GATE Overflow - Recent questions tagged tifr2013
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Powered by Question2AnswerTIFR2013-B-20
https://gateoverflow.in/25878/tifr2013-b-20
<p>Suppose $n$ processors are connected in a linear array as shown below. Each processor has a number. The processors need to exchange numbers so that the numbers eventually appear in ascending order (the processor $\rm P1$ should have the minimum value and the the processor $\rm Pn$ should have the maximum value).</p>
<p><img alt="" height="49" src="https://gateoverflow.in/?qa=blob&qa_blobid=3718674314406348016" width="515"></p>
<p>The algorithm to be employed is the following. Odd numbered processors and even numbered processors are activated alternate steps; assume that in the first step all the even numbered processors are activated. When a processor is activated, the number it holds is compared with the number held by its right-hand neighbour (if one exists) and the smaller of the two numbers is retained by the activated processor and the bigger stored in its right hand neighbour.
<br>
How long does it take for the processors to sort the values?</p>
<ol style="list-style-type:upper-alpha">
<li>$n \log n$ steps</li>
<li>$n^2$ steps</li>
<li>$n$ steps</li>
<li>$n^{1.5}$ steps</li>
<li>The algorithm is not guaranteed to sort</li>
</ol>Algorithmshttps://gateoverflow.in/25878/tifr2013-b-20Sun, 08 Nov 2015 13:58:11 +0000TIFR2013-B-19
https://gateoverflow.in/25872/tifr2013-b-19
<p>In a relational database there are three relations:</p>
<ul>
<li>Customers = $C$(CName),</li>
<li>Shops = $S$(SName),</li>
<li>Buys = $B$(CName, SName).</li>
</ul>
<p>Which of the following relational algebra expressions returns the names of shops that have no customers at all? [Here $\Pi$ is the projection operator.]</p>
<ol style="list-style-type: lower-alpha;">
<li>$\Pi _{S Name}B$</li>
<li>$S - B$</li>
<li>$S - \Pi _{S Name}B$</li>
<li>$S - \Pi _{S Name}((C \times S) - B)$</li>
<li>None of the above</li>
</ol>
Databaseshttps://gateoverflow.in/25872/tifr2013-b-19Sun, 08 Nov 2015 13:48:46 +0000TIFR2013-B-18
https://gateoverflow.in/25865/tifr2013-b-18
<p>Let $S$ be a set of numbers. For $x \in S$, the rank of $x$ is the number of elements in $S$ that are less than or equal to $x$. The procedure Select $(S, r)$ takes a set $S$ of numbers and a rank $r\left(1 \leq r \leq |S|\right)$ and returns the element in $S$ of rank $r$. The procedure MultiSelect$(S,R)$ takes a set of numbers $S$ and a list of ranks $R=\left\{r_{1} < r_{2} < ...<r_{k}\right\}$, and returns the list $\left\{x_{1} < x_{2} < ...<x_{k}\right\}$ of elements of $S$, such that the rank of $x_{i}$ is $r_{i}$. Suppose there is an implementation for Select $(S, r)$ that uses at most (constant ·$|S|$) binary comparisons between elements of $S$. The minimum number of comparisons needed to implement MultiSelect $(S,R)$ is</p>
<ol style="list-style-type: lower-alpha;">
<li>constant · $|S| \log |S|$</li>
<li>constant · $|S|$</li>
<li>constant · $|S||R|$</li>
<li>constant · $|R| \log |S|$</li>
<li>constant · $|S|(1 + \log |R|)$</li>
</ol>
Algorithmshttps://gateoverflow.in/25865/tifr2013-b-18Sun, 08 Nov 2015 13:34:45 +0000TIFR2013-B-17
https://gateoverflow.in/25860/tifr2013-b-17
<p>In a connected weighted graph with $n$ vertices, all the edges have distinct positive integer weights. Then, the maximum number of minimum weight spanning trees in the graph is</p>
<ol style="list-style-type:lower-alpha">
<li>$1$</li>
<li>$n$</li>
<li>equal to number of edges in the graph.</li>
<li>equal to maximum weight of an edge of the graph.</li>
