MCA Entrance Previous Year Question Bank

Let $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be a hyperbola where the sum of lengths of the transverse and conjugate axes is $4(2\sqrt{2}+\sqrt{14})$. If eccentricity $e=\frac{\sqrt{11}}{2}$, find $a^2+2b^2$.

Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{st}$ die is less than the number appeared on the $2^{nd}$ die, $B$ be the event that the number appeared on the $1^{st}$ die is even and that on the second die is odd, and $C$ be the event that the number appeared on the $1^{st}$ die is odd and that on the $2^{nd}$ is even. Then:

Let $H$ be the hyperbola, whose foci are $(1\pm \sqrt{2},0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is:

Let $T$ and $C$ respectively be the transverse and conjugate axes of the hyperbola $16x^2-y^2+64x+4y+44=0$. Then the area of the region above the parabola $x^2=y+4$, below the transverse axis $T$ and on the right of the conjugate axis $C$ is:

The vertices of a hyperbola $H$ are $(\pm 6,0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2}x+y=2\sqrt{2}$. If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis, then $d^2$ is equal to:

Let a tangent to the curve $y^2=24x$ meet the curve $xy=2$ at the points $A$ and $B$. Then the mid points of such line segment $AB$ lie on a parabola with the

The foci of a hyperbola are $(\pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2x+3y=6$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$ and $y$-axis are $a$ and $b$, respectively, then $|6a|+|5b|$ is equal to ______.

Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4,1)$ to the hyperbola $H:\frac{y^2}{25}-\frac{x^2}{16}=1$. If $Q$ is the point from which the tangents drawn to $H$ have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x$-axis, then $\frac{(PQ{)}^2}{\alpha \beta }$ is equal to ________.

Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of latus-rectum of the ellipse is _____.