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
2009 solved

10 maths questions

Q1 Sine and Cosine Rules Multi-step composite figure problem View
Let $ABC$ be a triangle $A \neq B$ and let $P \in (AB)$ be a point for which denote $m(\widehat{ACP}) = x$ and $m(\widehat{BCP}) = y$. Prove that $\frac{\sin A \sin B}{\sin(A-B)} = \frac{\sin x \sin y}{\sin(x-y)}$ if and only if $PA = PB$.
Q2 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f$ be a differentiable function on $[0, 2\pi]$ with $f'(x)$ increasing. Show that $\int_0^{2\pi} f(x) \cos x \, dx \geq 0$.
Q3 Areas Between Curves Maximize or Optimize Area View
A triangle has vertices $A$, $B$, $C$. A point $P$ is chosen on side $AB$, and lines through $P$ parallel to the other sides create smaller triangles $APQ$ and $BPR$ and a parallelogram $PQCR$. Find the minimum value of the maximum of the areas of triangles $APQ$ and $BPR$ as a fraction of the area of $ABC$.
Q4 Sequences and Series Recurrence Relations and Sequence Properties View
Find the general term $T_r$ of the sequence $2, 7, 14, 23, 34, \ldots$
Q5 Stationary points and optimisation Geometric or applied optimisation problem View
Find the maximum volume of a rectangular box (with a lid) that can be inscribed in a cylinder of radius $30$ cm and height $60$ cm.
Q6 Taylor series Extract derivative values from a given series View
Let $\log x = g(x) = x f(x)$. Find $f^{(n)}(1)$, the $n$-th derivative of $f$ evaluated at $x = 1$.
Q7 Circles Inscribed/Circumscribed Circle Computations View
Let $A, B, C, D, E$ be the vertices of a regular pentagon inscribed in a circle of radius $r$. Let $F$ be the midpoint of side $AB$. Find the circumradius $AO$ in terms of the side length $x = AB$.
Q8 Combinations & Selection Counting Integer Solutions to Equations View
Let $f(n)$ be the number of ways to write a positive integer as an ordered sum of three non-negative integers, where each integer is chosen from $\{0, 1, 2, \ldots, 2n-1\}$ (i.e., using $n$ colours with values $0$ to $2n-1$). Find $f(n)$.
Q9 Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
Let $P_0 = (0,0)$, $P_1 = (0,4)$, $P_2 = (4,0)$, $P_3 = (-4,-4)$, $P_4 = (2,4)$, $P_5 = (4,6)$ (or similar points). Find the region of all points closer to $P_0$ than to any of $P_1, P_2, P_3, P_4, P_5$, and compute its perimeter.
Q10 Number Theory Properties of Integer Sequences and Digit Analysis View
Find the $n$-th non-square positive integer, and show that it equals $n + \lfloor \sqrt{n} + \frac{1}{2} \rfloor$.