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
2025 mscds

11 maths questions

Q3 Composite & Inverse Functions Determine Domain or Range of a Composite Function View
3. What is the domain of the following real valued function?
$$f ( x ) = \log _ { 2 } \left( x ^ { 2 } - 5 x + 6 \right)$$
(a) $( - \infty , 2 )$
(b) $( 3 , \infty )$
(c) $( - \infty , 2 ) \cup ( 3 , \infty )$
(d) $( - \infty , \infty )$
Q4 Matrices True/False or Multiple-Select Conceptual Reasoning View
4. Let $A = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$ be a real matrix which satisfies $A ^ { - 1 } = A$. Which of the following statements is/are always true?
(a) $a + d = 0$
(b) $a d - b c = 1$
(c) $a d - b c \neq 0$
(d) $a ^ { 2 } + 2 b c + d ^ { 2 } = 2$
Q5 Number Theory Congruence Reasoning and Parity Arguments View
5. Let $\mathbb { Z }$ be the set of all integers. Let $A = \left\{ n \in \mathbb { Z } \mid n ^ { 2 } + 10 n + 21 \right.$ is divisible by 7 $\}$. Which of the following statements is/are true?
(a) $A = \{ n \in \mathbb { Z } \mid ( n \equiv 0 \bmod 7 ) \}$
(b) $A = \{ n \in \mathbb { Z } \mid ( n \equiv 0 \bmod 7 )$ or $( n \equiv 4 \bmod 7 ) \}$
(c) $A = \{ n \in \mathbb { Z } \mid ( n \equiv 1 \bmod 7 )$ or $( n \equiv 5 \bmod 7 ) \}$
(d) $A = \{ n \in \mathbb { Z } \mid ( n \equiv 2 \bmod 7 )$ or $( n \equiv 6 \bmod 7 ) \}$
Q6 Proof True/False Justification View
6. Let $n \geq 3$ be an integer, and let $x _ { 1 } , x _ { 2 } , \ldots , x _ { n }$ be variables which take real values with $0 \leq x _ { i } \leq 1$ for all $1 \leq i \leq n$. Let
$$\begin{aligned} A & = x _ { 1 } + x _ { 2 } + \ldots + x _ { n } \\ B & = x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 } + \ldots + x _ { n - 1 } x _ { n } + x _ { n } x _ { 1 } \end{aligned}$$
Which of the following statements is/are true.
(a) $A \geq B$ is always true.
(b) $B > A$ is true for some values of the $x _ { i }$ 's and $A > B$ is true for some values of the $x _ { i }$ 's.
(c) $A = B$ has a finite number of solutions
(d) $A = B$ has an infinite number of solutions.
Q7 Matrices True/False or Multiple-Select Conceptual Reasoning View
7. Let $B = \left( \left( b _ { i , j } \right) \right)$ be an $n \times n$ matrix. Let $p : \{ 1,2 , \ldots , n \} \mapsto \{ 1,2 , \ldots , n \}$ be a bijection (i.e. a one-to-one correspondence) and let a matrix $A = \left( \left( a _ { i , j } \right) \right)$ be defined by
$$a _ { i , j } = b _ { p ( i ) , p ( j ) } , \quad 1 \leq i , j \leq n .$$
Which of the following statement(s) is/are true for all choices of $B$ and $P$.
(a) $A$ admits an inverse if and only if $B$ admits an inverse.
(b) For any $x , y \in \mathbb { R } ^ { n } , A x = y$ admits a solution if and only if $B x = y$ admits a solution.
(c) $A$ and $B$ have the same trace.
(d) $A$ and $B$ have the same eigenvectors.
Q8 Stationary points and optimisation Find critical points and classify extrema of a given function View
8. Let $x$ be a variable that takes real values, and let $f ( x ) = x ^ { 3 } - 3 x$. Which of the following statements is/are true?
(a) $f ( x )$ has a local maximum at $x = - \sqrt { 3 }$
(b) $f ( x )$ has a local maximum at $x = - 1$
(c) $f ( x )$ has a local minimum at $x = \sqrt { 3 }$
(d) $f ( x )$ has a global minimum at $x = 1$
Q10 Combinations & Selection Counting Integer Solutions to Equations View
10. In how many ways can 10 identical chocolate bars be distributed among 5 children, in such a way that each child gets at least one chocolate bar?
(a) 50
(b) 126
(c) 252
(d) 3125
Q12 Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
12. Let $f ( x ) = \sqrt { x }$. We draw a tangent to the curve $y = f ( x )$ at the point on the curve whose $x$ coordinate is equal to 4 . Where does this tangent intersect the $X$-axis?
(a) $x = 4$
(b) $x = - 2$
(c) $x = - 4$
(d) $x = 2$
Q13 Matrices Determinant and Rank Computation View
13. Let $n$ be an integer, $n \geq 4$. $A$ is an $n \times n$ matrix with real entries. The matrix $B$ is obtained by the following sequence of operations on $A$. First, multiply each entry of $A$ by 2 . Then add 3 times the second column to the third column. Finally, swap the first and the fourth columns. If $\operatorname { det } ( A ) = 5$, which of the following statements are true?
(a) 10 divides $\operatorname { det } ( B )$
(b) $\operatorname { det } ( B ) = - 5$
(c) 100 divides $\operatorname { det } ( B )$
(d) $\operatorname { det } ( B ) = - 2 ^ { n } \cdot 5$
Q14 Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
14. Let $x , y , z$ be positive numbers such that $x ^ { 2 } + y ^ { 2 } = z ^ { 2 }$. Determine the value of the following expression:
$$\frac { \log _ { y + z } x + \log _ { z - y } x } { \left( \log _ { y + z } x \right) \left( \log _ { z - y } x \right) }$$
(a) Undefined.
(b) $\frac { 1 } { 2 }$
(c) $\frac { 1 } { 4 }$
(d) 2
Q15 Probability Definitions Verifying Statements About Probability Properties View
15. A game being offered in a casino consists of guessing the outcomes of two tosses of a fair coin. The gambler wins if she/he has correctly guessed at least one of the two tosses. To play a game, the gambler has to pay a fee of Rs. 80, and the winner gets a reward of Rs. 100 on winning the game (and nothing otherwise). Which of the following statements are correct?
(a) In the first 10 minutes on a given day exactly three gamblers play the game, one after the other. The probability that the casino owner makes a profit in the first 10 minutes equals $1 / 4$.
(b) One gambler plays the game three times. The probability that she wins exactly two of the three games is $27 / 64$.
(c) Three friends go together and play the game with each playing once. The probability that all three win equals $27 / 64$.
(d) If 1200 players play the game on a given day, the expected profit of the casino owner for the day equals Rs. 6000.