MCA Entrance Previous Year Question Bank

Let $S$ be the set of all $a\in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $P(b,c),b,c\in \mathbb{N}$, on the parabola $y^2=2ax$ and the lines $x=b,y=0$ is $16{\text{ unit}}^2$, then ${\sum }_{a\in S}a$ is equal to

A triangle is formed by X-axis, Y-axis and the line $3x+4y=60$. Then the number of points $P(a,b)$ which lie strictly inside the triangle, where $a$ is an integer and $b$ is a multiple of $a$, is

Let the plane $x+3y-2z+6=0$ meet the co-ordinate axes at the points $A,B,C$. If the orthocenter of the triangle $ABC$ is $(\alpha ,\beta ,\frac{6}{7})$, then $98(\alpha +\beta {)}^2$ is equal to ___.

In a triangle $ABC$, if $\cos A+2\cos B+\cos C=2$ and the lengths of the sides opposite to the angles $A$ and $C$ are $3$ and $7$ respectively, then $\cos A-\cos C$ is equal to

In the figure, ${\theta }_1+{\theta }_2=\frac{\pi }{2}$ and $\sqrt{3}(BE)=4(AB)$. If the area of $△CAB$ is $2\sqrt{3}-3$ unit${}^2$, when $\frac{{\theta }_2}{{\theta }_1}$ is the largest, then the perimeter (in unit) of $△CED$ is equal to

$A$ line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius :

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 squareunits, respectively. Then the area (in square units) of the DBCD is equal to:

Let the vertices $Q$ and $R$ of the triangle $PQR$ lie on the line $$\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$$$QR=5$ and the coordinates of the point $P$ be $(0,2,3)$. If the area of the triangle $PQR$ is $\frac{m}{n}$ then: