isi-entrance

Papers (32)
2026
UGA 30 UGB 8
2024
UGA 30 UGB 8
2023
UGA 30 UGB 8
2022
UGA 30 UGB 9
2021
UGA 30 UGB 8
2020
UGA 30 UGB 8
2019
UGA 30 UGB 8
2018
UGA 30 UGB 8
2017
UGA 30 UGB 8
2016
UGA 73 UGB 73
2015
UGA 39 UGB 39
2014
solved 22
2013
isi-entrance_2013.pdf 73
2012
solved 29
2011
solved 19
2010
solved 21
2009
solved 10
2007
solved 6
2006
solved 9
2005
solved 10
Unknown
subjective_collection 10
2014 solved

22 maths questions

Q1 Inequalities Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View
If $a, b, c, d$ are real numbers such that $a - b^2 \geq 1/4$, $b - c^2 \geq 1/4$, $c - d^2 \geq 1/4$, $d - a^2 \geq 1/4$, find the number of solutions $(a, b, c, d)$.
(A) 0 \quad (B) 1 \quad (C) 2 \quad (D) Infinitely many
Q2 Laws of Logarithms Express One Logarithm in Terms of Another View
If $\log_{12} 18 = a$, then $\log_{24} 16$ equals
(A) $\dfrac{8 - 4a}{5 - a}$ \quad (B) $\dfrac{4a - 8}{a - 5}$ \quad (C) $\dfrac{4 + 8a}{5 + a}$ \quad (D) None of these
Q3 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
Find the number of solutions of $\sec x + \tan x = 2\cos x$ in $[0, 2\pi]$.
(A) 0 \quad (B) 1 \quad (C) 2 \quad (D) 3
Q4 Permutations & Arrangements Forming Numbers with Digit Constraints View
Find the number of six-digit numbers using digits from $\{2, 3, 9\}$ (repetition allowed) that are divisible by 6.
(A) 80 \quad (B) 82 \quad (C) 81 \quad (D) 83
Q5 Integration by Substitution Substitution Combined with Symmetry or Companion Integral View
Evaluate $\displaystyle I = \int_{1/2014}^{2014} \frac{\tan^{-1} x}{x}\, dx$.
(A) $\dfrac{\pi}{2} \log(2014)$ \quad (B) $\pi \log(2014)$ \quad (C) $2\pi \log(2014)$ \quad (D) $\dfrac{\pi}{4} \log(2014)$
Q6 Straight Lines & Coordinate Geometry Reflection and Image in a Line View
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) $4x - 3y = 5$ \quad (B) $3x - 4y = 2$ \quad (C) $x - y = 1$ \quad (D) $2x - y = 3$
Q7 Modulus function Counting solutions satisfying modulus conditions View
Find the number of solutions of $|2x - [x]| = 4$, where $[x]$ denotes the greatest integer function.
(A) 2 \quad (B) 3 \quad (C) 4 \quad (D) Infinitely many
Q8 Sine and Cosine Rules Compute area of a triangle or related figure View
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) $\sin^2 36^\circ$ \quad (B) $\cos^2 36^\circ$ \quad (C) $\tan^2 36^\circ$ \quad (D) $\cos^2 54^\circ$
Q9 Complex Numbers Argand & Loci Geometric Properties of Triangles/Polygons from Affixes View
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) $z_1/2$ \quad (B) $z_2/2$ \quad (C) $(z_1 + z_2)/2$ \quad (D) $(z_1 - z_2)/2$
Q10 Combinations & Selection Counting Integer Solutions to Equations View
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 \quad (B) 280 \quad (C) 300 \quad (D) 320
Q11 Circles Inscribed/Circumscribed Circle Computations View
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) $a = 8r/5$ \quad (B) $a = 5r/8$ \quad (C) $a = 4r/3$ \quad (D) $a = 3r/4$
Q12 Number Theory Divisibility and Divisor Analysis View
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) $n = 20$ \quad (B) $n = 22$ \quad (C) $n = 24$ \quad (D) $n = 26$
Q13 Stationary points and optimisation Determine intervals of increase/decrease or monotonicity conditions View
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 \quad (B) always positive \quad (C) sometimes positive sometimes negative \quad (D) always zero
Q14 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
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) $\dfrac{16}{\sqrt{5}}$ \quad (B) $\dfrac{8}{\sqrt{5}}$ \quad (C) $\dfrac{32}{\sqrt{5}}$ \quad (D) $\dfrac{16}{\sqrt{5}}$
Q15 Number Theory Congruence Reasoning and Parity Arguments View
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 $a, b$ \quad (B) 3 divides $c$ \quad (C) $3^3$ divides $abc$ \quad (D) $3^4$ divides $abc$
Q17 Composite & Inverse Functions Find or Apply an Inverse Function Formula View
Let $f(x) = \dfrac{1}{x-2}$. Find the $x$-coordinates of the points where $f(x) = f^{-1}(x)$.
(A) $x = 1 \pm \sqrt{2}$ \quad (B) $x = 2 \pm \sqrt{2}$ \quad (C) $x = 1 \pm \sqrt{3}$ \quad (D) $x = 0, 4$
Q18 Number Theory GCD, LCM, and Coprimality View
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 \quad (B) 5 \quad (C) 6 \quad (D) 7
Q20 Small angle approximation View
For what values of $a$ does $\displaystyle\lim_{x \to 0} \frac{\sin^a x}{x} = 0$?
(A) $a \geq 1$ \quad (B) $a > 1$ \quad (C) $a \leq 1$ \quad (D) All real $a$
Q21 Areas by integration View
Let $R = \{(x, y) : x^2 + y^2 \leq 100,\ \sin(x+y) > 0\}$. Find the area of the region $R$.
(A) $25\pi$ \quad (B) $50\pi$ \quad (C) $100\pi$ \quad (D) $75\pi$
Q22 Sine and Cosine Rules Circumradius or incircle radius computation View
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) $\sqrt{3} : \sqrt{2}$ \quad (B) $\sqrt{2} : \sqrt{3}$ \quad (C) $1 : \sqrt{3}$ \quad (D) $\sqrt{3} : 1$
Q24 Combinations & Selection Counting Integer Solutions to Equations View
Find the number of ordered triples $(a, b, c)$ of positive integers such that $abc = 1000$.
(A) 90 \quad (B) 100 \quad (C) 110 \quad (D) 120
Q25 Chain Rule Limit Involving Derivative Definition of Composed Functions View
Let $f$ be a differentiable function with $f(3) \neq 0$. Evaluate $\displaystyle\lim_{x \to \infty} \left(\frac{f(3 + 1/x)}{f(3)}\right)^x$.
(A) $e^{f'(3)/f(3)}$ \quad (B) $e^{f(3)}$ \quad (C) $e^{f'(3)}$ \quad (D) 1