cmi-entrance

Papers (30)
2025
pgmath 18
2024
pgmath 4 ugmath 20
2023
pgmath 12 ugmath 16
2022
pgmath 18 ugmath_22may 16 ugmath_23may 16
2021
pgmath 11 ugmath 15
2020
pgmath 18 ugmath 16
2019
pgmath 20 ugmath 15
2018
pgmath 4 ugmath 6
2017
ugmath 15
2016
pgmath 10 ugmath 16
2015
pgmath 5 ugmath 13
2014
ugmath 9
2013
pgmath 11 ugmath 12
2012
pgmath 24 ugmath 14
2011
pgmath 15 ugmath 12
2010
pgmath 4 ugmath 19
2017 ugmath

15 maths questions

QA1 4 marks Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
Positive integers $a$ and $b$, possibly equal, are chosen randomly from among the divisors of 400. The numbers $a, b$ are chosen independently, each divisor being equally likely to be chosen. Find the probability that $\gcd(a, b) = 1$ and $\text{lcm}(a, b) = 400$.
QA2 4 marks Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View
Find the volume of the solid obtained when the region bounded by $y = \sqrt{x}$, $y = -x$ and the line $x = 9$ is revolved around the $x$-axis. (It may be useful to draw the specified region.)
QA3 4 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
10 mangoes are to be placed in 5 distinct boxes labeled $\mathrm{U}, \mathrm{V}, \mathrm{W}, \mathrm{X}, \mathrm{Y}$. A box may contain any number of mangoes including no mangoes or all the mangoes. What is the number of ways to distribute the mangoes so that exactly two of the boxes contain exactly two mangoes each?
QA4 4 marks Complex Numbers Arithmetic Solving Equations for Unknown Complex Numbers View
Find all complex solutions to the equation: $$x^{4} + x^{3} + 2x^{2} + x + 1 = 0.$$
QA5 4 marks Stationary points and optimisation Find concavity, inflection points, or second derivative properties View
Let $g$ be a function such that all its derivatives exist. We say $g$ has an inflection point at $x_0$ if the second derivative $g''$ changes sign at $x_0$ i.e., if $g''(x_0 - \epsilon) \times g''(x_0 + \epsilon) < 0$ for all small enough positive $\epsilon$.
(a) If $g''(x_0) = 0$ then $g$ has an inflection point at $x_0$. True or False?
(b) If $g$ has an inflection point at $x_0$ then $g''(x_0) = 0$. True or False?
(c) Find all values $x_0$ at which $x^{4}(x - 10)$ has an inflection point.
QA6 4 marks Sine and Cosine Rules Circumradius or incircle radius computation View
Consider the following construction in a circle. Choose points $A, B, C$ on the given circle such that $\angle ABC$ is $60^\circ$. Draw another circle that is tangential to the chords $AB$, $BC$ and to the original circle. Do the above construction in the unit circle to obtain a circle $S_1$. Repeat the process in $S_1$ to obtain another circle $S_2$. What is the radius of $S_2$?
QA7 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Write the values of the following.
(a) $\int_{-3}^{3} \left| 3x^{2} - 3 \right| dx$.
(b) $f'(1)$ where $f(t) = \int_{0}^{t} \left| 3x^{2} - 3 \right| dx$.
QA9 4 marks Differentiation from First Principles View
Consider the following function: $$f(x) = \begin{cases} x^{2} \cos\left(\frac{1}{x}\right), & x \neq 0 \\ a, & x = 0 \end{cases}$$ (a) Find the value of $a$ for which $f$ is continuous. Use this value of $a$ to calculate the following.
(b) $f'(0)$.
(c) $\lim_{x \rightarrow 0} f'(x)$.
QA10 4 marks Matrices Linear System and Inverse Existence View
For this question write your answers as a series of four letters (Y for Yes and N for No) in order. Is it possible to find a $2 \times 2$ matrix $M$ for which the equation $M\vec{x} = \vec{p}$ has:
(a) no solutions for some but not all $\vec{p}$; exactly one solution for all other $\vec{p}$?
(b) exactly one solution for some but not all $\vec{p}$; more than one solution for all other $\vec{p}$?
(c) no solutions for some but not all $\vec{p}$; more than one solution for all other $\vec{p}$?
(d) no solutions for some $\vec{p}$, exactly one solution for some $\vec{p}$ and more than one solution for some $\vec{p}$?
QB1 10 marks Addition & Double Angle Formulae Simplification of Trigonometric Expressions with Specific Angles View
Answer the following questions
(a) Evaluate $$\lim_{x \rightarrow 0^{+}} \left( x^{x^{x}} - x^{x} \right)$$ (b) Let $A = \frac{2\pi}{9}$, i.e., $A = 40$ degrees. Calculate the following $$1 + \cos A + \cos 2A + \cos 4A + \cos 5A + \cos 7A + \cos 8A$$ (c) Find the number of solutions to $e^{x} = \frac{x}{2017} + 1$.
QB2 15 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
Let $L$ be the line of intersection of the planes $x + y = 0$ and $y + z = 0$.
(a) Write the vector equation of $L$, i.e., find $(a, b, c)$ and $(p, q, r)$ such that $$L = \{(a, b, c) + \lambda(p, q, r) \mid \lambda \text{ is a real number.}\}$$ (b) Find the equation of a plane obtained by rotating $x + y = 0$ about $L$ by $45^\circ$.
QB3 15 marks Partial Fractions View
Let $p(x)$ be a polynomial of degree strictly less than 100 and such that it does not have $x^{3} - x$ as a factor. If $$\frac{d^{100}}{dx^{100}} \left( \frac{p(x)}{x^{3} - x} \right) = \frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$. Here $\frac{d^{100}}{dx^{100}}$ means taking the 100th derivative.
QB4 15 marks Number Theory Quadratic Diophantine Equations and Perfect Squares View
The domain of a function $f$ is the set of natural numbers. The function is defined as follows: $$f(n) = n + \lfloor \sqrt{n} \rfloor$$ where $\lfloor k \rfloor$ denotes the nearest integer smaller than or equal to $k$. For example, $\lfloor \pi \rfloor = 3$, $\lfloor 4 \rfloor = 4$. Prove that for every natural number $m$ the following sequence contains at least one perfect square $$m, f(m), f^{2}(m), f^{3}(m), \ldots$$ The notation $f^{k}$ denotes the function obtained by composing $f$ with itself $k$ times, e.g., $f^{2} = f \circ f$.
QB5 15 marks Number Theory Congruence Reasoning and Parity Arguments View
Each integer is colored with exactly one of three possible colors - black, red or white satisfying the following two rules: the negative of a black number must be colored white, and the sum of two white numbers (not necessarily distinct) must be colored black.
(a) Show that the negative of a white number must be colored black and the sum of two black numbers must be colored white.
(b) Determine all possible colorings of the integers that satisfy these rules.
QB6 15 marks Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View
You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.
(a) A line segment has its endpoints on opposite edges of the hexagon. Show that it passes through the center of the hexagon if and only if it divides the two edges in the same ratio.
(b) Suppose a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that center of the square is same as that of the hexagon.
(c) Suppose the side of the hexagon is of length 1. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.
(d) Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.