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
2011 solved

19 maths questions

Q1 Chain Rule Limit Evaluation Involving Composition or Substitution View
The limit $\lim \left[ \left\{ 1 - \cos \left( \sin ^ { 2 } a x \right) \right\} / x \right]$ as $x -> 0$
(a) Equals 1
(b) Equals a
(c) Equals 0
(d) Does not exist
Q2 Curve Sketching Variation Table and Monotonicity from Sign of Derivative View
The set of all $x$ for which the function $f ( x ) = \log _ { 1 / 2 } \left( x ^ { 2 } - 2 x - 3 \right)$ is defined and monotone increasing is
(a) $( - \infty , 1 )$
(b) $( - \infty , - 1 )$
(c) $( 1 , \infty )$
(d) $( 3 , \infty )$
Q3 Conic sections Chord Properties and Midpoint Problems View
Let a line with slope of $60 ^ { \circ }$ be drawn through the focus $F$ of the parabola $y ^ { 2 } = 8 ( x + 2 )$. If the two points of intersection of the line with the parabola are $A$ and $B$ and the perpendicular bisector of the chord $A B$ intersects the $x$-axis at the point $P$, then the length of the segment PF is
(a) $16 / 3$
(b) $8 / 3$
(c) $16 \sqrt{3} / 3$
(d) $8 \sqrt{3}$
Q4 Complex Numbers Argand & Loci Locus Identification from Modulus/Argument Equation View
Suppose $z$ is a complex number with $| z | < 1$. Let $w = ( 1 + z ) / ( 1 - z )$. Which of the following is always true? [$\operatorname { Re } ( w )$ is the real part of $w$ and $\operatorname { Im } ( w )$ is the imaginary part of $w$]
(a) $\operatorname { Re } ( w ) > 0$
(b) $\operatorname { Im } ( w ) \geq 0$
(c) $| w | \leq 1$
(d) $| w | \geq 1$
Q5 Number Theory Divisibility and Divisor Analysis View
Among all the factors of $4 ^ { 6 } 6 ^ { 7 } 21 ^ { 8 }$, the number of factors which are perfect squares is
(a) 240
(b) 360
(c) 400
(d) 640
Q6 Combinations & Selection Selection with Group/Category Constraints View
Let $A$ be the set $\{ 1,2 , \ldots , 20 \}$. Fix two disjoint subsets $S _ { 1 }$ and $S _ { 2 }$ of $A$, each with exactly three elements. How many 3-element subsets of $A$ are there, which have exactly one element common with $S _ { 1 }$ and at least one element common with $S _ { 2 }$?
(a) 51
(b) 102
(c) 135
(d) 153
Q7 Permutations & Arrangements Circular Arrangement View
In how many ways can 3 couples sit around a round table such that men and women alternate and none of the couples sit together?
(a) 1
(b) 2
(c) $5! / 3$
(d) None of these.
Q8 Curve Sketching Identifying the Correct Graph of a Function View
The equation $x ^ { 3 } + y ^ { 3 } = x y ( 1 + x y )$ represents
(a) Two parabolas intersecting at two points
(b) Two parabolas touching at one point
(c) Two non-intersecting hyperbolas
(d) One parabola passing through the origin.
Q9 Stationary points and optimisation Geometric or applied optimisation problem View
Consider the diagram below where $ABZP$ is a rectangle and $ABCD$ and $CXYZ$ are squares whose areas add up to 1. The maximum possible area of the rectangle $ABZP$ is
(a) $1 + 1 / \sqrt{2}$
(b) $2 - \sqrt{2}$
(c) $1 + \sqrt{2}$
(d) $( 1 + \sqrt{2} ) / 2$
Q10 Permutations & Arrangements Counting Functions with Constraints View
Let $A$ be the set $\{ 1,2 , \ldots , 6 \}$. How many functions from $A$ to $A$ are there such that the range of $f$ has exactly 5 elements?
