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 UGA

30 maths questions

Q1 Discrete Probability Distributions Multiple Choice: Direct Probability or Distribution Calculation View
Let $0 < x < \frac { 1 } { 6 }$ be a real number. When a certain biased dice is rolled, a particular face $F$ occurs with probability $\frac { 1 } { 6 } - x$ and and its opposite face occurs with probability $\frac { 1 } { 6 } + x$; the other four faces occur with probability $\frac { 1 } { 6 }$. Recall that opposite faces sum to 7 in any dice. Assume that the probability of obtaining the sum 7 when two such dice are rolled is $\frac { 13 } { 96 }$. Then, the value of $x$ is:
(A) $\frac { 1 } { 8 }$
(B) $\frac { 1 } { 12 }$
(C) $\frac { 1 } { 24 }$
(D) $\frac { 1 } { 27 }$.
Q2 Combinations & Selection Combinatorial Probability View
An office has 8 officers including two who are twins. Two teams, Red and Blue, of 4 officers each are to be formed randomly. What is the probability that the twins would be together in the Red team?
(A) $\frac { 1 } { 6 }$
(B) $\frac { 3 } { 7 }$
(C) $\frac { 1 } { 4 }$
(D) $\frac { 3 } { 14 }$
Q3 Permutations & Arrangements Linear Arrangement with Constraints View
Suppose Roger has 4 identical green tennis balls and 5 identical red tennis balls. In how many ways can Roger arrange these 9 balls in a line so that no two green balls are next to each other and no three red balls are together?
(A) 8
(B) 9
(C) 11
(D) 12
Q4 Permutations & Arrangements Permutation Properties and Enumeration (Abstract) View
The number of permutations $\sigma$ of $1,2,3,4$ such that $| \sigma ( i ) - i | < 2$ for every $1 \leq i \leq 4$ is
(A) 2
(B) 3
(C) 4
(D) 5.
Q5 Stationary points and optimisation Find critical points and classify extrema of a given function View
Let $f ( x )$ be a degree 4 polynomial with real coefficients. Let $z$ be the number of real zeroes of $f$, and $e$ be the number of local extrema (i.e., local maxima or minima) of $f$. Which of the following is a possible $( z , e )$ pair?
(A) $( 4,4 )$
(B) $( 3,3 )$
(C) $( 2,2 )$
(D) $( 0,0 )$
Q6 Number Theory Combinatorial Number Theory and Counting View
A number is called a palindrome if it reads the same backward or forward. For example, 112211 is a palindrome. How many 6-digit palindromes are divisible by 495?
(A) 10
(B) 11
(C) 30
(D) 45
Q7 Matrices True/False or Multiple-Select Conceptual Reasoning View
Let $A$ be a square matrix of real numbers such that $A ^ { 4 } = A$. Which of the following is true for every such $A$?
(A) $\operatorname { det } ( A ) \neq - 1$
(B) $A$ must be invertible.
(C) $A$ can not be invertible.
(D) $A ^ { 2 } + A + I = 0$ where $I$ denotes the identity matrix.
Q8 Sequences and series, recurrence and convergence Direct term computation from recurrence View
Consider the real-valued function $h : \{ 0,1,2 , \ldots , 100 \} \rightarrow \mathbb { R }$ such that $h ( 0 ) = 5 , h ( 100 ) = 20$ and satisfying $h ( i ) = \frac { 1 } { 2 } ( h ( i + 1 ) + h ( i - 1 ) )$, for every $i = 1,2 , \ldots , 99$. Then, the value of $h ( 1 )$ is:
(A) 5.15
(B) 5.5
(C) 6
(D) 6.15.
Q9 Combinations & Selection Lattice Path Counting View
An up-right path is a sequence of points $\mathbf { a } _ { 0 } = \left( x _ { 0 } , y _ { 0 } \right) , \mathbf { a } _ { 1 } = \left( x _ { 1 } , y _ { 1 } \right) , \mathbf { a } _ { 2 } = ( x _ { 2 } , y _ { 2 } ), \ldots$ such that $\mathbf { a } _ { i + 1 } - \mathbf { a } _ { i }$ is either $( 1,0 )$ or $( 0,1 )$. The number of up-right paths from $( 0,0 )$ to $( 100,100 )$ which pass through $( 1,2 )$ is:
(A) $3 \cdot \binom { 197 } { 99 }$
(B) $3 \cdot \binom { 100 } { 50 }$
(C) $2 \cdot \binom { 197 } { 98 }$
(D) $3 \cdot \binom { 197 } { 100 }$.
Q10 Taylor series Limit evaluation using series expansion or exponential asymptotics View
Let $f ( x ) = \frac { 1 } { 2 } x \sin x - ( 1 - \cos x )$. The smallest positive integer $k$ such that $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { k } } \neq 0$ is:
(A) 3
(B) 4
(C) 5
(D) 6.
