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
2019 UGB

8 maths questions

Q1 Number Theory Congruence Reasoning and Parity Arguments View
Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r > 1$, are of the form $n = 2^{l}$ for some $l \geq 0$.
Q2 Sequences and series, recurrence and convergence Sequence of functions convergence View
Let $f : (0, \infty) \rightarrow \mathbb{R}$ be defined by $$f(x) = \lim_{n \rightarrow \infty} \cos^{n}\left(\frac{1}{n^{x}}\right)$$
(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.
Q3 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View
Let $\Omega = \{ z = x + iy \in \mathbb{C} : |y| \leq 1 \}$. If $f(z) = z^{2} + 2$, then draw a sketch of $$f(\Omega) = \{ f(z) : z \in \Omega \}.$$ Justify your answer.
Q4 Differential equations Integral Equations Reducible to DEs View
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y} \int_{x-y}^{x+y} f(t)\, dt = f(x), \quad \text{for all } x \in \mathbb{R},\ y > 0$$ Show that there exist $a, b \in \mathbb{R}$ such that $f(x) = ax + b$ for all $x \in \mathbb{R}$.
Q5 Proof Existence Proof View
A subset $S$ of the plane is called convex if given any two points $x$ and $y$ in $S$, the line segment joining $x$ and $y$ is contained in $S$. A quadrilateral is called convex if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral $Q$ of area 1, there is a rectangle $R$ of area 2 such that $Q$ can be drawn inside $R$.
Q6 Taylor series Limit evaluation using series expansion or exponential asymptotics View
For all natural numbers $n$, let $$A_{n} = \sqrt{2 - \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}}} \quad (n \text{ many radicals})$$
(a) Show that for $n \geq 2$, $$A_{n} = 2\sin\frac{\pi}{2^{n+1}}$$
(b) Hence, or otherwise, evaluate the limit $$\lim_{n \rightarrow \infty} 2^{n} A_{n}$$
Q7 Number Theory Congruence Reasoning and Parity Arguments View
Let $f$ be a polynomial with integer coefficients. Define $$a_{1} = f(0),\quad a_{2} = f(a_{1}) = f(f(0)),$$ and $$a_{n} = f(a_{n-1}) \quad \text{for } n \geq 3$$ If there exists a natural number $k \geq 3$ such that $a_{k} = 0$, then prove that either $a_{1} = 0$ or $a_{2} = 0$.
Q8 Stationary points and optimisation Geometric or applied optimisation problem View
Consider the following subsets of the plane: $$C_{1} = \left\{(x, y) : x > 0,\ y = \frac{1}{x}\right\}$$ and $$C_{2} = \left\{(x, y) : x < 0,\ y = -1 + \frac{1}{x}\right\}$$ Given any two points $P = (x, y)$ and $Q = (u, v)$ of the plane, their distance $d(P, Q)$ is defined by $$d(P, Q) = \sqrt{(x - u)^{2} + (y - v)^{2}}$$ Show that there exists a unique choice of points $P_{0} \in C_{1}$ and $Q_{0} \in C_{2}$ such that $$d(P_{0}, Q_{0}) \leq d(P, Q) \quad \text{for all } P \in C_{1} \text{ and } Q \in C_{2}.$$