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
2018 UGB

8 maths questions

Q1 Standard trigonometric equations Locus or solution set characterization of a trigonometric relation View
Find all pairs $( x , y )$ with $x , y$ real, satisfying the equations: $$\sin \left( \frac { x + y } { 2 } \right) = 0 , \quad | x | + | y | = 1$$
Q2 Circles Area and Geometric Measurement Involving Circles View
Suppose that $P Q$ and $R S$ are two chords of a circle intersecting at a point $O$. It is given that $P O = 3 \mathrm {~cm}$ and $S O = 4 \mathrm {~cm}$. Moreover, the area of the triangle $P O R$ is $7 \mathrm {~cm} ^ { 2 }$. Find the area of the triangle $Q O S$.
Q3 Chain Rule Derivative of Composite Function in Applied/Modeling Context View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function such that for all $x \in \mathbb { R }$ and for all $t \geq 0$, $$f ( x ) = f \left( e ^ { t } x \right)$$ Show that $f$ is a constant function.
Q4 Standard Integrals and Reverse Chain Rule Differentiability and Properties of Integral-Defined Functions View
Let $f : ( 0 , \infty ) \rightarrow \mathbb { R }$ be a continuous function such that for all $x \in ( 0 , \infty )$, $$f ( 2 x ) = f ( x )$$ Show that the function $g$ defined by the equation $$g ( x ) = \int _ { x } ^ { 2 x } f ( t ) \frac { d t } { t } \text { for } x > 0$$ is a constant function.
Q5 Sequences and series, recurrence and convergence Proof by induction on sequence properties View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a differentiable function such that its derivative $f ^ { \prime }$ is a continuous function. Moreover, assume that for all $x \in \mathbb { R }$, $$0 \leq \left| f ^ { \prime } ( x ) \right| \leq \frac { 1 } { 2 }$$ Define a sequence of real numbers $\left\{ a _ { n } \right\} _ { n \in \mathbb { N } }$ by: $$\begin{gathered} a _ { 1 } = 1 , \\ a _ { n + 1 } = f \left( a _ { n } \right) \text { for all } n \in \mathbb { N } . \end{gathered}$$ Prove that there exists a positive real number $M$ such that for all $n \in \mathbb { N }$, $$\left| a _ { n } \right| \leq M$$
Q6 Proof Existence Proof View
Let $a \geq b \geq c > 0$ be real numbers such that for all $n \in \mathbb { N }$, there exist triangles of side lengths $a ^ { n } , b ^ { n } , c ^ { n }$. Prove that the triangles are isosceles.
Q7 Number Theory Congruence Reasoning and Parity Arguments View
Let $a , b , c \in \mathbb { N }$ be such that $$a ^ { 2 } + b ^ { 2 } = c ^ { 2 } \text { and } c - b = 1$$ Prove that ( i ) $a$ is odd, ( ii ) $b$ is divisible by 4 ,
(iii) $a ^ { b } + b ^ { a }$ is divisible by $c$.
Q8 Matrices Determinant and Rank Computation View
Let $n \geq 3$. Let $A = \left( \left( a _ { i j } \right) \right) _ { 1 \leq i , j \leq n }$ be an $n \times n$ matrix such that $a _ { i j } \in \{ 1 , - 1 \}$ for all $1 \leq i , j \leq n$. Suppose that $$\begin{aligned} & a _ { k 1 } = 1 \text { for all } 1 \leq k \leq n \text { and } \\ & \sum _ { k = 1 } ^ { n } a _ { k i } a _ { k j } = 0 \text { for all } i \neq j \end{aligned}$$ Show that $n$ is a multiple of 4.