<li>$n^{n-2}$</li>
</ol>Algorithmshttps://gateoverflow.in/25860/tifr2013-b-17Sun, 08 Nov 2015 13:20:46 +0000TIFR2013-B-16
https://gateoverflow.in/25859/tifr2013-b-16
<p>Consider a function $T_{k, n}: \left\{0, 1\right\}^{n}\rightarrow \left\{0, 1\right\}$ which returns $1$ if at least $k$ of its $n$ inputs are $1$. Formally, $T_{k, n}(x)=1$ if $\sum ^{n}_{1} x_{i}\geq k$. Let $y \in \left\{0, 1\right\}^{n}$ be such that $y$ has exactly $k$ ones. Then, the function $T_{k, n-1} \left(y_{1}, y_{2},....y_{i-1}, y_{i+1},...,y_{n}\right)$ (where $y_{i}$ is omitted) is equivalent to</p>
<ol style="list-style-type: lower-alpha;">
<li>$T_{k-1}, n(y)$</li>
<li>$T_{k, n}(y)$</li>
<li>$y_{i}$</li>
<li>$\neg y_{i}$</li>
<li>None of the above.</li>
</ol>
Set Theory & Algebrahttps://gateoverflow.in/25859/tifr2013-b-16Sun, 08 Nov 2015 13:13:21 +0000TIFR2013-B-15
https://gateoverflow.in/25798/tifr2013-b-15
<p>Let $G$ be an undirected graph with $n$ vertices. For any subset $S$ of vertices, the set of neighbours of $S$ consists of the union of $S$ and the set of vertices $S'$ that are connected to some vertex in $S$ by an edge of $G$. The graph $G$ has the nice property that every subset of vertices $S$ of size at most $n/2$ has at least $1.5 |S|$-many neighbours. What is the length of a longest path in $G$?</p>
<ol style="list-style-type: lower-alpha;">
<li>$O (1)$</li>
<li>$O (\log \log n)$ but not $O (1)$</li>
<li>$O (\log n)$ but not $O (\log \log n)$</li>
<li>$O \left(\sqrt{n}\right)$ but not $O (\log n)$</li>
<li>$O (n)$ but not $O \left(\sqrt{n}\right)$</li>
</ol>
Algorithmshttps://gateoverflow.in/25798/tifr2013-b-15Sat, 07 Nov 2015 20:04:42 +0000TIFR2013-B-14
https://gateoverflow.in/25794/tifr2013-b-14
<p>Assume a demand paged memory system where ONLY THREE pages can reside in the memory at a time. The following sequence gives the order in which the program references the pages.
<br>
$1, 3, 1, 3, 4, 2, 2, 4$
<br>
Assume that least frequently used page is replaced when necessary. If there is more than one least frequently used pages then the least recently used page among them is replaced. During the program’s execution, how many times will the pages $1,2,3$ and $4$ be brought to the memory?</p>
<ol style="list-style-type: lower-alpha;">
<li>$2,2,2,2$ times, respectively</li>
<li>$1,1,1,2$ times, respectively</li>
<li>$1,1,1,1$ times, respectively</li>
<li>$2,1,2,2$ times, respectively</li>
<li>None of the above</li>
</ol>
Operating Systemhttps://gateoverflow.in/25794/tifr2013-b-14Sat, 07 Nov 2015 19:54:47 +0000TIFR2013-B-13
https://gateoverflow.in/25775/tifr2013-b-13
<p>Given a binary tree of the following form and having $n$ nodes, the height of the tree is</p>
<p><img alt="" height="270" src="https://gateoverflow.in/?qa=blob&qa_blobid=10083534387550040754" width="289"></p>
<ol style="list-style-type: lower-alpha;">
<li>$\Theta \left(\log n\right)$</li>
<li>$\Theta \left(n\right)$</li>
<li>$\Theta \left(\sqrt{n}\right)$</li>
<li>$\Theta \left(n / \log n\right)$</li>
<li>None of the above.</li>
</ol>
DShttps://gateoverflow.in/25775/tifr2013-b-13Sat, 07 Nov 2015 17:20:06 +0000TIFR2013-B-12
https://gateoverflow.in/25774/tifr2013-b-12
<p>It takes $O(n)$ time to find the median in a list of $n$ elements, which are not necessarily in sorted order while it takes only $O(1)$ time to find the median in a list of $n$ sorted elements. How much time does it take to find the median of $2n$ elements which are given as two lists of $n$ sorted elements each?</p>
<ol style="list-style-type:lower-alpha">
<li>$O (1)$</li>
