4,y=0$ and $x=4$ is$\frac{64}{3}$6$\frac{16}{3}$\frac{32}{3}$
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328
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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}$72$
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374
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2 answers
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ISI2016-DCG-65
The value of $\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}90^{\circ}$ is$8$9$9.5$None of these
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609
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ISI2018-DCG-26
The area of the region bounded by the curves $y=\sqrt x,$ $2y+3=x$ and $x$-axis in the first quadrant is$9$\frac{27}{4}$36$18$
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332
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ISI2018-DCG-18
If $x+y=\pi, $ the expression $\cot \dfrac{x}{2}+\cot\dfrac{y}{2}$ can be written as$2 \: \text{cosec} \: x$\text{cosec} \: x + \text{cosec} \: y$2 \: \sin x$\sin x+\sin y$
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307
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1 answers
2 votes
ISI2018-DCG-20
The value of $\tan \left(\sin^{-1}\left(\frac{3}{5}\right)+\cot^{-1}\left(\frac{3}{2}\right)\right)$ is$\frac{1}{18}$\frac{11}{6}$\frac{13}{6}$\frac{17}{6}$
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391
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1 answers
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ISI2018-DCG-22
Let the sides opposite to the angles $A,B,C$ in a triangle $ABC$ be represented by $a,b,c$ respectively. If $(c+a+b)(a+b-c)=ab,$ then the angle $C$ is$\frac{\pi}{6}$\frac{\pi}{3}$\frac{\pi}{2}$\frac{2\pi}{3}$
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494
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1 answers
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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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ISI2017-DCG-29
The area (in square unit) of the portion enclosed by the curve $\sqrt{2x}+ \sqrt{2y} = 2 \sqrt{3}$ and the axes of reference is$2$4$6$8$
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ISI2016-DCG-63
If $\sin^{-1}\frac{1}{\sqrt{5}}$ and $\cos^{-1}\frac{3}{\sqrt{10}}$ lie in $\left[0,\frac{\pi}{2}\right],$ their sum is equal to$\frac{\pi}{6}$\frac{\pi}{3}$\sin^{-1}\frac{1}{\sqrt{50}}$\frac{\pi}{4}$
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ISI2016-DCG-5
If $\tan\: x=p+1$ and $\tan\; y=p-1,$ then the value of $2\:\cot\:(x-y)$ is$2p$p^{2}$(p+1)(p-1)$\frac{2p}{p^{2}-1}$
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556
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1 answers
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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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272
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ISI2016-DCG-61
The value of $\sin^{6}\frac{\pi}{81}+\cos^{6}\frac{\pi}{81}-1+3\sin^{2}\frac{\pi}{81}\:\cos^{2}\frac{\pi}{81}$ is$\tan^{6}\frac{\pi}{81}$0$-1$None of these
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303
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ISI2016-DCG-64
If $\cos2\theta=\sqrt{2}(\cos\theta-\sin\theta)$ 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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400
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ISI2014-DCG-20
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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354
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ISI2016-DCG-66
If $\sin(\sin^{-1}\frac{2}{5}+\cos^{-1}x)=1,$ then $x$ is$1$\frac{2}{5}$\frac{3}{5}$None of these
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336
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ISI2016-DCG-39
The medians $AD$ and $BE$ of the triangle with vertices $A(0,b),B(0,0)$ and $C(a,0)$ are mutually perpendicular if$b=\sqrt{2}a$a=\pm\sqrt{2}b$b=-\sqrt{2}a$b=a$
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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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ISI2015-MMA-32
If a square of side $a$ and an equilateral triangle of side $b$ are inscribed in a circle then $a/b$ equals$\sqrt{2/3}$\sqrt{3/2}$3/ \sqrt{2}$\sqrt{2}/3$
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ISI2014-DCG-27
Let $y^2-4ax+4a=0$ and $x^2+y^2-2(1+a)x+1+2a-3a^2=0$ be two curves. State which one of the following statements is true. ... are tangent to each otherThese two curves intersect orthogonally at one pointThese two curves do not intersect
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ISI2015-MMA-35
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
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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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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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326
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ISI2016-DCG-16
The set $\{(x,y)\: :\: \mid x\mid+\mid y\mid\:\leq\:1\}$ is represented by the shaded region in
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ISI2018-DCG-25
There are three circles of equal diameter ($10$ units each) as shown in the figure below. The straight line $PQ$ passes through the centres of all the three circles. The ... of the line segment $AB$ is$6$ units$7$ units$8$ units$9$ units
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ISI2016-DCG-15
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$
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ISI2016-DCG-38
The length of the chord on the straight line $3x-4y+5=0$ intercepted by the circle passing through the points $(1,2),(3,-4)$ and $(5,6)$ is$12$14$16$18$
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ISI2016-DCG-40
The equations $x=a\cos\theta+b\sin\theta$ and $y=a\sin\theta+b\cos\theta,(0\leq\theta\leq2\pi$ and $a,b$ are arbitrary constants$)$ representa circlea parabolaan ellipsea hyperbola
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ISI2016-DCG-41
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(x-21)$
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