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Recent questions tagged cmi2025-datascience

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CMI 2025 | Mathematics | Part A | Question: 5
Consider the map $f: \mathbb{R}^{2} \longrightarrow \mathbb{R}^{2},(x, y) \mapsto\left(-x-y^{2}, y+x^{2}\right)$. Pick the correct statement(s) from below.There exist inf...
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1 1 vote
1 1 answer
388
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CMI Data Science 2025 | Part B | Question: 5b
Let $M$ be an $n \times n$ matrix. We define an elementary row operation on $M$ to be one of the following:Interchanging some two rows of $M$.Multiplying a row in $M$ by ...
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1 1 answer
316
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CMI Data Science 2025 | Part B | Question: 4b
There are $18$ chocolates in a bag, of which $7$ are green, $6$ are blue, and $5$ are red. We pick chocolates one at a time from the bag without replacement.What is the p...
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1 1 vote
1 1 answer
435
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CMI Data Science 2025 | Part B | Question: 3b
$15$ balls are placed independently and uniformly at random into $15$ bins numbered from $1$ to $15.$ The probability that a ball ends up in a particular bin is $\frac{1}...
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1 1 answer
341
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CMI Data Science 2025 | Part B | Question: 2b
A toy company currently sells $1,000$ toys each month at a price of ₹$500$ per toy. To increase their sales, the company is considering to lower the price. Market researc...
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1 1 vote
1 1 answer
388
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CMI Data Science 2025 | Part A | Question: 1
Let $X=\{1,2,3\}$ and $Y=\{a, b, c, d\}$ be two finite sets. Which of the following statements regarding functions from $X \rightarrow Y$ is/are true?The number of one-to...
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2 2 votes
1 1 answer
411
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CMI Data Science 2025 | Part A | Question: 2
The two arguments to the function $\textsf{foo(A, n)}$ in the code below are: $\text{(i)}$ an integer array $\textsf{A}$ indexed from $0$, and $\text{(ii)}$ the number $\...
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CMI Data Science 2025 | Part A | Question: 3
What is the domain of the following real valued function?\[f(x)=\log _{2}\left(x^{2}-5 x+6\right).\]$(-\infty, 2)$$(3, \infty)$$(-\infty, 2) \cup(3, \infty)$$(-\infty, \i...
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3 3 votes
1 1 answer
382
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CMI Data Science 2025 | Part A | Question: 4
Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ be a real matrix which satisfies $A^{-1}=A$. Which of the following statements is/are always true?$a+d=0$$...
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2 2 votes
1 1 answer
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CMI Data Science 2025 | Part A | Question: 5
Let $\mathbb{Z}$ be the set of all integers. Let $A=\left\{n \in \mathbb{Z} \mid n^{2}+10 n+21 \text{ is divisible by }7\right\}$. Which of the following statements is/ar...
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CMI Data Science 2025 | Part A | Question: 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$$\begin{...
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2 2 votes
1 1 answer
355
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CMI Data Science 2025 | Part A | Question: 7
Let $B=\left(\left(b_{i, j}\right)\right)$ be an $n \times n$ matrix. Let $p:\{1,2, \ldots, n\} \mapsto\{1,2, \ldots, n\}$ be a bijection (i.e. a one-to-one corresponde...
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3 3 votes
3 3 answers
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CMI Data Science 2025 | Part A | Question: 8
Let $x$ be a variable that takes real values, and let $f(x)=x^{3}-3 x$. Which of the following statements is/are true?$f(x)$ has a local maximum at $x=-\sqrt{3}$$f(x)$ ha...
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1 1 vote
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CMI Data Science 2025 | Part A | Question: 9
A binary relation $R$ defined on a set $S$ is said to be antisymmetric if for $x, y \in S,$ $x R y$ and $y R x \Longrightarrow x=y$. Let $R_{1}, R_{2}$ be two binary rela...
