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
2022 UGB

9 maths questions

Q1 Permutations & Arrangements Distribution of Objects into Bins/Groups View
Consider a board having 2 rows and $n$ columns. Thus there are $2n$ cells in the board. Each cell is to be filled in by 0 or 1.
(a) In how many ways can this be done such that each row sum and each column sum is even?
(b) In how many ways can this be done such that each row sum and each column sum is odd?
Q2 Stationary points and optimisation Find critical points and classify extrema of a given function View
Consider the function $$f(x) = \sum_{k=1}^{m} (x-k)^4, \quad x \in \mathbb{R}$$ where $m > 1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.
Q3 Straight Lines & Coordinate Geometry Locus Determination View
Consider the parabola $C: y^2 = 4x$ and the straight line $L: y = x + 2$. Let $P$ be a variable point on $L$. Draw the two tangents from $P$ to $C$ and let $Q_1$ and $Q_2$ denote the two points of contact on $C$. Let $Q$ be the mid-point of the line segment joining $Q_1$ and $Q_2$. Find the locus of $Q$ as $P$ moves along $L$.
Q4 Roots of polynomials Existence or counting of roots with specified properties View
Let $P(x)$ be an odd degree polynomial in $x$ with real coefficients. Show that the equation $P(P(x)) = 0$ has at least as many distinct real roots as the equation $P(x) = 0$.
Q5 Number Theory Arithmetic Functions and Multiplicative Number Theory View
For any positive integer $n$, and $i = 1, 2$, let $f_i(n)$ denote the number of divisors of $n$ of the form $3k + i$ (including 1 and $n$). Define, for any positive integer $n$, $$f(n) = f_1(n) - f_2(n)$$ Find the values of $f\left(5^{2022}\right)$ and $f\left(21^{2022}\right)$.
Q6 Sequences and Series Recurrence Relations and Sequence Properties View
Consider a sequence $P_1, P_2, \ldots$ of points in the plane such that $P_1, P_2, P_3$ are non-collinear and for every $n \geq 4$, $P_n$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_1$ and $P_5$. Prove the following:
(a) The area of the triangle formed by the points $P_n, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity.
(b) The point $P_9$ lies on $L$.
Q7 Taylor series Limit evaluation using series expansion or exponential asymptotics View
Let $$P(x) = 1 + 2x + 7x^2 + 13x^3, \quad x \in \mathbb{R}$$ Calculate for all $x \in \mathbb{R}$, $$\lim_{n \to \infty} \left(P\left(\frac{x}{n}\right)\right)^n$$
Q8 Trigonometric equations in context View
Find the minimum value of $$|\sin x + \cos x + \tan x + \cot x + \sec x + \operatorname{cosec} x|$$ for real numbers $x$ not multiple of $\pi/2$.
Q9 Complex Numbers Arithmetic Modulus Inequalities and Bounds (Proof-Based) View
Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_1 + \ldots + z_n\right| \geq \frac{1}{k}\left(\left|z_1\right| + \ldots + \left|z_n\right|\right)$$ for every positive integer $n \geq 2$ and every choice $z_1, \ldots, z_n$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n = 2$. Then show that the same $k$ works for any $n \geq 2$.]