Recent questions and answers in Others

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3 answers

TIFR2021-A: 13
What are the last two digits of $7^{2021}$? $67$ $07$ $27$ $01$ $77$

What are the last two digits of $7^{2021}$? $67$ $07$ $27$ $01$ $77$

answered 4 days ago in Others Yashvir 51 views

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2 answers

UGCNET-Dec2005-II: 16
A schema describes : data elements records and files record relationship all of the above

A schema describes : data elements records and files record relationship all of the above

answered 6 days ago in Others himanshu dhawan 89 views

1 vote

1 answer

TIFR2021-B: 14
Consider the following greedy algorithm for colouring an $n$-vertex undirected graph $G$ with colours $c_{1}, c_{2}, \dots:$ consider the vertices of $G$ in any sequence and assign the chosen vertex the first colour that has not already been assigned to any of its neighbours. Let $m(n, r)$ be ... $m\left ( n, r \right ) = nr$ $m\left ( n, r \right ) = n\binom{r}{2}$

Consider the following greedy algorithm for colouring an $n$-vertex undirected graph $G$ with colours $c_{1}, c_{2}, \dots:$ consider the vertices of $G$ in any sequence and assign the chosen vertex the first colour that has not already been assigned to any of its neighbours. Let $m(n, r)$ be the minimum ... $m\left ( n, r \right ) = nr$ $m\left ( n, r \right ) = n\binom{r}{2}$

answered Apr 4 in Others Nikhil_dhama 26 views

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1 answer

TIFR2021-B: 13
Let $A$ be a $3 \times 6$ matrix with real-valued entries. Matrix $A$ has rank $3$. We construct a graph with $6$ vertices where each vertex represents distinct column in $A$, and there is an edge between two vertices if the two columns represented by the vertices ... graph is connected. There is a clique of size $3$. The graph has a cycle of length $4$. The graph is $3$-colourable.

Let $A$ be a $3 \times 6$ matrix with real-valued entries. Matrix $A$ has rank $3$. We construct a graph with $6$ vertices where each vertex represents distinct column in $A$, and there is an edge between two vertices if the two columns represented by the vertices are ... The graph is connected. There is a clique of size $3$. The graph has a cycle of length $4$. The graph is $3$-colourable.

answered Apr 4 in Others Nikhil_dhama 29 views

1 vote

1 answer

TIFR2021-B: 12
Let $G$ be an undirected graph. For any two vertices $u, v$ in $G$, let $\textrm{cut} (u, v)$ be the minimum number of edges that should be deleted from $G$ so that there is no path between $u$ and $v$ in the resulting graph. Let $a, b, c, d$ be $4$ vertices in $G$. Which of ... $\textrm{cut} (c,d) = 2$ $\textrm{cut} (b,d) = 2$, $\textrm{cut} (b,c) = 2$ and $\textrm{cut} (c,d) = 1$

Let $G$ be an undirected graph. For any two vertices $u, v$ in $G$, let $\textrm{cut} (u, v)$ be the minimum number of edges that should be deleted from $G$ so that there is no path between $u$ and $v$ in the resulting graph. Let $a, b, c, d$ be $4$ vertices in $G$. Which of the following ... $\textrm{cut} (c,d) = 2$ $\textrm{cut} (b,d) = 2$, $\textrm{cut} (b,c) = 2$ and $\textrm{cut} (c,d) = 1$

answered Apr 4 in Others Nikhil_dhama 28 views

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1 answer

TIFR2021-B: 10
Let $G$ be a connected bipartite simple graph (i.e., no parallel edges) with distinct edge weights. Which of the following statements on $\text{MST}$ (minimum spanning tree) need $\text{NOT}$ be true? $G$ hasa unique $\text{MST}$. Every $\text{MST}$ ... lightest edge. Every $\text{MST}$ in $G$ contains the third lightest edge. No $\text{MST}$ in $G$ contains the heaviest edge.