<li>$O \left(\log n\right)$ but not $O (1)$</li>
<li>$O (\sqrt{n})$ but not $O \left(\log n\right)$</li>
<li>$O (n)$ but not $O (\sqrt{n})$</li>
<li>$O \left(n \log n\right)$ but not $O (n)$</li>
</ol>Algorithmshttps://gateoverflow.in/25774/tifr2013-b-12Sat, 07 Nov 2015 16:53:56 +0000TIFR2013-B-11
https://gateoverflow.in/25772/tifr2013-b-11
<p>Which of the following statements is FALSE?</p>
<ol style="list-style-type: lower-alpha;">
<li>The intersection of a context free language with a regular language is context free.</li>
<li>The intersection of two regular languages is regular.</li>
<li>The intersection of two context free languages is context free</li>
<li>The intersection of a context free language and the complement of a regular language is context free.</li>
<li>The intersection of a regular language and the complement of a regular language is regular.</li>
</ol>
Theory of Computationhttps://gateoverflow.in/25772/tifr2013-b-11Sat, 07 Nov 2015 16:44:56 +0000TIFR2013-B-10
https://gateoverflow.in/25771/tifr2013-b-10
<p>Let $m, n$ be positive integers with $m$ a power of $2$. Let $s= 100 n^{2} \log m$. Suppose $S_{1}, S_{2},\dots ,S_{m}$ are subsets of ${1, 2, \dots, s}$ such that $|S_{i}|= 10 n \log m$ and $|S_{i} \cap S_{j}|\leq \log m$ for all $1 \leq i < j \leq m$. Such a collection of sets $S_{1},\dots ,S_{m}$ is an example of a so-called Nisan-Wigderson design. We now consider the set membership problem, where we have to store an arbitrary subset $T \subseteq \left\{1, 2,....,m\right\}, |T|=n$ as an array $A$ of $s$ bits so that given any integer $x, 1 \leq x \leq m$, we can discover whether $x \in T$ by reading only one bit of $A$. Consider the following strategy to solve this problem. Array $A$ is initialized to all zeroes. Given the set $T$ to be stored, we put a one in all the locations of $A$ indexed by the union $\cup_{t \in T}S_{t}$. Now, given the integer $x$, we read a random location in $A$ from $S_{x}$ and declare that $x \in T$ if the bit in that location is one. This strategy gives the correct answer with probability</p>
<ol style="list-style-type: lower-alpha;">
<li>$1$ if $x \in T$ and at most $0.1$ if $x ∉ T$.</li>
<li>At least $0.9$ if $x \in T$ and at most $0.1$ if $x ∉ T$.</li>
<li>At least $0.9$ if $x \in T$ and at least $0.9$ if $x ∉ T$.</li>
<li>$1$ if $x \in T$ and at least $0.9$ if $x ∉ T$.</li>
<li>At least $0.9$ if $x \in T$ and $1$ if $x ∉ T$. </li>
</ol>
Probabilityhttps://gateoverflow.in/25771/tifr2013-b-10Sat, 07 Nov 2015 16:31:42 +0000TIFR2013-B-9
https://gateoverflow.in/25675/tifr2013-b-9
<p>Suppose $n$ straight lines are drawn on a plane. When these lines are removed, the plane falls apart into several connected components called regions. $A$ region $R$ is said to be convex if it has the following property: whenever two points are in $R$, then the entire line segment joining them is in $R$. Suppose no two of the n lines are parallel. Which of the following is true?</p>
<ol style="list-style-type: lower-alpha;">
<li>$O (n)$ regions are produced, and each region is convex.</li>
<li>$O (n^{2})$ regions are produced but they need not all be convex.</li>
<li>$O (n^{2})$ regions are produced, and each region is convex.</li>
<li>$O (n \log n)$ regions are produced, but they need not all be convex.</li>
<li>All regions are convex but there may be exponentially many of them.</li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25675/tifr2013-b-9Fri, 06 Nov 2015 19:44:54 +0000TIFR2013-B-8
https://gateoverflow.in/25670/tifr2013-b-8
