# Recent questions tagged cmi2020-datascience 1
Consider the following program. Assume that $x$ and $y$ are integers. f(x, y) { if (y != 0) return (x * f(x,y-1)); else return 1; } What is $f(6,3)?$ $243$ $729$ $125$ $216$
2
Consider the matrices ... $|A|=|B|$ $\text{trace}(A)=\text{trace}(B)$ $|A|=-|B|$ $\text{trace}(AB)=\text{trace}(BA)$
3
Consider the following program. Assume that all variables are integers. Note that $x\%y$ computes the remainder after dividing $x$ and $y$. The division is an integer division. For example, $1/3$ will return zero while $10/3$ will return $3$. g(n) { result = 0; i = 1; repeat until( ... 2; result = result+(remainder*i); i = i * 10; } return result; } What is $g(25)?$ $11001$ $10011$ $11011$ $10101$
4
Which of the following limits are correct? $\displaystyle \lim_{x\rightarrow 0} \frac{x^2+2x}{2x}=1$ $\displaystyle \lim_{x\rightarrow 1/2 }\frac{2x^2+x-1}{2x-1}=\frac{3}{2}$ $\displaystyle \lim_{x\rightarrow \infty } 18x^3 – 12x^2 +1=\infty$ $\displaystyle \lim_{x\rightarrow -\infty} 8x^3 – 12x^2 +1=-\infty$
5
As per the data released by the US Department of Health, Education and Welfare, the number of Ph.D. degrees conferred in Earth Sciences from the year $1948$ to $1954$ is as given in Table $5$ ... or a rolling average is an average of a subset of data points. Choose the best answer. $900$ $9,000$ $9,00,000$ $90,00,000$
6
Suppose that $A$ is an $n \times n$ matrix with $n=10$ and $b$ is an $n \times 1$ vector. Suppose that the equation $Ax=b$ for an $n \times 1$ vector does not admit any solution. Which of the following conclusions can be drawn from the given information? $A^{-1}$ ... $n\times 1$ vector such that $Ax = c$ also does not admit a solution. Then the vector $c$ is a constant multiple of the vector $b$
7
Consider the following bar chart: Which of the following are true? Number of students who scored $A$ in Algebra is higher than the number of students who scored $A$ in Calculus Percentage of students who scored $A$ or $B$ in algebra is lower than the percentage of ... or $B$ in calculus Calculus is easier than algebra Considering this data, the average percentage of students scoring $A$ is $12\%$
8
Let $A=((a_{ij}))$ be a $7\times7$ matrix with $a_{i,i+1}=1$ for $1\leq i \leq 6$, $a_{7,1}=1$ and all the other elements of the matrix are zero. Which of the following statements are true? $|A|=1$ $\text{trace(A)}=0$ $A^{-1}=A$ $A^7 =I$, where $I$ is the identity matrix
9
Let $A$ and $B$ be events such that $P(A)=0.4, P(B)=0.5$ and $P(A\cup B)=0.7$. Which of the following are true? (For sets $A,B,A\Delta B=(A^c\cap B)\cup (A\cap B^c))$. $A$ and $B$ are mutually exclusive $A$ and $B$ are independent $P(A\Delta B)= 0.1$ $P(A^c \cup B^c)=0.8$
10
$\text{Description for the following question:}$ The lifespan of a battery in a car follows Gamma distribution with probability density function $f(x)=\frac{\beta^\alpha }{\Gamma(\alpha) } e^{-\beta x}x^{\alpha -1}, 0<x< \infty ,$ where $\alpha >0$ and $\beta >0$. The mean and variance of a ... $\beta =2$ $\mathbb E(X^2 )= \frac {\alpha}{\beta}(\frac {1+\alpha}{\beta })$ $\mathbb E(X^2 )=18$
11
Out of a large number of cars produced by the automaker, the percentage of batteries that will last for more than $8$ years is $[ \int^8_0 \frac{\beta^{\alpha}}{\Gamma (\alpha)} e^{-\beta x} x^{\alpha -1} dx ] \times 100\%$ ... $[ \int^8_0 \frac{x \beta^{\alpha}}{\Gamma (\alpha)} e^{-\beta x} x^{\alpha -1} dx ] \times 100\%$
12
How many squares are there on a $7\times 7$ chessboard? $49$ $204$ $203$ $140$
13
It is mid-semester exam week at $CMI$ and first-year students from both $M.Sc.$ Data Science $(DS)$ and $M.Sc.$ Computer Science $(CS)$ have their exams scheduled for Monday from $10$ a.m. to $1$ p.m. in Lecture Hall $1$. The first row in Lecture hall $1$ has six ... - be seated in this row, in such a way that two students from the same course do not sit next to each other? $36$ $48$ $72$ $96$
14
Suppose you roll two six-sided fair dice with faces numbered from $1$ to $6$ and take the sum of the two numbers that turn up. What is the probability that: the sum is $12;$ the sum is $12$, given that the sum is even; the sum is $12$, given that the sum is an even number greater than ... $\frac {1}{14}$, respectively $\frac {1}{36}, \frac {1}{16}$, and $\frac {1}{12}$, respectively
15
Let $f(x)$ be a real-valued function all of whose derivatives exist. Recall that a point $x_0$ in the domain is called an inflection point of $f(x)$ if the second derivative $f^ (x)$ changes sign at $x_0$ ... point $x_0 =6$ is the only inflection point $x_0 =0$ and $x_0 =6$, both are inflection points The function does not have an inflection point