Let $G$ be a connected bipartite simple graph (i.e., no parallel edges) with distinct edge weights. Which of the following statements on $\text{MST}$ (minimum spanning tree) need $\text{NOT}$ be true? $G$ hasa unique $\text{MST}$. Every $\text{MST}$ in $G$ ... second lightest edge. Every $\text{MST}$ in $G$ contains the third lightest edge. No $\text{MST}$ in $G$ contains the heaviest edge.

answered Apr 4 in Others Nikhil_dhama 23 views

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1 answer

TIFR2021-B: 1
Consider the following statements about propositional formulas. $\left ( p\wedge q \right )\rightarrow r$ and $\left ( p \rightarrow r \right )\wedge \left ( q\rightarrow r \right )$ are $\textit{not }$ ... on the values $p$ and $q$, $\text{(i)}$ can be either true or false, while $\text{(ii)}$ is always false.

Consider the following statements about propositional formulas. $\left ( p\wedge q \right )\rightarrow r$ and $\left ( p \rightarrow r \right )\wedge \left ( q\rightarrow r \right )$ are $\textit{not }$ ... Depending on the values $p$ and $q$, $\text{(i)}$ can be either true or false, while $\text{(ii)}$ is always false.

answered Apr 4 in Others Nikhil_dhama 27 views

1 vote

1 answer

TIFR2021-A: 14
Five married couples attended a party. In the party, each person shook hands with those they did not know. Everyone knows his or her spouse. At the end of the party, Shyamal, one of the attendees, listed the number of hands that other attendees including his spouse ... once in the list. How many persons shook hands with Shyamal at the party? $2$ $4$ $6$ $8$ Insufficient information

Five married couples attended a party. In the party, each person shook hands with those they did not know. Everyone knows his or her spouse. At the end of the party, Shyamal, one of the attendees, listed the number of hands that other attendees including his spouse shook. He ... $8$ once in the list. How many persons shook hands with Shyamal at the party? $2$ $4$ $6$ $8$ Insufficient information

answered Apr 4 in Others Nikhil_dhama 37 views

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2 answers

TIFR2021-A: 9
Fix $n\geq 6$. Consider the set $\mathcal{C}$ of binary strings $x_{1}, x_{2} \dots x_{n}$ of length $n$ such that the bits satisfy the following set of equalities, all modulo $2$: $x_{i}+x_{i+1}+x_{i+2}=0$ ... divisible by $3$ $\left | \mathcal{C} \right |=4$. If $n\geq 6$ is not divisible by $3$ then $\left | \mathcal{C} \right |=1$.

Fix $n\geq 6$. Consider the set $\mathcal{C}$ of binary strings $x_{1}, x_{2} \dots x_{n}$ of length $n$ such that the bits satisfy the following set of equalities, all modulo $2$: $x_{i}+x_{i+1}+x_{i+2}=0$ for all $1\leq i\leq n-2, x_{n-1}+x_{n}+x_{1}=0,$ ... $3$ $\left | \mathcal{C} \right |=4$. If $n\geq 6$ is not divisible by $3$ then $\left | \mathcal{C} \right |=1$.

answered Apr 4 in Others ankitgupta.1729 37 views

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1 answer

TIFR2021-A: 6
A matching in a graph is a set of edges such that no two edges in the set share a common vertex. Let $G$ be a graph on $n$ $\textit{vertices}$ in which there is a subset $M$ of $m$ $\textit{edges}$ which is a matching. Consider a random process where each vertex in the graph is independently ... $p^{2m}$ $\left ( 1-p^{2} \right )^{m}$ $1-\left ( 1-p\left ( 1-p \right ) \right )^{m}$

A matching in a graph is a set of edges such that no two edges in the set share a common vertex. Let $G$ be a graph on $n$ $\textit{vertices}$ in which there is a subset $M$ of $m$ $\textit{edges}$ which is a matching. Consider a random process where each vertex in the graph is independently selected ... $p^{2m}$ $\left ( 1-p^{2} \right )^{m}$ $1-\left ( 1-p\left ( 1-p \right ) \right )^{m}$

answered Apr 4 in Others Nikhil_dhama 33 views

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1 answer

TIFR2021-A: 2
What is the area of a rectangle with the largest perimeter that can be inscribed in the unit circle (i.e., all the vertices of the rectangle are on the circle with radius $1$)? $1$ $2$ $3$ $4$ $5$

What is the area of a rectangle with the largest perimeter that can be inscribed in the unit circle (i.e., all the vertices of the rectangle are on the circle with radius $1$)? $1$ $2$ $3$ $4$ $5$

answered Apr 3 in Others shashanks1999 87 views

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1 answer

CMI2020-A: 2
Consider the following regular expressions over alphabet$\{a,b\}$, where the notation $(a+b)^+$ means $(a+b)(a+b)^*$: $r_1=(a+b)^+a(a+b)^*$ $r_2=(a+b)^*b(a+b)^+$ Let $L_1$ and $L_2$ be the languages defined by $r_1$ and $r_2$, respectively. Which of the following regular expressions define $L_1\cap L_2$? ... $(a+b)^*a\;b(a+b)^*$ $(a+b)^*b(a+b)^*a(a+b)^*$ $(a+b)^*a(a+b)^*b(a+b)^*$

Consider the following regular expressions over alphabet$\{a,b\}$, where the notation $(a+b)^+$ means $(a+b)(a+b)^*$: $r_1=(a+b)^+a(a+b)^*$ $r_2=(a+b)^*b(a+b)^+$ Let $L_1$ and $L_2$ be the languages defined by $r_1$ and $r_2$, respectively. Which of the following regular expressions define $L_1\cap L_2$ ... $(a+b)^*a\;b(a+b)^*$ $(a+b)^*b(a+b)^*a(a+b)^*$ $(a+b)^*a(a+b)^*b(a+b)^*$

answered Mar 29 in Others Debdeep1998 28 views

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1 answer

CMI2020-A: 10
We have a procedure $P(n)$ that makes multiple calls to a procedure $Q(m)$, and runs in polynomial time in $n$. Unfortunately, a significant flaw was discovered in $Q(m)$, and it had to be replaced by $R(m)$, which runs in exponential time in $m$. Thankfully, $P$ is ... is proportional to $log\;n.$ $P(n)$ runs in polynomial time in $n$ if, for each call $Q(m),m \underline<log \;n.$

We have a procedure $P(n)$ that makes multiple calls to a procedure $Q(m)$, and runs in polynomial time in $n$. Unfortunately, a significant flaw was discovered in $Q(m)$, and it had to be replaced by $R(m)$, which runs in exponential time in $m$. Thankfully, $P$ is still correct ... $log\;n.$ $P(n)$ runs in polynomial time in $n$ if, for each call $Q(m),m \underline<log \;n.$

answered Mar 29 in Others Debdeep1998 11 views

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1 answer

GRAPH THEORY
consider the following directed graph,find the number of paths from A to I

consider the following directed graph,find the number of paths from A to I

answered Mar 26 in Others Kriti12kaushal 102 views

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2 answers

TIFR2021-B: 7
Which of the following regular expressions defines a language that is different from the other choices? $b^{\ast }\left ( a+b \right )^\ast a\left ( a+b \right )^ \ast ab^\ast \left ( a+b \right )^{\ast }$ ... $\left ( a+b \right )^{\ast }b^{\ast }a \left ( a+b\right )^{\ast }b^{\ast }\left ( a+b \right )^{\ast }$

Which of the following regular expressions defines a language that is different from the other choices? $b^{\ast }\left ( a+b \right )^\ast a\left ( a+b \right )^ \ast ab^\ast \left ( a+b \right )^{\ast }$ ... $\left ( a+b \right )^{\ast }b^{\ast }a \left ( a+b\right )^{\ast }b^{\ast }\left ( a+b \right )^{\ast }$

answered Mar 25 in Others jatinmittal199510 37 views

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1 answer

TIFR2021-B: 9
Let $L$ be a context-free language generated by the context-free grammar $G = (V, \Sigma, R, S)$ where $V$ is the finite set of variables, $\Sigma$ the finite set of terminals (disjoints from $V$), $R$ the finite set of rules and $S \in V$ the start variable. Consider the context-free ... ${L}'=L$ ${L}'=L^{\ast }$ ${L}'=\left \{ xx \mid x \in L \right \}$ None of the above

Let $L$ be a context-free language generated by the context-free grammar $G = (V, \Sigma, R, S)$ where $V$ is the finite set of variables, $\Sigma$ the finite set of terminals (disjoints from $V$), $R$ the finite set of rules and $S \in V$ the start variable. Consider the context-free grammar ${G}'$ ... ${L}'=LL$ ${L}'=L$ ${L}'=L^{\ast }$ ${L}'=\left \{ xx \mid x \in L \right \}$ None of the above

answered Mar 25 in Others Kanwae Kan 57 views

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1 answer

TIFR2021-B: 5
For a language $L$ over the alphabet $\{a, b\}$, let $\overline{L}$ denote the complement of $L$ and let $L^{\ast}$ denote the Kleene-closure of $L$. Consider the following sentences. $\overline{L}$ and $L^{\ast}$ are both context-free. $\overline{L}$ ... ? Both (i) and (iii) Only (i) Only (iii) Only (ii) None of the above

For a language $L$ over the alphabet $\{a, b\}$, let $\overline{L}$ denote the complement of $L$ and let $L^{\ast}$ denote the Kleene-closure of $L$. Consider the following sentences. $\overline{L}$ and $L^{\ast}$ are both context-free. $\overline{L}$ ... ? Both (i) and (iii) Only (i) Only (iii) Only (ii) None of the above

answered Mar 25 in Others jatinmittal199510 35 views

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1 answer

TIFR2021-B: 6
Consider the following pseudocode: procedure HowManyDash(n) if n=0 then print '-' else if n=1 then print '-' else HowManyDash(n-1) HowManyDash(n-2) end if end procedure How many ‘-’ does HowManyDash$(10)$ print? $9$ $10$ $55$ $89$ $1024$

Consider the following pseudocode: procedure HowManyDash(n) if n=0 then print '-' else if n=1 then print '-' else HowManyDash(n-1) HowManyDash(n-2) end if end procedure How many ‘-’ does HowManyDash$(10)$ print? $9$ $10$ $55$ $89$ $1024$

answered Mar 25 in Others jatinmittal199510 29 views

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1 answer

TIFR2021-B: 3
What is the prefix expression corresponding to the expression: $\left ( \left ( 9+8 \right ) \ast 7+\left ( 6\ast \left ( 5+4 \right ) \right )\ast 3\right )+2?$ You may assume that $\ast$ has precedence over $+$? $\ast + +\: 987 \ast \ast \: 6 + + \:5432$ ... $+ + \ast +\: 987 \ast \ast \: 6 + \:5432$ $+ \ast + \ast \: 987+ + \: 6 \ast \:5432$

What is the prefix expression corresponding to the expression: $\left ( \left ( 9+8 \right ) \ast 7+\left ( 6\ast \left ( 5+4 \right ) \right )\ast 3\right )+2?$ You may assume that $\ast$ has precedence over $+$? $\ast + +\: 987 \ast \ast \: 6 + + \:5432$ ... $+ + \ast +\: 987 \ast \ast \: 6 + \:5432$ $+ \ast + \ast \: 987+ + \: 6 \ast \:5432$

answered Mar 25 in Others jatinmittal199510 32 views

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1 answer

TIFR2021-A: 1
A box contains $5$ red marbles, $8$ green marbles, $11$ blue marbles, and $15$ yellow marbles. We draw marbles uniformly at random without replacement from the box. What is the minimum number of marbles to be drawn to ensure that out of the marbles drawn, at least $7$ are of the same colour? $7$ $8$ $23$ $24$ $39$

A box contains $5$ red marbles, $8$ green marbles, $11$ blue marbles, and $15$ yellow marbles. We draw marbles uniformly at random without replacement from the box. What is the minimum number of marbles to be drawn to ensure that out of the marbles drawn, at least $7$ are of the same colour? $7$ $8$ $23$ $24$ $39$

answered Mar 25 in Others vishnu_m7 141 views

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1 answer

TIFR2021-B: 2
Let $L$ be a singly-linked list $X$ and $Y$ be additional pointer variables such that $X$ points to the first element of $L$ and $Y$ points to the last element of $L$. Which of the following operations cannot be done in time that is bound above by a ... an element after the last element of $L$. Add an element before the first element of $L$. Interchange the first two elements of $L$.

Let $L$ be a singly-linked list $X$ and $Y$ be additional pointer variables such that $X$ points to the first element of $L$ and $Y$ points to the last element of $L$. Which of the following operations cannot be done in time that is bound above by a constant? ... Add an element after the last element of $L$. Add an element before the first element of $L$. Interchange the first two elements of $L$.

answered Mar 25 in Others jatinmittal199510 37 views

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1 answer

TIFR2021-A: 12
How many numbers in the range ${0, 1, \dots , 1365}$ have exactly four $1$'s in their binary representation? (Hint: $1365_{10}$ is $10101010101_{2}$, that is, $1365=2^{10} + 2^{8}+2^{6}+2^{4}+2^{2}+2^{0}.)$ In the following, the binomial coefficient $\binom{n}{k}$ counts the ... $\binom{11}{4}+\binom{9}{3}+\binom{7}{2}+\binom{5}{1}$ $1024$

How many numbers in the range ${0, 1, \dots , 1365}$ have exactly four $1$'s in their binary representation? (Hint: $1365_{10}$ is $10101010101_{2}$, that is, $1365=2^{10} + 2^{8}+2^{6}+2^{4}+2^{2}+2^{0}.)$ In the following, the binomial coefficient $\binom{n}{k}$ counts the number of ... $\binom{11}{4}+\binom{9}{3}+\binom{7}{2}+\binom{5}{1}$ $1024$

answered Mar 25 in Others jatinmittal199510 32 views

1 vote

1 answer

TIFR2021-A: 11
Find the following sum. $\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+\cdots+\frac{1}{40^{2}-1}$ $\frac{20}{41}$ $\frac{10}{41}$ $\frac{10}{21}$ $\frac{20}{21}$ $1$

Find the following sum. $\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+\cdots+\frac{1}{40^{2}-1}$ $\frac{20}{41}$ $\frac{10}{41}$ $\frac{10}{21}$ $\frac{20}{21}$ $1$

answered Mar 25 in Others jatinmittal199510 49 views

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1 answer

TIFR2021-A: 10
Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads than ketak? In the following, the binomial coefficient $\binom{n}{k}$ counts the number of $k$-element subsets of an $n$-element ... $\sum_{i=0}^{n}\frac{\binom{n}{i}}{2^{n}}$

Lavanya and Ketak each flip a fair coin (i.e., both heads and tails have equal probability of appearing) $n$ times. What is the probability that Lavanya sees more heads than ketak? In the following, the binomial coefficient $\binom{n}{k}$ counts the number of $k$-element subsets of an $n$ ... $\sum_{i=0}^{n}\frac{\binom{n}{i}}{2^{n}}$

answered Mar 25 in Others jatinmittal199510 37 views

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1 answer

TIFR2021-A: 7
Let $d$ be the positive square integers (that is, it is a square of some integer) that are factors of $20^{5} \times 21^{5}$. Which of the following is true about $d$? $50\leq d< 100$ $100\leq d< 150$ $150\leq d< 200$ $200\leq d< 300$ $300\leq d$

Let $d$ be the positive square integers (that is, it is a square of some integer) that are factors of $20^{5} \times 21^{5}$. Which of the following is true about $d$? $50\leq d< 100$ $100\leq d< 150$ $150\leq d< 200$ $200\leq d< 300$ $300\leq d$

answered Mar 25 in Others jatinmittal199510 33 views

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1 answer

TIFR2021-A: 5
Let $n, m$ and $k$ be three positive integers such that $n \geq m \geq k$. Let $S$ be a subset of $\left \{ 1, 2,\dots, n \right \}$ of size $k$. Consider sampling a function $f$ uniformly at random from the set of all functions mapping $\left \{ 1,\dots, n \right \}$ ... $1-\frac{k!\binom{m}{k}}{m^{k}}$ $1-\frac{k!\binom{n}{k}}{n^{k}}$ $1-\frac{k!\binom{n}{k}}{m^{k}}$

Let $n, m$ and $k$ be three positive integers such that $n \geq m \geq k$. Let $S$ be a subset of $\left \{ 1, 2,\dots, n \right \}$ of size $k$. Consider sampling a function $f$ uniformly at random from the set of all functions mapping $\left \{ 1,\dots, n \right \}$ ... $1-\frac{k!\binom{m}{k}}{m^{k}}$ $1-\frac{k!\binom{n}{k}}{n^{k}}$ $1-\frac{k!\binom{n}{k}}{m^{k}}$

answered Mar 25 in Others jatinmittal199510 40 views

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1 answer

TIFR2021-A: 4
What is the probability that at least two out of four people have their birthdays in the same month, assuming their birthdays are uniformly distributed over the twelve months? $\frac{25}{48}$ $\frac{5}{8}$ $\frac{5}{12}$ $\frac{41}{96}$ $\frac{55}{96}$

What is the probability that at least two out of four people have their birthdays in the same month, assuming their birthdays are uniformly distributed over the twelve months? $\frac{25}{48}$ $\frac{5}{8}$ $\frac{5}{12}$ $\frac{41}{96}$ $\frac{55}{96}$

answered Mar 25 in Others jatinmittal199510 63 views

1 vote

0 answers

TIFR2021-A: 3
Let $M$ be an $n \times m$ real matrix. Consider the following: Let $k_{1}$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \times k_{1}$ and $B$ is a $k_{1} \times m$ matrix. Let $k_{2}$ ... $k_{2}= k_{3}< k_{1}$ $k_{1}= k_{2}= k_{3}$ No general relationship exists among $k_{1}, k_{2}$ and $k_{3}$

Let $M$ be an $n \times m$ real matrix. Consider the following: Let $k_{1}$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \times k_{1}$ and $B$ is a $k_{1} \times m$ matrix. Let $k_{2}$ ... $k_{2}= k_{3}< k_{1}$ $k_{1}= k_{2}= k_{3}$ No general relationship exists among $k_{1}, k_{2}$ and $k_{3}$

asked Mar 25 in Others soujanyareddy13 44 views

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0 answers

TIFR2021-A: 8
Consider the sequence $y_{n}=\frac{1}{\int_{1}^{n}\frac{1}{\left ( 1+x/n \right )^{3}}dx}$ for $\text{n} = 2,3,4, \dots$ Which of the following is $\text{TRUE}$? The sequence $\{y_{n}\}$ does not have a limit as $n\rightarrow \infty$ ... $0$. The sequence $\{y_{n}\}$ first increases and then decreases as $\text{n}$ takes values $2, 3, 4, \dots$

Consider the sequence $y_{n}=\frac{1}{\int_{1}^{n}\frac{1}{\left ( 1+x/n \right )^{3}}dx}$ for $\text{n} = 2,3,4, \dots$ Which of the following is $\text{TRUE}$? The sequence $\{y_{n}\}$ does not have a limit as $n\rightarrow \infty$. $y_{n}\leq 1$ for all ... $0$. The sequence $\{y_{n}\}$ first increases and then decreases as $\text{n}$ takes values $2, 3, 4, \dots$

asked Mar 25 in Others soujanyareddy13 22 views

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0 answers

TIFR2021-A: 15
Let $P$ be a convex polygon with sides $5, 4, 4, 3$. For example, the following: Consider the shape in the plane that consists of all points within distance $1$ from some point in $P$. If $\ell$ is the perimeter of the shape, which of the following is always correct? ... from the given information. $20\leq \ell < 21$ $21\leq \ell< 22$ $22\leq \ell< 23$ $23\leq \ell< 24$

Let $P$ be a convex polygon with sides $5, 4, 4, 3$. For example, the following: Consider the shape in the plane that consists of all points within distance $1$ from some point in $P$. If $\ell$ is the perimeter of the shape, which of the following is always correct? $\ell$ cannot be determined from the given information. $20\leq \ell < 21$ $21\leq \ell< 22$ $22\leq \ell< 23$ $23\leq \ell< 24$

asked Mar 25 in Others soujanyareddy13 19 views

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0 answers

TIFR2021-B: 4
Consider the following two languages. ... in $\text{P}$. None of the above since we can answer this question only if we resolve the status of the $\text{NP}$ vs. $\text{P}$ question.

Consider the following two languages. $\begin{array}{rcl} \text{PRIME} & = & \{ 1^{n} \mid n \text{ is a prime number} \}, \\ \text{FACTOR} & = & \{ 1^{n}0 1^{a} 01^{b} \mid n \text{ has a factor in the range }[a,b] \} \end{array}$. What can you say ... $\text{P}$. None of the above since we can answer this question only if we resolve the status of the $\text{NP}$ vs. $\text{P}$ question.

asked Mar 25 in Others soujanyareddy13 28 views

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0 answers

TIFR2021-B: 8
Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements. If $AB = B$, the $A$ must be the identity matrix. If $A$ is an idempotent (i.e. $A^{2} = A$) nonsingular matrix, then $A$ ... $A$? $1, 2 $ and $3$ Only $2$ and $3$ Only $1$ and $2$ Only $1$ and $3$ Only $2$

Let $A$ and $B$ be two matrices of size $n \times n$ and with real-valued entries. Consider the following statements. If $AB = B$, the $A$ must be the identity matrix. If $A$ is an idempotent (i.e. $A^{2} = A$) nonsingular matrix, then $A$ must be the identity matrix. If $A^{-1} = A$, the ... $\text{MUST}$ be true of $A$? $1, 2 $ and $3$ Only $2$ and $3$ Only $1$ and $2$ Only $1$ and $3$ Only $2$

asked Mar 25 in Others soujanyareddy13 23 views

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0 answers

TIFR2021-B: 11
Suppose we toss a fair coin (i.e., both beads and tails have equal probability of appearing) repeatedly until the first time by which at least $\textit{two}$ heads and at least $\textit{two}$ tails have appeared in the sequence of tosses made. What is the expected number of coin tosses that we would have to make? $8$ $4$ $5.5$ $7.5$ $4.5$

Suppose we toss a fair coin (i.e., both beads and tails have equal probability of appearing) repeatedly until the first time by which at least $\textit{two}$ heads and at least $\textit{two}$ tails have appeared in the sequence of tosses made. What is the expected number of coin tosses that we would have to make? $8$ $4$ $5.5$ $7.5$ $4.5$

asked Mar 25 in Others soujanyareddy13 42 views

0 votes

0 answers

TIFR2021-B: 15
Let $A\left [ i \right ] : i=0, 1, 2, \dots , n-1$ be an array of $n$ distinct integers. We wish to sort $A$ in ascending order. We are given that each element in the array is at a position that is at most $k$ away from its position in the sorted array, that is, we are ... $t\left ( n \right ) = \Theta \left ( nk \right )$

Let $A\left [ i \right ] : i=0, 1, 2, \dots , n-1$ be an array of $n$ distinct integers. We wish to sort $A$ in ascending order. We are given that each element in the array is at a position that is at most $k$ ... $t\left ( n \right ) = \Theta \left ( nk \right )$

asked Mar 25 in Others soujanyareddy13 43 views

0 votes

1 answer

CMI-2018-DataScience-A: 4
A $n\times n$ matrix $A$ is said to be $symmetric$ if $A^T=A$. Suppose $A$ is an arbitrary $2\times 2$ matrix. Then which of the following matrices are symmetric (here $0$ denotes the $2\times 2$ matrix consisting of zeros): $A^TA$ $\begin{bmatrix} 0&A^T \\ A & 0 \end{bmatrix}$ $AA^T$ $\begin{bmatrix} A & 0 \\ 0 & A^T \end{bmatrix}$

A $n\times n$ matrix $A$ is said to be $symmetric$ if $A^T=A$. Suppose $A$ is an arbitrary $2\times 2$ matrix. Then which of the following matrices are symmetric (here $0$ denotes the $2\times 2$ matrix consisting of zeros): $A^TA$ $\begin{bmatrix} 0&A^T \\ A & 0 \end{bmatrix}$ $AA^T$ $\begin{bmatrix} A & 0 \\ 0 & A^T \end{bmatrix}$

answered Mar 7 in Others i_am_tatha 49 views

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CMI-2018-DataScience-A: 12
In an entrance examination with multiple choice questions, with each question having four options and a single correct answer, suppose that only $20\%$ candidates think they know the answer to one difficult question and only half of them know it ... same. If a candidate has correctly answered the question, what is the (conditional) probability that she knew the answer?