<p>Which one of the following languages over the alphabet ${0, 1}$ is regular?</p>
<ol style="list-style-type:lower-alpha">
<li>The language of balanced parentheses where $0, 1$ are thought of as $(,)$ respectively.</li>
<li>The language of palindromes, i.e. bit strings $x$ that read the same from left to right as well as right to left.</li>
<li>$L= \left \{ 0^{m^{2}}: 3 \leq m \right \}$</li>
<li>The Kleene closure $L^*$, where $L$ is the language in $(c)$ above.</li>
<li>$\left \{ 0^{m} 1^{n} | 1 \leq m \leq n\right \}$</li>
</ol>Theory of Computationhttps://gateoverflow.in/25670/tifr2013-b-8Fri, 06 Nov 2015 19:23:41 +0000TIFR2013-B-7
https://gateoverflow.in/25668/tifr2013-b-7
<p>Which of the following is not implied by $P=NP$?</p>
<ol style="list-style-type:lower-alpha">
<li>$3$SAT can be solved in polynomial time.</li>
<li>Halting problem can be solved in polynomial time.</li>
<li>Factoring can be solved in polynomial time.</li>
<li>Graph isomorphism can be solved in polynomial time.</li>
<li>Travelling salesman problem can be solved in polynomial time.</li>
</ol>
<p> </p>Algorithmshttps://gateoverflow.in/25668/tifr2013-b-7Fri, 06 Nov 2015 19:10:28 +0000TIFR2013-B-6
https://gateoverflow.in/25667/tifr2013-b-6
<p>Let $L$ and $L'$ be languages over the alphabet $\Sigma $. The left quotient of $L$ by $L'$ is</p>
<p>$L/L'\overset{\underset{\mathrm{}}{def}}{=} \left\{w ∈ \Sigma^* : wx ∈ L\text{ for some }x ∈ L'\right\}$</p>
<p>Which of the following is true?</p>
<ol style="list-style-type:lower-alpha">
<li>If $L/L'$ is regular then $L'$ is regular.</li>
<li>If $L$ is regular then $L/L'$ is regular.</li>
<li>If $L/L'$ is regular then $L$ is regular.</li>
<li>$L/L'$ is a subset of $L$.</li>
<li>If $L/L'$ and $L'$ are regular, then $L$ is regular.</li>
</ol>
<p> </p>Theory of Computationhttps://gateoverflow.in/25667/tifr2013-b-6Fri, 06 Nov 2015 18:56:57 +0000TIFR2013-B-5
https://gateoverflow.in/25666/tifr2013-b-5
<p>Given a weighted directed graph with $n$ vertices where edge weights are integers (positive, zero, or negative), determining whether there are paths of arbitrarily large weight can be performed in time</p>
<ol style="list-style-type:upper-alpha">
<li>$O(n)$</li>
<li>$O(n . \log(n))$ but not $O (n)$</li>
<li>$O(n^{1.5})$ but not $O (n \log n)$</li>
<li>$O(n^{3})$ but not $O(n^{1.5})$</li>
<li>$O(2^{n})$ but not $O(n^{3})$</li>
</ol>Algorithmshttps://gateoverflow.in/25666/tifr2013-b-5Fri, 06 Nov 2015 18:35:26 +0000TIFR2013-B-4
https://gateoverflow.in/25664/tifr2013-b-4
<p>A set $S$ together with partial order $\ll$ is called a well order if it has no infinite descending chains, i.e. there is no infinite sequence $x_1, x_2,\ldots$ of elements from $S$ such that $x_{i+1} \ll x_i$ and $x_{i+1} \neq x_i$ for all $i$.</p>
<p>Consider the set of all words (finite sequence of letters $a - z$), denoted by $W$, in dictionary order.</p>
<ol style="list-style-type: lower-alpha;">
<li>Between $``aa"$ and $``az"$ there are only $24$ words.</li>
<li>Between $``aa"$ and $``az"$ there are only $2^{24}$ words.</li>
<li>$W$ is not a partial order.</li>
<li>$W$ is a partial order but not a well order.</li>
<li>$W$ is a well order.</li>
</ol>
Set Theory & Algebrahttps://gateoverflow.in/25664/tifr2013-b-4Fri, 06 Nov 2015 18:24:35 +0000TIFR2013-B-3
https://gateoverflow.in/25659/tifr2013-b-3
<p>How many $4 \times 4$ matrices with entries from ${0, 1}$ have odd determinant?</p>
<p>Hint: Use modulo $2$ arithmetic.</p>
<ol style="list-style-type: lower-alpha;">
<li>$20160$</li>
<li>$32767$</li>
<li>$49152$</li>
<li>$57343$</li>
<li>$65520$</li>
</ol>
<p> </p>
