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Answers by soujanyareddy13

0 votes
1
CMI2015-B-01
Let $\Sigma=\{a,b\}.$ Given a language $L\underline\subset \Sigma^{\ast}$ and a word $w\in\Sigma^{\ast}$, define the languages: $Extend(L,w) :=\{xw\:|\:x\in L\}$ $Shrink(L,w) :=\{x\:|\:xw\in L\}$Show that if $L$ is regular, both $Extend(L,w)$ and $Shrink(L,w)$ are regular.
Let $\Sigma=\{a,b\}.$ Given a language $L\underline\subset \Sigma^{\ast}$ and a word $w\in\Sigma^{\ast}$, define the languages: $Extend(L,w) :=\{xw\:|\:x\in L\}$ $Shrink(L,w) :=\{x\:|\:xw\in L\}$Show that if $L$ is regular, both $Extend(L,w)$ and $Shrink(L,w)$ are regular.
answered 1 day ago in Theory of Computation 12 views
0 votes
2
CMI-2018-DataScience-B: 20
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ ... $A^4(1,3)=0$. Then which of the following are necessarily true? Give reasons. $m\underline> 6$
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ By convention, $A(i,i)=0$, i.e. a person ... $A^4(1,3)=0$. Then which of the following are necessarily true? Give reasons. $m\underline> 6$
answered 3 days ago in Others 15 views
0 votes
3
CMI-2018-DataScience-B: 19
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ ... $A^4(1,3)=0$. Then which of the following are necessarily true? Give reasons. $m\underline < 9$
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ By convention, $A(i,i)=0$, i.e. a person ... $A^4(1,3)=0$. Then which of the following are necessarily true? Give reasons. $m\underline < 9$
answered 3 days ago in Others 16 views
0 votes
4
CMI-2018-DataScience-B: 18
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ ... Then which of the following are necessarily true? Give reasons. $A^2(i,i)>0$ for all $i,\;1\underline< i \underline < m.$
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ By convention, $A(i,i)=0$, i.e. a person ... $A^2(i,i)>0$ for all $i,\;1\underline< i \underline < m.$
answered 3 days ago in Others 22 views
0 votes
5
CMI-2018-DataScience-B: 17
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ By ... $1$ and member $2$ have at least one friend in common.
$\text{Description for the following question:}$ A golf club has $m$ members with serial numbers $1,2,\dots ,m$. If members with serial numbers $i$ and $j$ are friends, then $A(i,j)=A(j,i)=1,$ otherwise $A(i,j)=A(j,i)=0.$ By convention, $A(i,i)=0$, i ... $A^4(1,3)=0$. Then which of the following are necessarily true? Give reasons. Member $1$ and member $2$ have at least one friend in common.
answered 3 days ago in Others 18 views
0 votes
6
CMI-2018-DataScience-B: 16
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A boolean value is a value from the set {$\text{True,False}$}. A $3$-ary boolean function is a ... $3$-ary boolean function $h$. How many neighbours does $h$ have?
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A boolean value is a value from the set {$\text{True,False}$}. A $3$-ary boolean function is a function that takes three ... $3$-ary boolean function $h$. How many neighbours does $h$ have?
answered 3 days ago in Others 35 views
0 votes
7
CMI-2018-DataScience-B: 15
A square piece of paper $ABCD$ of side length $1$ is folded along the segment that connects the upper right corner $B$ and the midpoint $Q$ of the left edge $AD$, as shown. What is the vertical distance between the base edge (segment $DC$) and the point $P$ (which was originally point $A$)?
A square piece of paper $ABCD$ of side length $1$ is folded along the segment that connects the upper right corner $B$ and the midpoint $Q$ of the left edge $AD$, as shown. What is the vertical distance between the base edge (segment $DC$) and the point $P$ (which was originally point $A$)?
answered 3 days ago in Others 21 views
0 votes
8
CMI-2018-DataScience-B: 14
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar charts ... in the same proportion across the five states, then what are the sales of chocolate in Tamil Nadu, in lakhs of rupees?
