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
2015 ugmath

13 maths questions

QB2 15 marks Proof Direct Proof of an Inequality View
Let $p$, $q$ and $r$ be real numbers with $p^2 + q^2 + r^2 = 1$.
(a) Prove the inequality $3p^2 q + 3p^2 r + 2q^3 + 2r^3 \leq 2$.
(b) Also find the smallest possible value of $3p^2 q + 3p^2 r + 2q^3 + 2r^3$. Specify exactly when the smallest and the largest possible value is achieved.
QB3 15 marks Number Theory Congruence Reasoning and Parity Arguments View
(a) Show that there are exactly 2 numbers $a$ in $\{2, 3, \ldots, 9999\}$ for which $a^2 - a$ is divisible by 10000. Find these values of $a$.
(b) Let $n$ be a positive integer. For how many numbers $a$ in $\{2, 3, \ldots, n^2 - 1\}$ is $a^2 - a$ divisible by $n^2$? State your answer suitably in terms of $n$ and justify.
QB4 12 marks Stationary points and optimisation Properties of differentiable functions (abstract/theoretical) View
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a twice differentiable function, where $\mathbb{R}$ denotes the set of real numbers. Suppose that for all real numbers $x$ and $y$, the function $f$ satisfies
$$f'(x) - f'(y) \leq 3|x - y|$$
Answer the following questions. No credit will be given without full justification.
(a) Show that for all $x$ and $y$, we must have $\left|f(x) - f(y) - f'(y)(x - y)\right| \leq 1.5(x - y)^2$.
(b) Find the largest and smallest possible values for $f''(x)$ under the given conditions.
QB5 12 marks Number Theory GCD, LCM, and Coprimality View
For an arbitrary integer $n$, let $g(n)$ be the GCD of $2n + 9$ and $6n^2 + 11n - 2$. What is the largest positive integer that can be obtained as the value of $g(n)$? If $g(n)$ can be arbitrarily large, state so explicitly and prove it.
Q1 4 marks Probability Definitions Multiple Choice: Direct Probability or Distribution Calculation View
Ten people sitting around a circular table decide to donate some money for charity. You are told that the amount donated by each person was the average of the money donated by the two persons sitting adjacent to him/her. One person donated Rs. 500. Choose the correct option for each of the following two questions. Write your answers as a sequence of two letters (a/b/c/d).
What is the total amount donated by the 10 people?
(a) exactly Rs. 5000
(b) less than Rs. 5000
(c) more than Rs. 5000
(d) not possible to decide among the above three options.
What is the maximum amount donated by an individual?
(a) exactly Rs. 500
(b) less than Rs. 500
(c) more than Rs. 500
(d) not possible to decide among the above three options.
Q3 4 marks Number Theory Prime Number Properties and Identification View
A positive integer $n$ is called a magic number if it has the following property: if $a$ and $b$ are two positive numbers that are not coprime to $n$ then $a + b$ is also not coprime to $n$. For example, 2 is a magic number, because sum of any two even numbers is also even. Which of the following are magic numbers? Write your answers as a sequence of four letters (Y for Yes and N for No) in correct order.
(i) 129
(ii) 128
(iii) 127
(iv) 100.
Q4 4 marks Completing the square and sketching Piecewise differentiability and continuity conditions View
Let $A$, $B$ and $C$ be unknown constants. Consider the function $f(x)$ defined by
$$\begin{aligned} f(x) &= Ax^2 + Bx + C \text{ when } x \leq 0 \\ &= \ln(5x + 1) \text{ when } x > 0 \end{aligned}$$
Write the values of the constants $A$, $B$ and $C$ such that $f''(x)$, i.e., the double derivative of $f$, exists for all real $x$. If this is not possible, write ``not possible''. If some of the constants cannot be uniquely determined, write ``not unique'' for each such constant.