Linear Algebrahttps://gateoverflow.in/25659/tifr2013-b-3Fri, 06 Nov 2015 17:27:16 +0000TIFR2013-B-2
https://gateoverflow.in/25657/tifr2013-b-2
<p>Consider polynomials in a single variable $x$ of degree $d$. Suppose $d < n/2$. For such a polynomial $p(x)$, let $C_{p}$ denote the $n$-tuple $(P\left ( i \right ))_{1 \leq i \leq n}$. For any two such distinct polynomials $p, q,$ the number of coordinates where the tuples $C_{p}, C_{q}$ differ is.</p>
<ol style="list-style-type: upper-alpha;">
<li>At most $d$</li>
<li>At most $n - d$</li>
<li>Between $d$ and $n - d$</li>
<li>At least $n - d$</li>
<li>None of the above.</li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25657/tifr2013-b-2Fri, 06 Nov 2015 17:19:42 +0000TIFR2013-B-1
https://gateoverflow.in/25508/tifr2013-b-1
<p>Let $G= (V, E)$ be a simple undirected graph on $n$ vertices. A colouring of $G$ is an assignment of colours to each vertex such that endpoints of every edge are given different colours. Let $\chi (G)$ denote the chromatic number of $G$, i.e. the minimum numbers of colours needed for a valid colouring of $G$. A set $B\subseteq V$ is an independent set if no pair of vertices in $B$ is connected by an edge. Let $a(G)$ be the number of vertices in a largest possible independent set in $G$. In the absence of any further information about $G$ we can conclude.</p>
<ol style="list-style-type: upper-alpha;">
<li>$\chi (G)\geq a(G)$</li>
<li>$\chi (G)\leq a(G)$</li>
<li>$a(G)\geq n/\chi (G)$</li>
<li>$a(G)\leq n/\chi (G)$</li>
<li>None of the above.</li>
</ol>Graph Theoryhttps://gateoverflow.in/25508/tifr2013-b-1Thu, 05 Nov 2015 14:47:13 +0000TIFR2013-A-20
https://gateoverflow.in/25502/tifr2013-a-20
<p>Consider a well functioning clock where the hour, minute and the seconds needles are exactly at zero. How much time later will the minutes needle be exactly one minute ahead ($1/60$ th of the circumference) of the hours needle and the seconds needle again exactly at zero?</p>
<p>Hint: When the desired event happens both the hour needle and the minute needle have moved an integer multiple of $1/60$ th of the circumference.</p>
<ol style="list-style-type: lower-alpha;">
<li>$144$ minutes</li>
<li>$66$ minutes</li>
<li>$96$ minutes</li>
<li>$72$ minutes</li>
<li>$132$ minutes </li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25502/tifr2013-a-20Thu, 05 Nov 2015 14:26:46 +0000TIFR2013-A-19
https://gateoverflow.in/25500/tifr2013-a-19
<p>Consider a sequence of numbers $(\epsilon _{n}: n= 1, 2,...)$, such that $\epsilon _{1}=10$ and</p>
<p>$\epsilon _{n+1}=\frac{20\epsilon _{n}}{20+\epsilon _{n}}$</p>
<p>for $n\geq 1$. Which of the following statements is true?</p>
<p>Hint: Consider the sequence of reciprocals.</p>
<ol style="list-style-type: lower-alpha;">
<li>The sequence $(\epsilon _{n}: n= 1, 2,...)$ converges to zero.</li>
<li>$\epsilon _{n}\geq 1$ for all $n$</li>
<li>The sequence $(\epsilon _{n}: n= 1, 2,...)$ is decreasing and converges to 1.</li>
<li>The sequence $(\epsilon _{n}: n= 1, 2,...)$ is decreasing and then increasing. Finally it converges to 1.</li>
<li>None of the above.</li>
</ol>
<p> </p>
Numerical Abilityhttps://gateoverflow.in/25500/tifr2013-a-19Thu, 05 Nov 2015 14:16:43 +0000TIFR2013-A-18
https://gateoverflow.in/25498/tifr2013-a-18
<p>Consider three independent uniformly distributed (taking values between $0$ and $1$) random variables. What is the probability that the middle of the three values (between the lowest and the highest value) lies between $a$ and $b$ where $0 ≤ a < b ≤ 1$.</p>
<ol style="list-style-type: lower-alpha;">
