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
2024 mscds

10 maths questions

Q2 Sign Change & Interval Methods View
2. Which of the following statements are true?
(a) Let $f ( x ) = x ^ { 3 } + x - 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ - 1,0 )$.
(b) Let $f ( x ) = x ^ { 3 } + x - 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ 0,1 )$.
(c) Let $f ( x ) = x ^ { 3 } + x + 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ 0,1 )$.
(d) Let $f ( x ) = x ^ { 3 } + x + 1$ for $x \in \mathbb { R }$. Then the equation $f ( x ) = 0$ has at least one root in $[ - 1,0 )$.
3. 26 children participated in a chess tournament. A child got two points for winning, zero for losing and one point for a draw. Each child played against every other child. After the tournament was over no child had an odd score. There were no draws in the entire tournament and no two players had the same score. For convenience assume that the children are named $A , B , C , \ldots , Z$, by the 26 letters of the English alphabet in increasing order of scores, so $A$ has lowest score, $Z$ has highest score. Thus we have $A < B < C < \cdots < X < Y < Z$. Pick all statements which are true.
(a) K lost to Q
(b) K lost to B
(c) M lost to N
(d) If L lost to M then N lost to M.
4. 14 teams participate in a volleyball tournament. Each team plays the other exactly once. There are no draws. Assume the teams are labeled by $j , 1 \leq j \leq 14$. Let $x _ { j }$ denote the number of games team $j$ wins and $y _ { j }$ the number of games that team $j$ lost. Pick the correct alternative(s):
(a) $\sum _ { j } x _ { j } ^ { 2 } = \sum _ { j } y _ { j } ^ { 2 }$.
(b) $\sum _ { j } x _ { j } ^ { 2 } > \sum _ { j } y _ { j } ^ { 2 }$.
(c) $\sum _ { j } x _ { j } = \sum _ { j } y _ { j }$.
(d) $\sum _ { j } \left| x _ { j } \right| = \sum _ { j } \left| y _ { j } \right|$.
Q5 Inequalities Identify Always-True Inequality from Options View
5. Which of the following statements is/are true for real numbers $x , y$ ?
(a) If $x ^ { 2 } = y ^ { 2 }$ then $x = y$.
(b) If $x ^ { 3 } = y ^ { 3 }$ then $x = y$.
(c) If $x < y$ then $x ^ { 2 } < y ^ { 2 }$.
(d) If $x < y$ then $x ^ { 3 } < y ^ { 3 }$.
Q6 Matrices True/False or Multiple-Select Conceptual Reasoning View
6. Suppose that $A$ is an $n \times n$ matrix with $n \geq 6$. For an $n \times 1$ vector $b$ consider the equations
$$\begin{aligned} & A x = b , \text { for an } n \times 1 \text { vector } x \\ & A ^ { T } x = b \text { for an } n \times 1 \text { vector } x \end{aligned}$$
where $A ^ { T }$ is the transpose of the matrix $A$. Which of the following statements are correct?
(a) If equation (??) admits a solution for all $b$, then $A ^ { - 1 }$ exists.
(b) If equation (??) admits a solution for all $b$, then equation (??) also admits a solution for all $b$.
(c) If equation (??) admits a solution for some $b$, then $A ^ { - 1 }$ exists.
(d) If equation (??) admits a solution for some $b$, then equation (??) also admits a solution for that $b$.
Q7 Inequalities Identify Always-True Inequality from Options View
7. Which of the following inequalities are correct?
(a) $1 + x \leq e ^ { x }$ for all $x \in \mathbb { R }$
(b) $2 ^ { n } \leq 2 n$ ! for all positive integers $n$
(c) $\left( 1 + x ^ { 2 } \right) ^ { n } \leq ( 1 + x ) ^ { 2 n }$ for all $x \in ( 0 , \infty )$
(d) $\sin ( x ) \leq \tan ( x )$ for all $x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$
Q8 Matrices True/False or Multiple-Select Conceptual Reasoning View