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar charts shows the detail of ... sold in the same proportion across the five states, then what are the sales of chocolate in Tamil Nadu, in lakhs of rupees?
answered 3 days ago in Others 17 views
0 votes
9
CMI-2018-DataScience-B: 13
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar ... the detail of breakdown for strawberry flavor by states in lakhs of rupees. What are the total sales of the chocolate flavor?
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar chart shows the detail of breakdown for strawberry flavor by states in lakhs of rupees. What are the total sales of the chocolate flavor?
answered 3 days ago in Others 25 views
0 votes
10
CMI-2018-DataScience-B: 12
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar ... detail of breakdown for strawberry flavor by states in lakhs of rupees. What is the total revenue of the ice cream company?
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar chart shows the detail of breakdown for strawberry flavor by states in lakhs of rupees. What is the total revenue of the ice cream company?
answered 3 days ago in Others 17 views
0 votes
11
CMI-2018-DataScience-B: 11
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar ... the detail of breakdown for strawberry flavor by states in lakhs of rupees. What are the total sales of the strawberry flavor?
$\text{Description for the following question:}$ An Ice-cream company mainly operates in the five southern states of India. The pie chart shows the breakdown of revenues (in percentages) for the ice cream company over the last summer. The bar chart shows the detail of breakdown for strawberry flavor by states in lakhs of rupees. What are the total sales of the strawberry flavor?
answered 3 days ago in Others 35 views
0 votes
12
CMI-2018-DataScience-B: 10
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of ... credit than the bank loses the entire loan amount. What is the expected revenue of the bank from a loan of $Rs. 100,000?$
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of successes out of $n$ ... on his/her credit than the bank loses the entire loan amount. What is the expected revenue of the bank from a loan of $Rs. 100,000?$
answered 3 days ago in Others 21 views
0 votes
13
CMI-2018-DataScience-B: 9
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number ... expected number of success is $\mathbb{E}(X)=np$. For the situation in the previous problem, what is the expected number defaults?
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of successes out of ... $\mathbb{E}(X)=np$. For the situation in the previous problem, what is the expected number defaults?
answered 3 days ago in Others 25 views
0 votes
14
CMI-2018-DataScience-B: 8
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the ... credit default in a year. You can assume that whether a given debtor will default or not is independent of the behavior of other debtors.
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. $\text{Description for the following question:}$ Suppose $X$ is the number of successes ... one credit default in a year. You can assume that whether a given debtor will default or not is independent of the behavior of other debtors.
answered 3 days ago in Others 26 views
0 votes
15
CMI-2018-DataScience-B: 7
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A computer password requires you to use exactly $1$ uppercase letter, $3$ ... $33$ special characters that can be used). In how many ways can you create such a password$?$
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A computer password requires you to use exactly $1$ uppercase letter, $3$ lowercase letters, $3$ digits and $2$ special characters (there are $33$ special characters that can be used). In how many ways can you create such a password$?$
answered 3 days ago in Others 20 views
0 votes
16
CMI-2018-DataScience-B: 6
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A $4$-digit number is represented as $abcd$ ... $a\neq0.$ Suppose the number $dcba$, obtained by reversing the digits of $abcd$, is $9$ times $abcd$. Find the number $abcd$.
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A $4$-digit number is represented as $abcd$ i.e. $a\times 10^3 +b\times 10^2 +c\times10+d,$ where $a\neq0.$ Suppose the number $dcba$, obtained by reversing the digits of $abcd$, is $9$ times $abcd$. Find the number $abcd$.
answered 3 days ago in Others 17 views
0 votes
17
CMI-2018-DataScience-B: 5
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A function $f$ from the set $A$ to itself is said to have a fixed point if $f(i)=i$ ... $A$ is the set $\{a,b,c,d\}$. Find the number of bijective functions from the set $A$ to itself having no fixed point.
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. A function $f$ from the set $A$ to itself is said to have a fixed point if $f(i)=i$ for some $i$ in $A$. Suppose $A$ is the set $\{a,b,c,d\}$. Find the number of bijective functions from the set $A$ to itself having no fixed point.
answered 3 days ago in Others 31 views
0 votes
18
CMI-2018-DataScience-B: 4
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. In computing, a floating point operation (flop) is any one of the following operations ... . How does this number change if both the matrices are upper triangular?