Q6 4 marks Circles Tangent Lines and Tangent Lengths View
Fill in the blanks. Let $C_1$ be the circle with center $(-8, 0)$ and radius 6. Let $C_2$ be the circle with center $(8, 0)$ and radius 2. Given a point $P$ outside both circles, let $\ell_i(P)$ be the length of a tangent segment from $P$ to circle $C_i$. The locus of all points $P$ such that $\ell_1(P) = 3\ell_2(P)$ is a circle with radius \_\_\_ and center at (\_\_\_, \_\_\_).
Q7 4 marks Binomial Theorem (positive integer n) Determine the Limit of a Sequence via Geometric Series View
(i) By the binomial theorem $(\sqrt{2} + 1)^{10} = \sum_{i=0}^{10} C_i (\sqrt{2})^i$, where $C_i$ are appropriate constants. Write the value of $i$ for which $C_i (\sqrt{2})^i$ is the largest among the 11 terms in this sum.
(ii) For every natural number $n$, let $(\sqrt{2} + 1)^n = p_n + \sqrt{2} q_n$, where $p_n$ and $q_n$ are integers. Calculate $\lim_{n \rightarrow \infty} \left(\frac{p_n}{q_n}\right)^{10}$.
Q8 4 marks Permutations & Arrangements Forming Numbers with Digit Constraints View
The format for car license plates in a small country is two digits followed by three vowels, e.g. 04 IOU. A license plate is called ``confusing'' if the digit 0 (zero) and the vowel O are both present on it. For example $04\,IOU$ is confusing but $20\,AEI$ is not. (i) How many distinct number plates are possible in all? (ii) How many of these are not confusing?
Q9 4 marks Trig Graphs & Exact Values Inverse trigonometric equation View
Recall that $\sin^{-1}$ is the inverse function of $\sin$, as defined in the standard fashion. (Sometimes $\sin^{-1}$ is called $\arcsin$.) Let $f(x) = \sin^{-1}(\sin(\pi x))$. Write the values of the following. (Some answers may involve the irrational number $\pi$. Write such answers in terms of $\pi$.)
(i) $f(2.7)$
(ii) $f'(2.7)$
(iii) $\int_0^{2.5} f(x)\, dx$
(iv) the smallest positive $x$ at which $f'(x)$ does not exist.
Q10 4 marks Complex Numbers Arithmetic Modulus Inequalities and Triangle Inequality Applications View
Answer the three questions below. To answer (i) and (ii), replace ? with exactly one of the following four options: $<$, $=$, $>$, not enough information to compare.
(i) Suppose $z_1, z_2$ are complex numbers. One of them is in the second quadrant and the other is in the third quadrant. Then $\left|\left|z_1\right| - \left|z_2\right|\right| \quad ? \quad \left|z_1 + z_2\right|$.
(ii) Complex numbers $z_1$, $z_2$ and $0$ form an equilateral triangle. Then $\left|z_1^2 + z_2^2\right| \quad ? \quad \left|z_1 z_2\right|$.
(iii) Let $1, z_1, z_2, z_3, z_4, z_5, z_6, z_7$ be the complex 8th roots of unity. Find the value of $\prod_{i=1,\ldots,7}(1 - z_i)$, where the symbol $\Pi$ denotes product.
Q11 4 marks Probability Definitions Combinatorial Conditional Probability (Counting-Based) View
There are four distinct balls labelled $1, 2, 3, 4$ and four distinct bins labelled A, B, C, D. The balls are picked up in order and placed into one of the four bins at random. Let $E_i$ denote the event that the first $i$ balls go into distinct bins. Calculate the following probabilities.
(i) $\Pr[E_4]$
(ii) $\Pr[E_4 \mid E_3]$
(iii) $\Pr[E_4 \mid E_2]$
(iv) $\Pr[E_3 \mid E_4]$.
Notation: $\Pr[X] =$ the probability of event $X$ taking place. $\Pr[X \mid Y] =$ the probability of event $X$ taking place, given that event $Y$ has taken place.