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2 2 answers
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CMI Data Science 2025 | Part A | Question: 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?$50$$126$$252$$3125$
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CMI Data Science 2025 | Part A | Question: 11
Which of the following is/are logically equivalent to $\neg(P \Longrightarrow Q)?$$\neg P \vee Q$$\neg P \wedge Q$$Q \Longrightarrow P$$P \wedge \neg Q$
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295
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CMI Data Science 2025 | Part A | Question: 12
Let $f(x)=\sqrt{x}$. We draw a tangent to the curve $y=f(x)$ at the point on the curve whose $x$ coordinate is equal to $4$. Where does this tangent intersect the $X$-axi...
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4 4 votes
2 2 answers
404
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CMI Data Science 2025 | Part A | Question: 13
Let $n$ be an integer, $n \geq 4$. $A$ is an $n \times n$ matrix with real entries. The matrix $B$ is obtained by the following sequence of operations on $A$. First, mult...
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2 2 votes
2 2 answers
459
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CMI Data Science 2025 | Part A | Question: 14
Let $x,$ $y,$ $z$ be positive numbers such that $x^{2}+y^{2}=z^{2}$. Determine the value of the following expression:$$\frac{\log _{y+z} x+\log _{z-y} x}{\left(\log _{y+z...
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1 1 answer
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CMI Data Science 2025 | Part A | Question: 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 ...
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2 2 answers
381
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CMI Data Science 2025 | Part A | Question: 16
In the following code the operator $\textsf{%}$ denotes the remainder after integer division. That is: for positive integers $\textsf{a,b}$ the value $\textsf{a % b}$ is ...
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2 2 answers
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CMI Data Science 2025 | Part A | Question: 17
Alpha and Beta are inhabitants of an island of knights and knaves, where knights always tell the truth and knaves always lie. Alpha and Beta are alone at a beach when Al...
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2 2 votes
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CMI Data Science 2025 | Part A | Question: 18
The two arguments to the function $\textsf{Mystery(A, n)}$ in the code below are: $\text{(i)}$ an integer array $\textsf{A}$ indexed from $0$, and $\text{(ii)}$ the numbe...
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1 1 vote
1 1 answer
269
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CMI Data Science 2025 | Part A | Question: 19
The two arguments to the function $\textsf{Mystery(A, n)}$ in the code below are: $\text{(i)}$ an integer array $\textsf{A}$ indexed from $0$, and $\text{(ii)}$ the numbe...
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1 1 vote
2 2 answers
400
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CMI Data Science 2025 | Part A | Question: 20
The two arguments to the function $\textsf{Mystery(A, n)}$ in the code below are: $\text{(i)}$ an integer array $\textsf{A}$ indexed from $0,$ and $\text{(ii)}$ the numbe...
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1 1 vote
1 1 answer
257
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CMI Data Science 2025 | Part B | Question: 1
A girl writes five consecutive positive integers on a blackboard. She then erases one of them. The sum of the remaining four numbers is $2025.$ What number did she erase?
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1 1 vote
2 2 answers
379
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CMI Data Science 2025 | Part B | Question: 2a
A toy company currently sells $1,000$ toys each month at a price of ₹$500$ per toy. To increase their sales, the company is considering to lower the price. Market researc...
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4 4 votes
2 2 answers
421
421 views
CMI Data Science 2025 | Part B | Question: 3a
$15$ balls are placed independently and uniformly at random into $15$ bins numbered from $1$ to $15.$ The probability that a ball ends up in a particular bin is $\frac{1}...
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3 3 votes
1 1 answer
340
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CMI Data Science 2025 | Part B | Question: 4a
There are $18$ chocolates in a bag, of which $7$ are green, $6$ are blue, and $5$ are red. We pick chocolates one at a time from the bag without replacement.What is the p...
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2 2 votes
1 1 answer
343
343 views
CMI Data Science 2025 | Part B | Question: 5a
Let $M$ be an $n \times n$ matrix. We define an elementary row operation on $M$ to be one of the following:Interchanging some two rows of $M$.Multiplying a row in $M$ by ...
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