<li>$3 (1 - b) a (b - a)$</li>
<li>$3 (b - a) - (b^{2}- a^{2})/2)$</li>
<li>$6 (1 - b) a (b - a)$</li>
<li>$(1 - b) a (b - a)$</li>
<li>$6 ((b^{2}- a^{2})/ 2 - (b^{3} - a^{3})/3)$.</li>
</ol>
Probabilityhttps://gateoverflow.in/25498/tifr2013-a-18Thu, 05 Nov 2015 13:56:12 +0000TIFR2013-A-17
https://gateoverflow.in/25497/tifr2013-a-17
<p>A stick of unit length is broken into two at a point chosen at random. Then, the larger part of the stick is further divided into two parts in the ratio $4:3$. What is the probability that the three sticks that are left CANNOT form a triangle?</p>
<ol style="list-style-type: lower-alpha;">
<li>$1/4$</li>
<li>$1/3$</li>
<li>$5/6$</li>
<li>$1/2$</li>
<li>$\log_{e}(2)/2$</li>
</ol>
Probabilityhttps://gateoverflow.in/25497/tifr2013-a-17Thu, 05 Nov 2015 13:46:36 +0000TIFR2013-A-16
https://gateoverflow.in/25496/tifr2013-a-16
<p>The minimum of the function $f(x) = x \log_{e}(x)$ over the interval $[1/2, \infty )$ is</p>
<ol style="list-style-type: lower-alpha;">
<li>$0$</li>
<li>$-e$</li>
<li>$-\log_{e}(2)/2$</li>
<li>$-1/e$</li>
<li>None of the above</li>
</ol>
Calculushttps://gateoverflow.in/25496/tifr2013-a-16Thu, 05 Nov 2015 13:41:17 +0000TIFR2013-A-15
https://gateoverflow.in/25438/tifr2013-a-15
<p>Let $\DeclareMathOperator{S}{sgn}
<br>
\S (x)= \begin{cases}
<br>
+1 & \text{if } x \geq 0 \\
<br>
-1 & \text{if } x < 0
<br>
\end{cases}$</p>
<p>What is the value of the following summation?</p>
<p>$$\sum_{i=0}^{50} \S \Bigl ( (2i - 1) (2i - 3) \dots (2i - 99) \Bigr )$$</p>
<ol style="list-style-type: lower-alpha;">
<li>$0$</li>
<li>$-1$</li>
<li>$+1$</li>
<li>$25$</li>
<li>$50$</li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25438/tifr2013-a-15Wed, 04 Nov 2015 20:08:21 +0000TIFR2013-A-14
https://gateoverflow.in/25437/tifr2013-a-14
<p>An unbiased die is thrown $n$ times. The probability that the product of numbers would be even is</p>
<ol style="list-style-type: lower-alpha;">
<li>$1/(2n)$</li>
<li>$1/[(6n)!]$</li>
<li>$1 - 6^{-n}$</li>
<li>$6^{-n}$</li>
<li>None of the above.</li>
</ol>
Probabilityhttps://gateoverflow.in/25437/tifr2013-a-14Wed, 04 Nov 2015 19:54:17 +0000TIFR2013-A-13
https://gateoverflow.in/25435/tifr2013-a-13
<p>Doctors $A$ and $B$ perform surgery on patients in stages $III$ and $IV$ of a disease. Doctor $A$ has performed a $100$ surgeries (on $80$ stage $III$ and $20$ stage $IV$ patients) and $80$ out of her $100$ patients have survived ($78$ stage $III$ and $2$ stage $IV$ survivors). Doctor $B$ has also performed $100$ surgeries (on $50$ stage $III$ and $50$ stage $IV$ patients). Her success rate is $60/100$ ($49$ stage $III$ survivors and $11$ stage $IV$ survivors).$A$ patient has been advised that she is equally likely to be suffering from stage $III$ or stage $IV$ of this disease. Which doctor would you recommend to this patient and why?</p>
<ol style="list-style-type: lower-alpha;">
<li>Doctor $A$ since she has a higher success rate</li>
<li>Doctor $A$ since she specializes in stage III patients and the success of surgery in stage $IV$ patients is anyway too low</li>
<li>Doctor $B$ since she has performed more stage $IV$ surgeries</li>
<li>Doctor $B$ since she appears to be more successful</li>
<li>There is not enough data since the choice depends on the stage of the disease the patient is suffering from.</li>
</ol>
Probabilityhttps://gateoverflow.in/25435/tifr2013-a-13Wed, 04 Nov 2015 19:40:54 +0000TIFR2013-A-12
https://gateoverflow.in/25434/tifr2013-a-12
<p>Among numbers $1$ to $1000$ how many are divisible by $3$ or $7$?</p>
<ol style="list-style-type:lower-alpha">