16
Which of the following are true? $\frac {2019}{2020} < \frac {2020}{2021}$ $x+\frac{1}{x} \geq 2$ for all $x>0$ $2^{60} >5^{24}$ $2^{314} <31^{42}$
17
The identity $\frac{1}{(1-2r)}=\displaystyle\sum^{\infty} _{k=0} (2r)^k$ is true if and only if $r\neq \frac{1}{2}$ if and only if $0\leq r < \frac{1}{2}$ if and only if $-\frac{1}{2} \leq r<\frac{1}{2}$ if and only if $-\frac{1}{2}<r<\frac{1}{2}$
18
The sum and product of the roots of the polynomial $9x^2+171x-81$ are, respectively: $-19$ and $-9$ $19$ and $9$ $-9$ and $19$ $9$ and $-19$
19
Choose the conclusions that follow logically from the statements given below. Nobody who really appreciates A.R.Rahman fails to subscribe to his YouTube channel. Owls are hopelessly ignorant of music. No one who is hopelessly ignorant of music ever subscribes to A.R ... Rahman Owls are not really appreciated by A.R.Rahman Anyone who really appreciates A.R.Rahman is not hopelessly ignorant of music
20
Which of the following inequalities are true? $e^x\geq(1+x)$ for $x\geq 0$ $e^x\leq(1+x)$ for $x<0$ $\text{In}(x)<(1+x)$ for $x>0$ $e^x<x^2$ for all real numbers $x$
1 vote
21
For any string $\text{str, length(str)}$ returns the length of the string, $\text{append(str1, str2)}$ concatenates $\text{str1}$ with another string $\text{str2}$, and $\text{trim(str)}$ removes any spaces that exist at the end of the string $\text{str}$. The function $\text{reverse(str, i, j)}$ ... n; i=i+1) { if(str[i] is ' ') { reverse(str, j, i-1); j = i + 1; } } trim(str); return str; }
1 vote
22
Consider the matrix $A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Find $A^n,$ in terms of $n,$ for $n\geq2.$
23
The following graph shows the performance of students in an exam. The marks scored by every student are a multiple of five. The $j^{th}$-percentile $u^*$ for a discrete data $x_1,x_2, ,x_n$ ... presented in the graph, answer the following questions. Compute the $10^{th}$ percentile of marks. Is the median score higher than the mean score?
24
In the figure shown below, the circle has diameter $5$. Moreover, $AB$ is parallel to $DE.$ If $DE=3$ and $AB=6,$ what is the area of triangle $ABC?$
25
A permutation $\sigma$ is a bijection from the set $[n]=\{1,2,\dots ,n\}$ to itself. We denote it using the notation $\begin{pmatrix}1 & 2 & & n \\ {\sigma(1)} & {\sigma(2)} &\dots & {\sigma(n)} \end{pmatrix},$ e.g. if $n=3$ ... $\sigma =\begin{pmatrix} 1 & 2 & 3&4&5&6 \\ 2& 3&4&5&1&6 \end{pmatrix}, \; \tau=\begin{pmatrix} 1 & 2 & 3&4&5&6 \\ 4& 1&3&2&6&5 \end{pmatrix}$
26
A permutation $\sigma$ is a bijection from the set $[n]=\{1,2, ,n\}$ to itself. We denote it using the notation $\begin{pmatrix}1 & 2 & & n \\ {\sigma(1)} & {\sigma(2)} & & {\sigma(n)} \end{pmatrix},$ ... $|A_{\sigma} |$ and $|A_\tau|?$ Can you relate these with the signs of permutations $\sigma$ and $\tau ?$
27
The case fatality rate ($CFR$) of a disease is the ratio of the number of deaths from the disease to the total number of people diagnosed with the disease ( patients ), and is usually expressed as a percentage. It has been reported that the $CFR$ of ... probability that an elderly Pandamic-$20$ patient in Gondwanaland survives the disease if they were put on a ventilator as part of the treatment?
Owing to a defect in a certain machine which makes $N95$ masks, there is a $0.1\%$ probability that a mask it makes is $\text{not}$ effective in preventing airbone viruses from being inhaled. What is the probability that the first $1000$ masks that the machine ... probability that among the first one crore $(10^7)$ masks that the machine produces, there is at least one mask which is not effective?
The International Chess Federation is organizing an online chess tournament in which $20$ of the world's top players will take part. Each player will play exactly one game against each other player. The tournament is spread over three weeks; it starts at $9$ a.m. on ... of Week $3,$ there are at least two players who would have completed the same number of games in the tournament till that point.
Your class has a textbook and a final exam. Let $P,Q$ and $R$ be the following propositions: $P:$ You get an $A$ on the final exam. $Q:$ You do every exercise in the book. $R:$ You get an $A$ in the class. Translate the following assertions into propositional formulas ... final. You get an $A$ on the final, but you don't do every exercise in this book; nevertheless, you get an $A$ in this class.