8. Which of the following statements are valid for all $n \times n$ matrices $A , B$ :
(a) $\left( A ^ { T } A \right) ^ { T } = A A ^ { T }$.
(b) If $A , B$ are invertible, then inverse of $A B$ is $A ^ { - 1 } B ^ { - 1 }$.
(c) $( A + B ) ^ { T } = A ^ { T } + B ^ { T }$
(d) $A x = B x$ for some $n \times 1$ vector $x$ implies that $A y = B y$ for all $n \times 1$ vectors $y$.
Q9 Polynomial Division & Manipulation View
9. Let $x$ be a variable that takes real values, and let $f ( x ) , g ( x ) , h ( x )$ be three polynomials with real-valued coefficients. Further, let $f ( x )$ be a polynomial of degree $1 , g ( x )$ a polynomial of degree 2 , and $h ( x )$ a polynomial of degree 3 . Which of the following statements is/are always true for any three polynomials of these types?
(a) The graphs of $f ( x )$ and $g ( x )$ intersect at one or more points.
(b) The graphs of $f ( x )$ and $h ( x )$ intersect at one or more points.
(c) The graphs of $g ( x )$ and $h ( x )$ intersect at one or more points.
Q10 Connected Rates of Change Volume/Height Related Rates for Containers and Solids View
10. A spherical ball of ice of radius $20 m$ is dropped in a vat of hot water. The ice melts in such a way that (i) the shape of the ball remains spherical, and (ii) the radius of the ball decreases at a constant rate of $0.5 m s ^ { - 1 }$. At what rate does the volume of the ice ball decrease, when the radius of the ball is $15 m$ ?
(a) $100 \pi m ^ { 3 } s ^ { - 1 }$
(b) $225 \pi m ^ { 3 } s ^ { - 1 }$
(c) $450 \pi m ^ { 3 } s ^ { - 1 }$
(d) $600 \pi m ^ { 3 } s ^ { - 1 }$
Q11 Curve Sketching Range and Image Set Determination View
11. Let $x$ be a real-valued variable. What is the range of the function $f ( x ) = \sin ^ { 2 } x - \sin x + 2$ ?
(a) $[ 0,2 ]$
(b) $[ 1,2 ]$
(c) $[ 1,4 ]$
(d) $\left[ \frac { 7 } { 4 } , 4 \right]$
Q17 Probability Definitions Conditional Probability and Bayes' Theorem View
17. One day, Captain Haddock receives a mysterious letter with a confusing paragraph. Captain Haddock and Tintin are investigating the matter. There are two possible suspects: Professor Calculus and Thomson $\&$ Thompson. Based on their past experience:
  • The probability that Professor Calculus sends a letter is $60 \%$, while the probability that Thomson $\&$ Thompson send a letter is $40 \%$.
  • When Professor Calculus sends letters, there is an $80 \%$ probability that the letter contains a confusing paragraph.
  • When Thomson \& Thompson send letters, there is a $5 \%$ probability that the letter contains a confusing paragraph.

What is the probability that the letter was sent by Professor Calculus?
(a) 0.96
(b) 0.80
(c) 0.50
(d) 0.48
Q18 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
18. Let $f$ be a function on the positive real numbers such that $f ( x y ) = f ( x ) + f ( y )$. If $f ( 2024 ) = 2$ then which of the following statement(s) is/ are true?
(a) $f \left( \frac { 1 } { 2024 } \right) = 1$
(b) $f \left( \frac { 1 } { 2024 } \right) = - 1$
(c) $f \left( \frac { 1 } { 2024 } \right) = - 2$
(d) $f \left( \frac { 1 } { 2024 } \right) = 2$
The following description is for questions 19 and 20.
A perfect shuffle of a deck of cards divides the deck into two equal parts and then interleaves the cards from each half, starting with the first card of the first half.
For instance, if we shuffle a deck of cards containing 10 cards arranged $[ 1,2,3,4,5,6,7,8,9,10 ]$ we first create two equal decks with cards $[ 1,2,3,4,5 ]$ and $[ 6,7,8,9,10 ]$ and then interleave them to get a new deck $[ 1,6,2,7,3,8,4,9,5,10 ]$.