<li>$333$</li>
<li>$142$</li>
<li>$475$</li>
<li>$428$</li>
<li>None of the above.</li>
</ol>Numerical Abilityhttps://gateoverflow.in/25434/tifr2013-a-12Wed, 04 Nov 2015 19:29:40 +0000TIFR2013-A-11
https://gateoverflow.in/25433/tifr2013-a-11
<p>Let there be a pack of $100$ cards numbered $1$ to $100$. The $i^{th}$ card states: "There are at most $i - 1$ true cards in this pack". Then how many cards of the pack contain TRUE statements?</p>
<ol style="list-style-type: lower-alpha;">
<li>0</li>
<li>1</li>
<li>100</li>
<li>50</li>
<li>None of the above.</li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25433/tifr2013-a-11Wed, 04 Nov 2015 19:25:43 +0000TIFR2013-A-10
https://gateoverflow.in/25432/tifr2013-a-10
<p>Three men and three rakhsasas arrive together at a ferry crossing to find a boat with an oar, but no boatman. The boat can carry one or at the most two persons, for example, one man and one rakhsasas, and each man or rakhsasas can row. But if at any time, on any bank, (including those who maybe are in the boat as it touches the bank) rakhsasas outnumber men, the former will eat up the latter. If all have to go to the other side without any mishap, what is the minimum number of times that the boat must cross the river?</p>
<ol style="list-style-type:lower-alpha">
<li>7</li>
<li>9</li>
<li>11</li>
<li>13</li>
<li>15</li>
</ol>Numerical Abilityhttps://gateoverflow.in/25432/tifr2013-a-10Wed, 04 Nov 2015 19:17:54 +0000TIFR2013-A-9
https://gateoverflow.in/25431/tifr2013-a-9
<p>There are $n$ kingdoms and $2n$ champions. Each kingdom gets $2$ champions. The number of ways in which this can be done is:</p>
<ol style="list-style-type: upper-alpha;">
<li>$\frac{\left ( 2n \right )!}{2^{n}}$</li>
<li>$\frac{\left ( 2n \right )!}{n!}$</li>
<li>$\frac{\left ( 2n \right )!}{2^{n} . n!}$</li>
<li>$\frac{n!}{2}$</li>
<li>None of the above.</li>
</ol>Combinatoryhttps://gateoverflow.in/25431/tifr2013-a-9Wed, 04 Nov 2015 19:01:20 +0000TIFR2013-A-8
https://gateoverflow.in/25430/tifr2013-a-8
<p>Find the sum of the infinite series</p>
<p> $\frac{1}{1\times 3 \times 5} + \frac{1}{3\times 5\times 7} + \frac{1}{5\times 7 \times 9} + \frac{1}{7\times 9 \times 11} + ......$</p>
<ol style="list-style-type: lower-alpha;">
<li>$\infty $</li>
<li>$\frac{1}{2}$</li>
<li>$\frac{1}{6}$</li>
<li>$\frac{1}{12}$</li>
<li>$\frac{1}{14}$</li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25430/tifr2013-a-8Wed, 04 Nov 2015 18:53:18 +0000TIFR2013-A-7
https://gateoverflow.in/25429/tifr2013-a-7
<p>For any complex number $z$, $arg$ $z$ defines its phase, chosen to be in the interval $0\leq arg z < 360^{∘}$. If $z_{1}, z_{2}$ and $z_{3}$ are three complex numbers with the same modulus but different phases ($arg z_{3} < arg z_{2} < arg z_{1} < 180^{∘}$), then the quantity</p>
<p>$\frac{arg \left(z_{1}/z_{2}\right)}{arg \left[(z_{1}-z_{3})/(z_{2}-z_{3})\right]}$</p>
<p>is a constant, and has the value</p>
<ol style="list-style-type: lower-alpha;">
<li>2</li>
<li>$\frac{1}{3}$</li>
<li>1</li>
<li>3</li>
<li>$\frac{1}{2}$</li>
</ol>
<p> </p>
Numerical Abilityhttps://gateoverflow.in/25429/tifr2013-a-7Wed, 04 Nov 2015 18:33:18 +0000TIFR2013-A-6
https://gateoverflow.in/25390/tifr2013-a-6
<p>You are lost in the National park of Kabrastan. The park population consists of tourists and Kabrastanis. Tourists comprise two-thirds of the population the park, and give a correct answer to requests for directions with probability $3/4$. The air of Kabrastan has an amnesaic quality however, and so the answers to repeated questions to tourists are independent, even if the question and the person are the same. If u ask a Kabrastani for directions, the answer is always wrong.</p>