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. In computing, a floating point operation (flop) is any one of the following operations performed by a computer ... $c_{ij}=\displaystyle\sum^5 _{k=1} a_{ik} b_{kj}$. How does this number change if both the matrices are upper triangular?
answered 3 days ago in Others 21 views
0 votes
19
CMI-2018-DataScience-B: 3
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Find $A^{10}$ where $A$ is the matrix $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$. Justify your answer.
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Find $A^{10}$ where $A$ is the matrix $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$. Justify your answer.
answered 3 days ago in Others 20 views
0 votes
20
CMI-2018-DataScience-B: 2
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Suppose $A,B$ and $C$ are $m\times m$ matrices. What does the following algorithm compute? (Here $A(i,j)$ ... .) for i=1 to m for j=1 to m for k=1 to m C(i,j)=A(i,k)*B(k,j)+C(i,j) end end end
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Suppose $A,B$ and $C$ are $m\times m$ matrices. What does the following algorithm compute? (Here $A(i,j)$ denotes the $(i.j)^{th}$ entry of matrix $A$.) for i=1 to m for j=1 to m for k=1 to m C(i,j)=A(i,k)*B(k,j)+C(i,j) end end end
answered 3 days ago in Others 35 views
0 votes
21
CMI-2018-DataScience-B: 1
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Let $N=\{1,2,3,...\}$ be the set of natural integers and let $f:N\times N \mapsto N$ be defined by $f(m,n)=(2m-1)*2^n.$Is $f$ injective? Is $f$ surjective? Give reasons.
For numerical answers, the following forms are acceptable: fractions, decimals, symbolic e.g.:$\left( \begin{array}{c} n \\ r \end{array} \right)^n P_r , n!$ etc. Let $N=\{1,2,3,...\}$ be the set of natural integers and let $f:N\times N \mapsto N$ be defined by $f(m,n)=(2m-1)*2^n.$Is $f$ injective? Is $f$ surjective? Give reasons.
answered 3 days ago in Others 25 views
0 votes
22
CMI-2019-DataScience-B: 18
$\text{Description of the following question:}$ The break-up of the colour of cars sold by an Indian company in a given year is provided in the pie-chart, see the Figure(1). The bar chart shows the number of grey coloured cars sold in ... cars were sold in Chennai. What is the largest possible percentage of brown cars sold in Chennai, rounded off to the nearest integer?
$\text{Description of the following question:}$ The break-up of the colour of cars sold by an Indian company in a given year is provided in the pie-chart, see the Figure(1). The bar chart shows the number of grey coloured cars sold in different cities, ... of all cars were sold in Chennai. What is the largest possible percentage of brown cars sold in Chennai, rounded off to the nearest integer?
answered 3 days ago in Others 17 views
0 votes
23
CMI-2019-DataScience-B: 17
$\text{Description of the following question:}$ The break-up of the colour of cars sold by an Indian company in a given year is provided in the pie-chart, see the Figure(1). The bar chart shows the number of grey coloured cars sold in different cities ... grey cars sold in Delhi is the same as the proportion of red cars sold in Delhi, how many red cars were sold in Delhi?
$\text{Description of the following question:}$ The break-up of the colour of cars sold by an Indian company in a given year is provided in the pie-chart, see the Figure(1). The bar chart shows the number of grey coloured cars sold in different cities, see the Figure( ... of grey cars sold in Delhi is the same as the proportion of red cars sold in Delhi, how many red cars were sold in Delhi?
answered 3 days ago in Others 17 views
0 votes
24
CMI-2019-DataScience-B: 16
$\text{Description of the following question:}$ The break-up of the colour of cars sold by an Indian company in a given year is provided in the pie-chart, see the Figure(1). The bar chart shows the number of grey coloured cars sold in different cities, see the Figure(2). How many blue cars were sold in that year?
$\text{Description of the following question:}$ The break-up of the colour of cars sold by an Indian company in a given year is provided in the pie-chart, see the Figure(1). The bar chart shows the number of grey coloured cars sold in different cities, see the Figure(2). How many blue cars were sold in that year?
answered 3 days ago in Others 14 views
0 votes
25
CMI-2019-DataScience-B: 15
Consider the following pseudocode for a function that operates on an $N$ element array $A[1],A[2],\dots,A[N]$ of integers. function mystery (A[1...N]) { int i,j,position,tmp; for i=1 to N { position=i; for j=i+1 to N { if(A[j]<A[ ... Explain what effect the function has on the input array $A$. If $N=100$, how many times is the comparison $A[i]<A[position]$ checked?
