MCA Entrance Previous Year Question Bank

Let $PQ$ be a focal chord of the parabola $y^2=4x$ such that it subtends an angle of $\frac{\pi }{2}$ at the point $(3,0)$. Let the line segment $PQ$ be also a focal chord of the ellipse E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a^2 > b^2. If $e$ is the eccentricity of the ellipse $E$, then the value of $\frac{1}{e^2}$ is equal to :

The line $x=8$ is the directrix of the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with focus $(2,0)$. If the tangent to $E$ at point $P$ in the first quadrant passes through $(0,4\sqrt{3})$ and intersects the $x$-axis at $Q$, then $(3PQ{)}^2$ is equal to:

If the maximum distance of normal of the ellipse \frac{x^2}{4} + \frac{y^2}{b^2} = 1, b < 2, from the origin is $1$, then the eccentricity of the ellipse is :

If the tangent at a point $P$ on the parabola $y^2=3x$ is parallel to the line $x+2y=1$ and the tangents at the points $Q$ and $R$ on the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are perpendicular to the line $x-y=2$, then the area of the triangle $PQR$ is:

Let $C$ be the largest circle centred at $(2,0)$ and inscribed in the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If $(1,\alpha )$ lies on $C$, then $10{\alpha }^2$ is equal to

Let a tangent to the curve $9x^2+16y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $AB$ is:

Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along the $x$-axis. If its minor axis subtends an angle $60^{\circ }$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to ________.

Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$-axis at the points $A$ and $B$, respectively. Let the circle $C$ be drawn taking $AB$ as a diameter and the line $x=2\sqrt{5}$ intersect $C$ at the points $P$ and $Q$. If the tangents at the points $P$ and $Q$ on the circle intersect at the point $(\alpha ,\beta )$, then ${\alpha }^2-{\beta }^2$ is equal to:

Let $P\left(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}\right)$, $Q$, $R$, and $S$ be four points on the ellipse $9x^2+4y^2=36$. Let $PQ$ and $RS$ be mutually perpendicular and pass through the origin. If $\frac{1}{(PQ{)}^2}+\frac{1}{(RS{)}^2}=\frac{p}{q}$, where $p$ and $q$ are coprime, then $p+q$ is equal to

If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^2+4y^2=36$ is $r$, then $12r^2$ is equal to: