Arun Sharma Solutions for Exercise 1: CA(1999-2008, 2017)

An equilateral triangle $\mathrm{BPC}$ is drawn inside a square $\mathrm{ABCD}.$ What is the value of the angle $\mathrm{APD}$ in degrees?

In a triangle $\mathrm{ABC},$ the lengths of the sides $\mathrm{AB}$ and AC equal $17.5 \mathrm{cm}$ and $9 \mathrm{cm}$ respectively. Let $\mathrm{D}$ be a point on the line segment $\mathrm{BC}$ such that $\mathrm{AD}$ is perpendicular to $\mathrm{BC}.$ If $\mathrm{AD}=3 \mathrm{cm},$ then what is the radius (in $\mathrm{cm}$ ) of the circle circumscribing the triangle $\mathrm{ABC} ?$

Consider obtuse-angled triangles with sides $8 \mathrm{cm},$ $15 \mathrm{cm}$ and $\mathrm{x} \mathrm{cm}.$ If $\mathrm{x}$ is an integer then how many such triangles exist?

A ball of diameter $4 \mathrm{cm}$ is kept on top of a hollow cylinder standing vertically. The height of the cylinder is $3 \mathrm{cm},$ while its volume is $9{\mathrm{\pi cm}}^3.$ Then the vertical distance, in $\mathrm{cm},$ of the topmost point of the ball from the base of the cylinder is 

If three sides of a rectangular park have a total length $400 \mathrm{ft},$ then the area of the park is maximum when the length (in $\mathrm{ft}$ ) of its longer side is

$\mathrm{ABCD}$ is a quadrilateral inscribed in a circle with centre $\mathrm{O}.$ If $\angle \mathrm{COD}=120$ degrees and $\angle \mathrm{BAC}=30$ degrees, then the value of $\angle \mathrm{BCD}$ (in degrees) is

Let $\mathrm{P}$ be an interior point of a right-angled isosceles triangle $\mathrm{ABC}$ with hypotenuse $\mathrm{AB}.$ If the perpendicular distance of $\mathrm{P}$ from each of $\mathrm{AB}, \mathrm{BC},$ and $\mathrm{CA}$ is $4(\sqrt{2}-1)\mathrm{cm},$ then the area, in sq $\mathrm{cm},$ of the triangle $\mathrm{ABC}$ is

The points $(2, 5)$ and $(6, 3)$ are two end points of a diagonal of a rectangle. If the other diagonal has the equation $\mathrm{y}=3\mathrm{x}+\mathrm{c},$ then $\mathrm{c}$ is