Consider the following pseudocode for a function that operates on an $N$ element array $A[1],A[2],\dots,A[N]$ of integers. function mystery (A[1...N]) { int i,j,position,tmp; for i=1 to N { position=i; for j=i+1 to N { if(A[j]<A[position]) { position=j; ... tmp; } } Explain what effect the function has on the input array $A$. If $N=100$, how many times is the comparison $A[i]<A[position]$ checked?
answered 3 days ago in Others 11 views
0 votes
26
CMI-2019-DataScience-B: 14
A thief picks a wallet and starts running at a speed of $5\; m/s$ with zero acceleration. In $5$ seconds, a policewoman notices the alarm and starts following the thief with a speed of $3\; m/s$ with a uniform acceleration of $1\; m/s^2$. How many seconds ... $v_0$ m/s and uniform acceleration $a\; m/s^2$ will cover a distance of $(at^2/2+v_0t)$ meters in $t$ seconds.)
A thief picks a wallet and starts running at a speed of $5\; m/s$ with zero acceleration. In $5$ seconds, a policewoman notices the alarm and starts following the thief with a speed of $3\; m/s$ with a uniform acceleration of $1\; m/s^2$. How many seconds will it take the ... velocity $v_0$ m/s and uniform acceleration $a\; m/s^2$ will cover a distance of $(at^2/2+v_0t)$ meters in $t$ seconds.)
answered 3 days ago in Others 12 views
0 votes
27
CMI-2019-DataScience-B: 13
How many positive integers are factors of 2310?
How many positive integers are factors of 2310?
answered 3 days ago in Others 12 views
0 votes
28
CMI-2019-DataScience-B: 12
Let $n,k$ be positive integers. The expansion of $(x_1+\dots+x_k)^n$ is given by $(x_1+\dots+x_k)^n=\sum\frac{n!}{n_1!n_2!\dots n_k!}x_1^{n_1}x_2^{n_2}\dots x_k^{n_k},$ where the sum is taken over all sequences $n_1,n_2,\dots,n_k$ of non-negative integers such that $n_1,n_2+\dots+n_k=n$. What is the coefficient of $x^5$ in the expansion of $(1+3x+2x^2)^4$?
Let $n,k$ be positive integers. The expansion of $(x_1+\dots+x_k)^n$ is given by $(x_1+\dots+x_k)^n=\sum\frac{n!}{n_1!n_2!\dots n_k!}x_1^{n_1}x_2^{n_2}\dots x_k^{n_k},$ where the sum is taken over all sequences $n_1,n_2,\dots,n_k$ of non-negative integers such that $n_1,n_2+\dots+n_k=n$. What is the coefficient of $x^5$ in the expansion of $(1+3x+2x^2)^4$?
answered 3 days ago in Others 12 views
0 votes
29
CMI-2019-DataScience-B: 11
A thin piece of metal of length $20$ cm and width $16$ cm is to be used to construct an open-topped box. A square will be cut from each corner and the sides will be folded up. What size corner should be cut so that the volume of the box is maximized?
A thin piece of metal of length $20$ cm and width $16$ cm is to be used to construct an open-topped box. A square will be cut from each corner and the sides will be folded up. What size corner should be cut so that the volume of the box is maximized?
answered 3 days ago in Others 14 views
0 votes
30
CMI-2019-DataScience-B: 10
Let $p(x)$ be a polynomial with integer coefficients. Let $n$ be a positive integer and suppse $a$ and $b$ are two integers such that $a \equiv b(\text{mod}\;n)$. Is it true that $p(a)\equiv p(b)(\text{mod}\;n)$? Justify your answer.
Let $p(x)$ be a polynomial with integer coefficients. Let $n$ be a positive integer and suppse $a$ and $b$ are two integers such that $a \equiv b(\text{mod}\;n)$. Is it true that $p(a)\equiv p(b)(\text{mod}\;n)$? Justify your answer.
answered 3 days ago in Others 13 views
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