# Recent questions tagged curves

1
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. These two curves intersect at two points These two curves are tangent to each other These two curves intersect orthogonally at one point These two curves do not intersect
2
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 $0^{\circ}$ $30^{\circ}$ $45^{\circ}$ $90^{\circ}$
3
The area under the curve $x^2+3x-4$ in the positive quadrant and bounded by the line $x=5$ is equal to $59 \frac{1}{6}$ $61 \frac{1}{3}$ $40 \frac{2}{3}$ $72$
4
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 member of $\mathcal{G}$ intersect each member of $\mathcal{F}$ orthogonally. Then $\mathcal{G}$ is given by $xy=k$ $x^2+y^2=k^2$ $y^2+2x^2=k^2$ $x^2-y^2+2yk=k^2$
5
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 $t=0$ and $t=2 \pi$ is $2 \pi$ $2 \sqrt{2 \pi}$ $\sqrt{2 \pi}$ none of the above
6
The length of the curve $x=t^3$, $y=3t^2$ from $t=0$ to $t=4$ is $5 \sqrt{5}+1$ $8(5 \sqrt{5}+1)$ $5 \sqrt{5}-1$ $8(5 \sqrt{5}-1)$
7
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 the point $(\pi/2, 0)$ when $t=0$, then the equation of the curve in rectangular co-ordinates is $y=1/2 \cos ^2 x$ $y=\sin 2x$ $y=\cos 2x+1$ $y=\sin ^2 x-1$
8
The shaded region in the following diagram represents the relation $y\:\leq\: x$ $\mid \:y\mid \:\leq\: \mid x\:\mid$ $y\:\leq\: \mid x\:\mid$ $\mid \:y\mid\: \leq\: x$
9
The set $\{(x,y)\: :\: \mid x\mid+\mid y\mid\:\leq\:1\}$ is represented by the shaded region in
10
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$)$ represent a circle a parabola an ellipse a hyperbola
11
If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2},$ then the equation of the hyperbola is $y^{2}-x^{2}=32$ $x^{2}-y^{2}=16$ $y^{2}-x^{2}=16$ $x^{2}-y^{2}=32$
The piecewise linear function for the following graph is $f(x)=\begin{cases} = x,x\leq-2 \\ =4,-2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = x-2,x\leq-2 \\ =4,-2<x<3 \\ = x-1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2x,x\leq-2 \\ =x,-2<x<3 \\ = x+1,x\geq 3\end{cases}$ $f(x)=\begin{cases} = 2-x,x\leq-2 \\ =4,-2<x<3 \\ = x+1,x\geq 3\end{cases}$
The area bounded by $y=x^{2}-4,y=0$ and $x=4$ is $\frac{64}{3}$ $6$ $\frac{16}{3}$ $\frac{32}{3}$
The area of the region bounded by the curves $y=\sqrt x,$ $2y+3=x$ and $x$-axis in the first quadrant is $9$ $\frac{27}{4}$ $36$ $18$