(a) 240
(b) 720
(c) 1800
(d) 10800
Q11 Circles Circles Tangent to Each Other or to Axes View
Let $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C _ { 1 }$ touches $C _ { 2 }$ and $C _ { 2 }$ touches $C _ { 3 }$. If the area of the smallest circle is 1 unit, then area of the largest circle is
(a) $\{ ( \sqrt{2} + 1 ) / ( \sqrt{2} - 1 ) \} ^ { 4 }$
(b) $( 1 + \sqrt{2} ) ^ { 2 }$
(c) $( 2 + \sqrt{2} ) ^ { 2 }$
(d) $2 ^ { 4 }$
Q12 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Let $[ x ]$ denote the largest integer less than or equal to $x$. Then $\int_0^{n^{1/k}} \left[ x ^ { k } + n \right] dx$ equals
(a) $n ^ { 2 } + \sum_{i=1}^{n} i ^ { 1 / k }$
(b) $2 n ^ { ( 1 + k ) / k } - \sum_{i=1}^{n} i ^ { 1 / k }$
(c) $2 n ^ { ( 1 + k ) / k } - \sum_{i=1}^{n-1} i ^ { 1 / k }$
(d) None of these.
Q13 Stationary points and optimisation Find critical points and classify extrema of a given function View
Consider the function $f ( x ) = x ( x - 1 ) e ^ { 2 x }$ if $x \leq 0$ $f ( x ) = x ( 1 - x ) e ^ { - 2 x }$ if $x > 0$ Then $f ( x )$ attains its maximum value at
(a) $1 - 1 / \sqrt{2}$
(b) $1 + 1 / \sqrt{2}$
(c) $- 1 / \sqrt{2}$
(d) $1 / \sqrt{2}$
Q14 Proof True/False Justification View
Consider the function $f ( x ) = x ^ { n } ( 1 - x ) ^ { n } / n !$, where $n \geq 1$ is a fixed integer. Let $f ^ { ( k ) }$ denote the $k$-th derivative of $f$. Which of the following is true for all $k \geq 1$?
(a) $f ^ { ( k ) } ( 0 )$ and $f ^ { ( k ) } ( 1 )$ are integers.
(b) $f ^ { ( k ) } ( 0 )$ is an integer, but not $f ^ { ( k ) } ( 1 )$
(c) $f ^ { ( k ) } ( 1 )$ is an integer, but not $f ^ { ( k ) } ( 0 )$
(d) Neither $f ^ { ( k ) } ( 1 )$ nor $f ^ { ( k ) } ( 0 )$ is an integer.
Q15 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of solutions of the equation $\sin ( \cos \theta ) = \theta$, $- 1 \leq \theta \leq 1$, is
(a) 0
(b) 1
(c) 2
(d) 3
Q16 Vectors Introduction & 2D Area Computation Using Vectors View
Suppose $ABCD$ is a parallelogram and $P, Q$ are points on the sides $BC$ and $CD$ respectively, such that $PB = \alpha BC$ and $DQ = \beta DC$. If the area of the triangles $ABP$, $ADQ$, $PCQ$ are 15, 15 and 4 respectively, then the area of $APQ$ is
(a) 14
(b) 15
(c) 16
(d) 18.
Q18 Vectors 3D & Lines Volume of a 3D Solid View
A regular tetrahedron has all its vertices on a sphere of radius $R$. Then the length of each edge of the tetrahedron is
(a) $( \sqrt{2} / \sqrt{3} ) R$
(b) $( \sqrt{3} / 2 ) R$
(c) $( 4 / 3 ) R$
(d) $( 2 \sqrt{2} / \sqrt{3} ) R$
Q20 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View
Let $f ( x ) = ( \tan x ) ^ { 3 / 2 } - 3 \tan x + \sqrt{\tan x}$. Consider the three integrals $I _ { 1 } = \int_0^1 f ( x ) \, dx$; $I _ { 2 } = \int_{0.3}^{1.3} f ( x ) \, dx$ and $I _ { 3 } = \int_{0.5}^{1.5} f ( x ) \, dx$. Then,
(a) $I _ { 1 } > I _ { 2 } > I _ { 3 }$
(b) $I _ { 2 } > I _ { 1 } > I _ { 3 }$
(c) $I _ { 3 } > I _ { 1 } > I _ { 2 }$
(d) $I _ { 1 } > I _ { 3 } > I _ { 2 }$
Q21 Complex numbers 2 Roots of Unity and Cyclotomic Properties View
Let $a < b < c$ be three real numbers and $w$ denote a complex cube root of unity. If $\left( a + bw + cw ^ { 2 } \right) ^ { 3 } + \left( a + bw ^ { 2 } + cw \right) ^ { 3 } = 0$, then which of the following must be true?
(a) $a + b + c = 0$
(b) $abc = 0$
(c) $ab + bc + ca = 0$
(d) $b = ( c + a ) / 2$.