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
2017 UGB

8 maths questions

Q1 Number Theory Congruence Reasoning and Parity Arguments View
Let the sequence $\left\{ a _ { n } \right\} _ { n \geq 1 }$ be defined by
$$a _ { n } = \tan ( n \theta )$$
where $\tan ( \theta ) = 2$. Show that for all $n , a _ { n }$ is a rational number which can be written with an odd denominator.
Q2 Circles Chord Length and Chord Properties View
Consider a circle of radius 6 as given in the diagram below. Let $B$, $C , D$ and $E$ be points on the circle such that $B D$ and $C E$, when extended, intersect at $A$. If $A D$ and $A E$ have length 5 and 4 respectively, and $D B C$ is a right angle, then show that the length of $B C$ is $\frac { 12 + 9 \sqrt { 15 } } { 5 }$.
Q3 Differentiation from First Principles View
Suppose $f : \mathbb { R } \rightarrow \mathbb { R }$ is a function given by
$$f ( x ) = \begin{cases} 1 & \text { if } x = 1 \\ e ^ { \left( x ^ { 10 } - 1 \right) } + ( x - 1 ) ^ { 2 } \sin \left( \frac { 1 } { x - 1 } \right) & \text { if } x \neq 1 \end{cases}$$
(a) Find $f ^ { \prime } ( 1 )$.
(b) Evaluate $\lim _ { u \rightarrow \infty } \left[ 100 u - u \sum _ { k = 1 } ^ { 100 } f \left( 1 + \frac { k } { u } \right) \right]$.
Q4 Areas by integration View
Let $S$ be the square formed by the four vertices $( 1,1 ) , ( 1 , - 1 ) , ( - 1,1 )$, and $( - 1 , - 1 )$. Let the region $R$ be the set of points inside $S$ which are closer to the centre than to any of the four sides. Find the area of the region $R$.
Q5 Number Theory Properties of Integer Sequences and Digit Analysis View
Let $g : \mathbb { N } \rightarrow \mathbb { N }$ with $g ( n )$ being the product of the digits of $n$.
(a) Prove that $g ( n ) \leq n$ for all $n \in \mathbb { N }$.
(b) Find all $n \in \mathbb { N }$, for which $n ^ { 2 } - 12 n + 36 = g ( n )$.
Q6 Number Theory Quadratic Diophantine Equations and Perfect Squares View
Let $p _ { 1 } , p _ { 2 } , p _ { 3 }$ be primes with $p _ { 2 } \neq p _ { 3 }$, such that $4 + p _ { 1 } p _ { 2 }$ and $4 + p _ { 1 } p _ { 3 }$ are perfect squares. Find all possible values of $p _ { 1 } , p _ { 2 } , p _ { 3 }$.
Q7 Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View
Let $A = \{ 1,2 , \ldots , n \}$. For a permutation $P = ( P ( 1 ) , P ( 2 ) , \cdots , P ( n ) )$ of the elements of $A$, let $P ( 1 )$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i , j \in A$:
  • if $i < j < P ( 1 )$, then $j$ appears before $i$ in $P$; and
  • if $P ( 1 ) < i < j$, then $i$ appears before $j$ in $P$.
Q8 Proof Direct Proof of a Stated Identity or Equality View
Let $k , n$ and $r$ be positive integers.
(a) Let $Q ( x ) = x ^ { k } + a _ { 1 } x ^ { k + 1 } + \cdots + a _ { n } x ^ { k + n }$ be a polynomial with real coefficients. Show that the function $\frac { Q ( x ) } { x ^ { k } }$ is strictly positive for all real $x$ satisfying
$$0 < | x | < \frac { 1 } { 1 + \sum _ { i = 1 } ^ { n } \left| a _ { i } \right| }$$
(b) Let $P ( x ) = b _ { 0 } + b _ { 1 } x + \cdots + b _ { r } x ^ { r }$ be a non-zero polynomial with real coefficients. Let $m$ be the smallest number such that $b _ { m } \neq 0$. Prove that the graph of $y = P ( x )$ cuts the $x$-axis at the origin (i.e. $P$ changes sign at $x = 0$) if and only if $m$ is an odd integer.