Q11 Measures of Location and Spread View
Nine students in a class gave a test for 50 marks. Let $S _ { 1 } \leq S _ { 2 } \leq \cdots \leq S _ { 5 } \leq \cdots \leq S _ { 8 } \leq S _ { 9 }$ denote their ordered scores. Given that $S _ { 1 } = 20$ and $\sum _ { i = 1 } ^ { 9 } S _ { i } = 250$, let $m$ be the smallest value that $S _ { 5 }$ can take and $M$ be the largest value that $S _ { 5 }$ can take. Then the pair $( m , M )$ is given by
(A) $( 20,35 )$
(B) $( 20,34 )$
(C) $( 25,34 )$
(D) $( 25,50 )$.
Q12 Combinations & Selection Counting Arrangements with Run or Pattern Constraints View
Let 10 red balls and 10 white balls be arranged in a straight line such that 10 each are on either side of a central mark. The number of such symmetrical arrangements about the central mark is
(A) $\frac { 10 ! } { 5 ! 5 ! }$
(B) $10 !$
(C) $\frac { 10 ! } { 5 ! }$
(D) $2 \cdot 10 !$
Q13 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View
If $z = x + i y$ is a complex number such that $\left| \frac { z - i } { z + i } \right| < 1$, then we must have
(A) $x > 0$
(B) $x < 0$
(C) $y > 0$
(D) $y < 0$.
Q14 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
Let $S = \left\{ x - y \mid x , y \text{ are real numbers with } x ^ { 2 } + y ^ { 2 } = 1 \right\}$. Then the maximum number in the set $S$ is
(A) 1
(B) $\sqrt { 2 }$
(C) $2 \sqrt { 2 }$
(D) $1 + \sqrt { 2 }$.
Q15 Uniform Distribution View
In a factory, 20 workers start working on a project of packing consignments. They need exactly 5 hours to pack one consignment. Every hour 4 new workers join the existing workforce. It is mandatory to relieve a worker after 10 hours. Then the number of consignments that would be packed in the initial 113 hours is
(A) 40
(B) 50
(C) 45
(D) 52.
Q16 Stationary points and optimisation Geometric or applied optimisation problem View
Let $A B C D$ be a rectangle with its shorter side $a > 0$ units and perimeter $2 s$ units. Let $P Q R S$ be any rectangle such that vertices $A , B , C$ and $D$ respectively lie on the lines $P Q , Q R , R S$ and $S P$. Then the maximum area of such a rectangle $P Q R S$ in square units is given by
(A) $s ^ { 2 }$
(B) $2 a ( s - a )$
(C) $\frac { s ^ { 2 } } { 2 }$
(D) $\frac { 5 } { 2 } a ( s - a )$.
Q17 Number Theory Quadratic Diophantine Equations and Perfect Squares View
The number of pairs of integers $( x , y )$ satisfying the equation $x y ( x + y + 1 ) = 5 ^ { 2018 } + 1$ is:
(A) 0
(B) 2
(C) 1009
(D) 2018.
Q18 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
Let $p ( n )$ be the number of digits when $8 ^ { n }$ is written in base 6, and let $q ( n )$ be the number of digits when $6 ^ { n }$ is written in base 4. For example, $8 ^ { 2 }$ in base 6 is 144, hence $p ( 2 ) = 3$. Then $\lim _ { n \rightarrow \infty } \frac { p ( n ) q ( n ) } { n ^ { 2 } }$ equals:
(A) 1
(B) $\frac { 4 } { 3 }$
(C) $\frac { 3 } { 2 }$
(D) 2.
Q19 Standard trigonometric equations Locus or solution set characterization of a trigonometric relation View
For a real number $\alpha$, let $S _ { \alpha }$ denote the set of those real numbers $\beta$ that satisfy $\alpha \sin ( \beta ) = \beta \sin ( \alpha )$. Then which of the following statements is true?
(A) For any $\alpha , S _ { \alpha }$ is an infinite set.
(B) $S _ { \alpha }$ is a finite set if and only if $\alpha$ is not an integer multiple of $\pi$.
(C) There are infinitely many numbers $\alpha$ for which $S _ { \alpha }$ is the set of all real numbers.
(D) $S _ { \alpha }$ is always finite.
Q20 Matrices Matrix Power Computation and Application View
If $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & i \end{array} \right)$ and $A ^ { 2018 } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, then $a + d$ equals:
(A) $1 + i$
(B) 0
(C) 2
(D) 2018.
