isi-entrance

Papers (32)
2026
UGA 30 UGB 8
2024
UGA 26 UGB 8
2023
UGA 22 UGB 7
2022
UGA 26 UGB 9
2021
UGA 27 UGB 8
2020
UGA 24 UGB 7
2019
UGA 21 UGB 7
2018
UGA 24 UGB 8
2017
UGA 27 UGB 8
2016
UGA 63 UGB 67
2015
UGA 38 UGB 35
2014
solved 19
2013
isi-entrance_2013.pdf 68
2012
solved 28
2011
solved 18
2010
solved 16
2009
solved 10
2007
solved 6
2006
solved 7
2005
solved 8
Unknown
subjective_collection 10
2006 solved

7 maths questions

Q1 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
If $x^{2/3} + y^{2/3} = a^{1/3}$, find the equation of the tangent to the curve at a point, and show that the length of the tangent intercepted between the axes is constant.
Q2 Proof Nature of roots given coefficient constraints View
a) Show that if $a$ and $b$ are irrational numbers that are roots of a quadratic with rational coefficients, then $(a-b)^2$ is not a perfect square of any rational number.
b) i) If $a = r \pm \sqrt{s}$ is a quadratic surd, find a rational $x$ such that $a + x$ is irrational but $a_n = (r + (r^2 - s)) \pm \sqrt{s} \notin \mathbb{Q}$. If $a$ is not a surd, take $x = -a$.
ii) Find $y$ such that the required condition holds.
Q3 Number Theory Proof Involving Combinatorial or Number-Theoretic Structure View
Show that $n^4 + 4^n$ is composite for all integers $n > 1$.
Q4 Proof Multi-step composite figure problem View
In a triangle, $E$ is the midpoint of $AC$. Let $\angle BCE = \angle ABE$. Prove that $AB + BD = CD$ (where $D$ is the midpoint of $BC$), i.e., $AB + BD = l_1 + l_2$.
Q7 Binomial Theorem (positive integer n) Direct Proof of an Inequality View
Prove that $2^n < \dbinom{2n}{n} < \dfrac{2^n}{\prod_{j=0}^{n-1}\left(1 - \frac{j}{n}\right)}$ for all positive integers $n$.
Q8 Curve Sketching Exponential Inequality Solving View
Find all values of $c$ for which the equation $\log_2 x = cx$ has solutions.
Q10 Sequences and series, recurrence and convergence Closed-form expression derivation View
Let $f(n)$ satisfy the recurrence $f(n) + f(n-1) = nf(n-1) + (n-1)f(n-2)$ with $f(0) = 1$, $f(1) = 0$. Find a closed form for $f(n)$.