A ray of light is incident along the line $x = 2y$ and hits a mirror. The angle of incidence equals the angle of reflection. Find the equation of the reflected ray passing through the point $(2, 1)$. (A) $4x - 3y = 5$ (B) $3x - 4y = 2$ (C) $x - y = 1$ (D) $2x - y = 3$
A regular pentagon is inscribed in a circle of radius $r$ and another regular pentagon is circumscribed about the same circle. Find the ratio of the area of the inscribed pentagon to the area of the circumscribed pentagon. (A) $\sin^2 36^\circ$ (B) $\cos^2 36^\circ$ (C) $\tan^2 36^\circ$ (D) $\cos^2 54^\circ$
Let $z_1$ be a purely imaginary complex number and $z_2$ be any complex number such that $|z_1 + z_2| = |z_1 - z_2|$. Find the circumcentre of the triangle with vertices $0, z_1, z_2$. (A) $z_1/2$ (B) $z_2/2$ (C) $(z_1 + z_2)/2$ (D) $(z_1 - z_2)/2$
In how many ways can 20 identical chocolates be distributed among 8 students such that each student gets at least one chocolate and exactly 2 students get at least 2 chocolates? (A) 308 (B) 280 (C) 300 (D) 320
A circle of radius $r$ is inscribed in a circular sector. The chord of the sector has length $a$. If the circle touches the chord and the two radii of the sector, find the relation between $a$ and $r$. (A) $a = 8r/5$ (B) $a = 5r/8$ (C) $a = 4r/3$ (D) $a = 3r/4$
Let $P = \{2^a \cdot 3^b : 0 \leq a, b \leq 5\}$. Find the largest $n$ such that $2^n$ divides some element of $P$. (A) $n = 20$ (B) $n = 22$ (C) $n = 24$ (D) $n = 26$
Let $f(x) = ax^3 + bx^2 + cx + d$ be a strictly increasing function with $a > 0$. Define $g(x) = f'(x) - 6ax - 2b + 6a$. Then $g(x)$ is (A) always negative (B) always positive (C) sometimes positive sometimes negative (D) always zero
Let $A = (h, k)$, $B = (2, 6)$, $C = (5, 2)$ be vertices of a triangle with area 12. Find the minimum distance from $A$ to the origin. (A) $\dfrac{16}{\sqrt{5}}$ (B) $\dfrac{8}{\sqrt{5}}$ (C) $\dfrac{32}{\sqrt{5}}$ (D) $\dfrac{16}{\sqrt{5}}$
Let $a, b, c$ be positive integers with $a^2 + b^2 = c^2$. Which of the following must be true? (A) 3 divides exactly one of $a, b$ (B) 3 divides $c$ (C) $3^3$ divides $abc$ (D) $3^4$ divides $abc$
Let $N$ be the smallest positive integer such that among any $N$ consecutive integers, at least one is coprime to $374 = 2 \times 11 \times 17$. Find $N$. (A) 4 (B) 5 (C) 6 (D) 7
In a triangle $ABC$, the circumradius is $r$ and $BC = r/2$. The circumcentre $O$ lies on $AD$ where $D$ is the midpoint of $BC$. Find the ratio $BC : AD$. (A) $\sqrt{3} : \sqrt{2}$ (B) $\sqrt{2} : \sqrt{3}$ (C) $1 : \sqrt{3}$ (D) $\sqrt{3} : 1$