cmi-entrance

Papers (31)
2025
mscds 11
2024
mscds 10 pgmath 1 ugmath 20
2023
pgmath 3 ugmath 16
2022
pgmath 2 ugmath_22may 16 ugmath_23may 16
2021
pgmath 1 ugmath 15
2020
ugmath 15
2019
pgmath 2 ugmath 10
2018
pgmath 2 ugmath 14
2017
pgmath 1 ugmath 13
2016
pgmath 2 ugmath 11
2015
pgmath 1 ugmath 13
2014
ugmath 9
2013
pgmath 8 ugmath 10
2012
pgmath 2 ugmath 14
2011
pgmath 1 ugmath 12
2010
pgmath 4 ugmath 19
2018 ugmath

14 maths questions

QA1 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
Consider an equilateral triangle $ABC$ with altitude 3 centimeters. A circle is inscribed in this triangle, then another circle is drawn such that it is tangent to the inscribed circle and the sides $AB$, $AC$. Infinitely many such circles are drawn; each tangent to the previous circle and the sides $AB$, $AC$. Find the sum of the areas of all these circles.
QA2 4 marks Exponential Functions MCQ on Function Properties View
Consider the following function defined for all real numbers $x$ $$f(x) = \frac{2018}{100 + e^{x}}$$ How many integers are there in the range of $f$?
QA3 4 marks Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View
List every solution of the following equation. You need not simplify your answer(s). $$\sqrt[3]{x+4} - \sqrt[3]{x} = 1$$
QA4 4 marks Indefinite & Definite Integrals Substitution Combined with Symmetry or Companion Integral View
Compute the following integral $$\int_{0}^{\frac{\pi}{2}} \frac{\mathrm{~d}x}{(\sqrt{\sin x} + \sqrt{\cos x})^{4}}.$$
QA5 4 marks Proof Quadratic Diophantine Equations and Perfect Squares View
List in increasing order all positive integers $n \leq 40$ such that $n$ cannot be written in the form $a^{2} - b^{2}$, where $a$ and $b$ are positive integers.
QA6 4 marks Complex numbers 2 Solving Polynomial Equations in C View
Consider the equation $$z^{2018} = 2018^{2018} + i$$ where $i = \sqrt{-1}$.
(a) How many complex solutions does this equation have?
(b) How many solutions lie in the first quadrant?
(c) How many solutions lie in the second quadrant?
QA7 4 marks Roots of polynomials Determine coefficients or parameters from root conditions View
Let $x^{3} + ax^{2} + bx + 8 = 0$ be a cubic equation with integer coefficients. Suppose both $r$ and $-r$ are roots of this equation, where $r > 0$ is a real number. List all possible pairs of values $(a, b)$.
QA8 4 marks Combinations & Selection Selection with Arithmetic or Divisibility Conditions View
How many non-congruent triangles are there with integer lengths $a \leq b \leq c$ such that $a + b + c = 20$?
QA9 4 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
Consider a sequence of polynomials with real coefficients defined by $$p_{0}(x) = \left(x^{2}+1\right)\left(x^{2}+2\right) \cdots\left(x^{2}+1009\right)$$ with subsequent polynomials defined by $p_{k+1}(x) := p_{k}(x+1) - p_{k}(x)$ for $k \geq 0$. Find the least $n$ such that $$p_{n}(1) = p_{n}(2) = \cdots = p_{n}(5000)$$
QA10 4 marks Function Transformations View
Recall that $\arcsin(t)$ (also known as $\sin^{-1}(t)$) is a function with domain $[-1,1]$ and range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. Consider the function $f(x) := \arcsin(\sin(x))$ and answer the following questions as a series of four letters (T for True and F for False) in order.
(a) The function $f(x)$ is well defined for all real numbers $x$.
(b) The function $f(x)$ is continuous wherever it is defined.
(c) The function $f(x)$ is differentiable wherever it is continuous.
QB1 10 marks Proof Proof Involving Combinatorial or Number-Theoretic Structure View
Answer the following questions
(a) A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_{1}, \ldots, a_{k}$, each $a_{i} > 1$, such that $$\frac{1}{a_{1}} + \cdots + \frac{1}{a_{k}} = 1.$$ Show that if $k$ is stable then $k+1$ is also stable. Using this or otherwise, find all stable numbers.
(b) Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^{*}(y) := \max_{x \in A}\{yx - f(x)\}$$ whenever the above maximum is finite.
For the function $f(x) = -\ln(x)$, determine the set of points for which $f^{*}$ is defined and find an expression for $f^{*}(y)$ involving only $y$ and constants.
QB2 15 marks Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View
Answer the following questions
(a) Find all real solutions of the equation $$\left(x^{2}-2x\right)^{x^{2}+x-6} = 1$$ Explain why your solutions are the only solutions.
(b) The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} - \sqrt[3]{6\sqrt{3}-10}$$
QB3 15 marks Proof Proof by Induction or Recursive Construction View
Let $f$ be a function on nonnegative integers defined as follows $$f(2n) = f(f(n)) \quad \text{and} \quad f(2n+1) = f(2n)+1$$
(a) If $f(0) = 0$, find $f(n)$ for every $n$.
(b) Show that $f(0)$ cannot equal 1.
(c) For what nonnegative integers $k$ (if any) can $f(0)$ equal $2^{k}$?
QB4 15 marks Sine and Cosine Rules Multi-step composite figure problem View
Let $ABC$ be an equilateral triangle with side length 2. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k < 1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$ with $CB' = AC' = k$. Line segments are drawn from points $A', B', C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Show that $PQR$ is an equilateral triangle with side length $\dfrac{4(1-k)}{\sqrt{k^{2}-2k+4}}$.