MCA Entrance Previous Year Question Bank

The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2y+1=0$ and $2x-y-1=0$ and whose orthocenter is $(\frac{7}{3},\frac{7}{3})$ is:

The distance between the two points $A$ and $A^{\prime }$ which lie on $y=2$ such that both the line segments $AB$ and $A^{\prime }B$ (where $B$ is the point $(2,3)$) subtend angle $\frac{\pi }{4}$ at the origin, is equal to:

A straight line cuts off intercepts $OA=a,OB=b$. If the perpendicular from origin makes an angle of $\frac{\pi }{6}$ with positive $y$-axis and area of $\Delta OAB$ is $\frac{98}{3}\sqrt{3}$, then $a^2-b^2$ is:

A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2x$ and $x^2+y^2=4x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r^2$ is equal to:

A light ray emits from the origin making an angle $30^{\circ }$ with the positive $x$-axis. After getting reflected by the line $x+y=1$, if this ray intersects $x$-axis at $Q$, then the abscissa of $Q$ is:

Let $B$ and $C$ be two points on the line $y+x=0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y-2x=2$ such that $\Delta ABC$ is an equilateral triangle. Then, the area of the $\Delta ABC$ is:

Consider the triangles with vertices $A(2,1)$, $B(0,0)$ and $C(t,4)$, $t\in [0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$, respectively, then $6\alpha +21\beta$ is equal to ________.

If $(\alpha ,\beta )$ is the orthocenter of the triangle $ABC$ with vertices $A(3,-7)$, $B(-1,2)$ and $C(4,5)$, then $9\alpha -6\beta +60$ is equal to:

Let $S$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda }{0}=\frac{y-3}{4}=\frac{z+6}{1}\text{and}\frac{x+\lambda }{3}=\frac{y}{-4}=\frac{z-6}{0}\text{is }13.$ Then, $8\left|{\sum }_{\lambda \in S}\lambda \right|$ is equal to:

Let $(\alpha ,\beta )$ be the centroid of the triangle formed by the lines $15x-y=82$, $6x-5y=-4$ and $9x+4y=17$. Then, $\alpha +2\beta$ and $2\alpha -\beta$ are the roots of the equation: