MCA Entrance Previous Year Question Bank

Let P : y^2 = 4ax, a > 0 be a parabola with focus $S$. Let the tangents to the parabola $P$ make an angle of $\frac{\pi }{4}$ with the line $y=3x+5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A,B$ and $S$ are collinear is:

If the $x$-intercept of a focal chord of the parabola $y^2=8x+4y+4$ is $3$, then the length of this chord is equal to:

Let $y=f(x)$ represent a parabola with focus $(-\frac{1}{2},0)$ and directrix $y=-\frac{1}{2}$. Then $S=\{x\in \mathbb{R}:{\tan }^{-1}(\sqrt{f(x)})+{\sin }^{-1}(\sqrt{f(x)+1})=\frac{\pi }{2}\}$:

The parabolas $a{x}^2+2bx+cy=0$ and $d{x}^2+2ex+fy=0$ intersect on the line $y=1$. If $a,b,c,d,e,f$ are positive real numbers and $a,b,c$ are in G.P., then:

If $P(h,k)$ be a point on the parabola $x=4y^2$, which is nearest to the point $Q(0,33)$, then the distance of $P$ from the directrix of the parabola $y^2=4(x+y)$ is equal to:

The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to parabola $y^2=6x$. The locus of its circumcentre is:

If the area enclosed by the parabola $P_1:2y=5x^2$ and $P_2:x^2-y+6=0$ is equal to the area enclosed by $P_1$ and y = \alpha x, \alpha > 0, then ${\alpha }^3$ is equal to:

The distance of the point $(6,-2\sqrt{2})$ from the common tangent y = mx + c, m > 0, of the curves $x=2y^2$ and $x=1+y^2$ is:

Let $PQ$ be a focal chord of the parabola $y^2=36x$ of length $100$, making an acute angle with the positive $x$-axis. Let the ordinate of $P$ be positive and $M$ be the point on the line segment $PQ$ such that $PM:MQ=3:1$. Then, which of the following points does NOT lie on the line passing through $M$ and perpendicular to the line $PQ$?

Let the tangent to the parabola $y^2=12x$ at the point $(3,\alpha )$ be perpendicular to the line $2x+2y=3$. Then, the square of distance of the point $(6,-4)$ from the normal to the hyperbola ${\alpha }^2{x}^2-9y^2=9{\alpha }^2$ at its point $(\alpha -1,\alpha +2)$ is equal to _____ .