In an entrance examination with multiple choice questions, with each question having four options and a single correct answer, suppose that only $20\%$ candidates think they know the answer to one difficult question and only half of them know it correctly and the ... tick the same. If a candidate has correctly answered the question, what is the (conditional) probability that she knew the answer?

answered Mar 2 in Others Nihal Singh 44 views

2 votes

1 answer

TIFR2016-A-3
Consider the following set of $3n$ linear equations in $3n$ ... subspace of $\mathbb{R}^{3n}$ of dimension n $S$ is a subspace of $\mathbb{R}^{3n}$ of dimension $n-1$ $S$ has exactly $n$ elements

Consider the following set of $3n$ linear equations in $3n$ variables: $\begin{matrix} x_1-x_2=0 & x_4-x_5 =0 & \dots & x_{3n-2}-x_{3n-1}=0 \\ x_2-x_3=0 & x_5-x_6=0 & & x_{3n-1}-x_{3n}=0 \\ x_1-x_3=0 & x_4-x_6 =0 & & x_{3n-2}=x_{3n}=0 \end{matrix}$ ... $S$ is a subspace of $\mathbb{R}^{3n}$ of dimension n $S$ is a subspace of $\mathbb{R}^{3n}$ of dimension $n-1$ $S$ has exactly $n$ elements

answered Feb 20 in Others ankitgupta.1729 157 views

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2 answers

CMI-2018-DataScience-A: 3
Let $x=\begin{bmatrix} 3& 1 & 2 \end{bmatrix}$. Which of the following statements are true? $x^Tx$ is a $3\times 3$ matrix $xx^T$ is a $3\times 3$ matrix $xx^T$ is a $1\times 1$ matrix $xx^T=x^Tx$

Let $x=\begin{bmatrix} 3& 1 & 2 \end{bmatrix}$. Which of the following statements are true? $x^Tx$ is a $3\times 3$ matrix $xx^T$ is a $3\times 3$ matrix $xx^T$ is a $1\times 1$ matrix $xx^T=x^Tx$

answered Feb 16 in Others Hira Thakur 71 views

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2 answers

CMI-2018-DataScience-A: 2
Let ... $|A|=|B|$ $|C|=|D|$ $|B|=-|C|$ $|A|=-|D|$

Let $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}, C=\begin{bmatrix} 4 & 5 & 6 \\ 1 & 2 & 3 \\ 7 & 8 & 9 \end{bmatrix}$ and $D=\begin{bmatrix} -1 & 2 & 3 \\ 4 & -5 & 6 \\ 7 & 8 & -9 \end{bmatrix}$. Which of the following statements are true? $|A|=|B|$ $|C|=|D|$ $|B|=-|C|$ $|A|=-|D|$

answered Feb 16 in Others Hira Thakur 72 views

0 votes

1 answer

CMI-2019-DataScience-A: 4
Let $A=\begin{bmatrix} 1& 1& 1\\0&2&2\\0&0&3 \end{bmatrix}, B=\begin{bmatrix} 5&5&5\\0&10&10\\0&0&15\end{bmatrix}, C=\begin{bmatrix} 3&0&0\\3&6&0\\3&6&9 \end{bmatrix}$. Which of the following statements are true? $|A|=|B|$ $|B|=125|A|$ $|C|=27|A|$ $|C|=\frac{|A|}{3}$

Let $A=\begin{bmatrix} 1& 1& 1\\0&2&2\\0&0&3 \end{bmatrix}, B=\begin{bmatrix} 5&5&5\\0&10&10\\0&0&15\end{bmatrix}, C=\begin{bmatrix} 3&0&0\\3&6&0\\3&6&9 \end{bmatrix}$. Which of the following statements are true? $|A|=|B|$ $|B|=125|A|$ $|C|=27|A|$ $|C|=\frac{|A|}{3}$

answered Feb 16 in Others Hira Thakur 20 views

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