Q21 Proof True/False Justification View
Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : \mathbb { R } \rightarrow \mathbb { R }$ be two functions. Consider the following two statements: $\mathbf { P ( 1 ) }$: If $\lim _ { x \rightarrow 0 } f ( x )$ exists and $\lim _ { x \rightarrow 0 } f ( x ) g ( x )$ exists, then $\lim _ { x \rightarrow 0 } g ( x )$ must exist. $\mathbf { P ( 2 ) }$: If $f , g$ are differentiable with $f ( x ) < g ( x )$ for every real number $x$, then $f ^ { \prime } ( x ) < g ^ { \prime } ( x )$ for all $x$. Then, which one of the following is a correct statement?
(A) Both $\mathrm { P } ( 1 )$ and $\mathrm { P } ( 2 )$ are true.
(B) Both $P ( 1 )$ and $P ( 2 )$ are false.
(C) $\mathrm { P } ( 1 )$ is true and $\mathrm { P } ( 2 )$ is false.
(D) $\mathrm { P } ( 1 )$ is false and $\mathrm { P } ( 2 )$ is true.
Q22 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of solutions of the equation $\sin ( 7 x ) + \sin ( 3 x ) = 0$ with $0 \leq x \leq 2 \pi$ is
(A) 9
(B) 12
(C) 15
(D) 18.
Q23 Conditional Probability Bayes' Theorem with Production/Source Identification View
A bag contains some candies, $\frac { 2 } { 5 }$ of them are made of white chocolate and the remaining $\frac { 3 } { 5 }$ are made of dark chocolate. Out of the white chocolate candies, $\frac { 1 } { 3 }$ are wrapped in red paper, the rest are wrapped in blue paper. Out of the dark chocolate candies, $\frac { 2 } { 3 }$ are wrapped in red paper, the rest are wrapped in blue paper. If a randomly selected candy from the bag is found to be wrapped in red paper, then what is the probability that it is made up of dark chocolate?
(A) $\frac { 2 } { 3 }$
(B) $\frac { 3 } { 4 }$
(C) $\frac { 3 } { 5 }$
(D) $\frac { 1 } { 4 }$
Q24 Probability Definitions Combinatorial Counting (Non-Probability) View
A party is attended by twenty people. In any subset of four people, there is at least one person who knows the other three (we assume that if $X$ knows $Y$, then $Y$ knows $X$). Suppose there are three people in the party who do not know each other. How many people in the party know everyone?
(A) 16
(B) 17
(C) 18
(D) Cannot be determined from the given data.
Q25 Number Theory Quadratic Diophantine Equations and Perfect Squares View
The sum of all natural numbers $a$ such that $a ^ { 2 } - 16 a + 67$ is a perfect square is:
(A) 10
(B) 12
(C) 16
(D) 22.
Q26 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
The sides of a regular hexagon $A B C D E F$ are extended by doubling them (for example, $B A$ extends to $B A ^ { \prime }$ with $B A ^ { \prime } = 2 B A$) to form a bigger regular hexagon $A ^ { \prime } B ^ { \prime } C ^ { \prime } D ^ { \prime } E ^ { \prime } F ^ { \prime }$. Then, the ratio of the areas of the bigger to the smaller hexagon is:
(A) 2
(B) 3
(C) $2 \sqrt { 3 }$
(D) $\pi$.
Q27 Straight Lines & Coordinate Geometry Perspective, Projection, and Applied Geometry View
Between 12 noon and 1 PM, there are two instants when the hour hand and the minute hand of a clock are at right angles. The difference in minutes between these two instants is:
(A) $32 \frac { 8 } { 11 }$
(B) $30 \frac { 8 } { 11 }$
(C) $32 \frac { 5 } { 11 }$
(D) $30 \frac { 5 } { 11 }$.
Q28 Discriminant and conditions for roots Condition for repeated (equal/double) roots View
For which values of $\theta$, with $0 < \theta < \pi / 2$, does the quadratic polynomial in $t$ given by $t ^ { 2 } + 4 t \cos \theta + \cot \theta$ have repeated roots?
(A) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 18 }$
(B) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 12 }$
(C) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 18 }$
(D) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 12 }$
Q29 Complex Numbers Argand & Loci Geometric Properties of Triangles/Polygons from Affixes View
Let $\alpha , \beta , \gamma$ be complex numbers which are the vertices of an equilateral triangle. Then, we must have:
(A) $\alpha + \beta + \gamma = 0$
(B) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 0$
(C) $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \alpha \beta + \beta \gamma + \gamma \alpha = 0$
(D) $( \alpha - \beta ) ^ { 2 } + ( \beta - \gamma ) ^ { 2 } + ( \gamma - \alpha ) ^ { 2 } = 0$
Q30 Permutations & Arrangements Distribution of Objects into Bins/Groups View
Assume that $n$ copies of unit cubes are glued together side by side to form a rectangular solid block. If the number of unit cubes that are completely invisible is 30, then the minimum possible value of $n$ is:
(A) 204
(B) 180
(C) 140
(D) 84.