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
2006 solved

9 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 Discriminant and conditions for roots 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 Proof Proof Involving Combinatorial or Number-Theoretic Structure View
Show that $n^4 + 4^n$ is composite for all integers $n > 1$.
Q4 Sine and Cosine Rules 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., $AB + BD = l_1 + l_2$.
Q5 Sine and Cosine Rules Compute area of a triangle or related figure View
Three triangles are formed by drawing lines from the vertices of a triangle $ABC$ to the opposite sides, each making equal angles with the sides. Let $\Delta_1, \Delta_2, \Delta_3$ be the areas of the three smaller triangles formed at the vertices with the circumradius equal to 1.
a) Express the total area $\Delta = \Delta_1 + \Delta_2 + \Delta_3$ in terms of $A, B, C$.
b) Find the angles $A, B, C$ that maximize $\Delta$.
c) Verify that the maximum occurs for an isosceles triangle and prove $C = A$ by calculus.
Q7 Proof 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 Exponential Equations & Modelling Exponential Inequality Solving View
Find all values of $c$ for which the equation $\log_2 x = cx$ has solutions.
Q9 Number Theory Properties of Integer Sequences and Digit Analysis View
Find a 4-digit number $N$ such that $N = 4M$, where $M$ is the number obtained by reversing the digits of $N$.
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 $f(n)$.