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
subjective_collection

10 maths questions

Q1 Combinations & Selection Counting Integer Solutions to Equations View
How many natural numbers less than $10^{8}$ are there, whose sum of digits equals 7?
Q2 Principle of Inclusion/Exclusion View
Find the number of positive integers less than or equal to 6300 which are not divisible by 3, 5 and 7.
Q3 Discriminant and conditions for roots Range of a rational function via discriminant View
If $c$ is a real number with $0 < c < 1$, then show that the values taken by the function $y = \frac{x^2 + 2x + c}{x^2 + 4x + 3c}$, as $x$ varies over real numbers, range over all real numbers.
Q4 Number Theory GCD, LCM, and Coprimality View
Let $X = \{0,1,2,3,\ldots,99\}$. For $a, b$ in $X$, we define $a * b$ to be the remainder obtained by dividing the product $ab$ by 100. For example, $9 * 18 = 62$ and $7 * 5 = 35$. Let $x$ be an element in $X$. An element $y$ in $X$ is called the inverse of $x$ if $x * y = 1$. Find all elements of $X$ that have inverses and write down their inverses.
Q5 Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
Evaluate $\lim_{n \rightarrow \infty} \left\{\frac{1}{n+1} + \frac{1}{n+2} + \ldots + \frac{1}{n+n}\right\}$.
Q6 Circles Circle-Related Locus Problems View
Tangents are drawn to a given circle from a point on a given straight line, which does not meet the given circle. Prove that the locus of the mid-point of the chord joining the two points of contact of the tangents with the circle is a circle.
Q7 Curve Sketching Sketching a Curve from Analytical Properties View
Draw the graph (on plain paper) of $f(x) = \min\{|x|-1, |x-1|-1, |x-2|-1\}$.
Q8 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
Let $\{C_n\}$ be an infinite sequence of circles lying in the positive quadrant of the $XY$-plane, with strictly decreasing radii and satisfying the following conditions. Each $C_n$ touches both $X$-axis and the $Y$-axis. Further, for all $n \geq 1$, the circle $C_{n+1}$ touches the circle $C_n$ externally. If $C_1$ has radius 10 cm, then show that the sum of the areas of all these circles is $\frac{25\pi}{3\sqrt{2}-4}$ sq. cm.
Q9 Simultaneous equations View
Consider the system of equations $x + y = 2$, $ax + y = b$. Find conditions on $a$ and $b$ under which
  1. [(i)] the system has exactly one solution;
  2. [(ii)] the system has no solution;
  3. [(iii)] the system has more than one solution.
Q10 Sequences and series, recurrence and convergence Closed-form expression derivation View
Let $\{x_n\}$ be a sequence such that $x_1 = 2$, $x_2 = 1$ and $2x_n - 3x_{n-1} + x_{n-2} = 0$ for $n > 2$. Find an expression for $x_n$.