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
2020 UGB

8 maths questions

Q1 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
Let $i$ be a root of the equation $x^{2} + 1 = 0$ and let $\omega$ be a root of the equation $x^{2} + x + 1 = 0$. Construct a polynomial
$$f(x) = a_{0} + a_{1}x + \ldots + a_{n}x^{n}$$
where $a_{0}, a_{1}, \ldots, a_{n}$ are all integers such that $f(i + \omega) = 0$.
Q2 Discriminant and conditions for roots Condition for repeated (equal/double) roots View
Let $a$ be a fixed real number. Consider the equation
$$(x + 2)^{2}(x + 7)^{2} + a = 0, \quad x \in \mathbb{R},$$
where $\mathbb{R}$ is the set of real numbers. For what values of $a$, will the equation have exactly one double-root?
Q3 Straight Lines & Coordinate Geometry Locus Determination View
Let $A$ and $B$ be variable points on $x$-axis and $y$-axis respectively such that the line segment $AB$ is in the first quadrant and of a fixed length $2d$. Let $C$ be the mid-point of $AB$ and $P$ be a point such that
(a) $P$ and the origin are on the opposite sides of $AB$ and,
(b) $PC$ is a line segment of length $d$ which is perpendicular to $AB$.
Find the locus of $P$.
Q4 Sequences and series, recurrence and convergence Convergence proof and limit determination View
Let a real-valued sequence $\left\{x_{n}\right\}_{n \geq 1}$ be such that
$$\lim_{n \rightarrow \infty} n x_{n} = 0$$
Find all possible real values of $t$ such that $\lim_{n \rightarrow \infty} x_{n}(\log n)^{t} = 0$.
Q5 Stationary points and optimisation Geometric or applied optimisation problem View
Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius 1 is regular (i.e., has equal sides).
Q6 Implicit equations and differentiation Eliminate parameter from implicit family and derive ODE View
Prove that the family of curves
$$\frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} = 1$$
satisfies
$$\frac{dy}{dx}\left(a^{2} - b^{2}\right) = \left(x + y\frac{dy}{dx}\right)\left(x\frac{dy}{dx} - y\right).$$
Q7 Number Theory GCD, LCM, and Coprimality View
Consider a right-angled triangle with integer-valued sides $a < b < c$ where $a, b, c$ are pairwise co-prime. Let $d = c - b$. Suppose $d$ divides $a$. Then
(a) Prove that $d \leq 2$.
(b) Find all such triangles (i.e. all possible triplets $a, b, c$) with perimeter less than 100.
Q8 Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View
A finite sequence of numbers $(a_{1}, \ldots, a_{n})$ is said to be alternating if
$$a_{1} > a_{2}, \quad a_{2} < a_{3}, \quad a_{3} > a_{4}, \quad a_{4} < a_{5}, \ldots$$ $$\text{or} \quad a_{1} < a_{2}, \quad a_{2} > a_{3}, \quad a_{3} < a_{4}, \quad a_{4} > a_{5}, \ldots$$
How many alternating sequences of length 5, with distinct numbers $a_{1}, \ldots, a_{5}$ can be formed such that $a_{i} \in \{1, 2, \ldots, 20\}$ for $i = 1, \ldots, 5$?