<p>Suppose you ask a randomly chosen passer by whether the exit from the park is East or West. The answer is East. You then ask the same person again, and the reply is again East. What is the probability of East being correct?</p>
<ol style="list-style-type: lower-alpha;">
<li>$1/4$</li>
<li>$1/3$</li>
<li>$1/2$</li>
<li>$2/3$</li>
<li>$3/4$</li>
</ol>
Probabilityhttps://gateoverflow.in/25390/tifr2013-a-6Wed, 04 Nov 2015 13:52:44 +0000TIFR2013-A-5
https://gateoverflow.in/25387/tifr2013-a-5
<p>The late painter Maqbool Fida Husain once coloured the surface of a huge hollow steel sphere, of radius $1$ metre, using just two colours, Red and Blue. As was his style however, both the red and blue areas were a bunch of highly irregular disconnected regions. The late sculptor Ramkinkar Baij then tried to fit in a cube inside the sphere, the eight vertices of the cube touching only red coloured parts of the surface of the sphere. Assume $\pi=3.14$ for solving this problem. Which of the following is true?</p>
<ol style="list-style-type: lower-alpha;">
<li>Baij is bound to succeed if the area of the red part is $10 sq. metres$;</li>
<li>Baij is bound to fail if the area of the red part is $10 sq. metres$;</li>
<li>Baij is bound to fail if the area of the red part is $11 sq. metres$;</li>
<li>Baij is bound to succeed if the area of the red part is $11 sq. metres$;</li>
<li>None of the above.</li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25387/tifr2013-a-5Wed, 04 Nov 2015 13:38:41 +0000TIFR2013-A-4
https://gateoverflow.in/25386/tifr2013-a-4
<p>A biased coin is tossed repeatedly. Assume that the outcomes of different tosses are independent and probability of heads is $2/3$ in each toss. What is the probability of obtaining an even number of heads in $5$ tosses, zero being treated as an even number?</p>
<ol style="list-style-type: lower-alpha;">
<li>$121/243$</li>
<li>$122/243$</li>
<li>$124/243$</li>
<li>$125/243$</li>
<li>$128/243$</li>
</ol>
Probabilityhttps://gateoverflow.in/25386/tifr2013-a-4Wed, 04 Nov 2015 13:21:58 +0000TIFR2013-A-3
https://gateoverflow.in/25384/tifr2013-a-3
<p>Three candidates, Amar, Birendra and Chanchal stand for the local election. Opinion polls are conducted and show that fraction $a$ of the voters prefer Amar to Birendra, fraction $b$ prefer Birendra to Chanchal and fraction $c$ prefer Chanchal to Amar. Which of the following is impossible?</p>
<ol style="list-style-type:lower-alpha">
<li>$(a, b, c) = (0.51, 0.51, 0.51);$</li>
<li>$(a, b, c) =(0.61, 0.71, 0.67);$</li>
<li>$(a, b, c) = (0.68, 0.68, 0.68);$</li>
<li>$(a, b, c) = (0.49, 0.49, 0.49);$</li>
<li>None of the above.</li>
</ol>Mathematical Logichttps://gateoverflow.in/25384/tifr2013-a-3Wed, 04 Nov 2015 13:09:25 +0000TIFR2013-A-2
https://gateoverflow.in/25383/tifr2013-a-2
<p>Consider the following two types of elections to determine which of two parties $A$ and $B$ forms the next government in the 2014 Indian elections. Assume for simplicity an Indian population of size $545545 (=545 * 1001)$. There are only two parties $A$ and $B$ and every citizen votes.</p>
<p>TYPE C: The country is divided into $545$ constituencies and each constituency has $1001$ voters. Elections are held for each constituency and a party is said to win a constituency if it receive a majority of the vote in that constituency. The party that wins the most constituencies forms the next government.</p>
<p>TYPE P: There are no constituencies in this model. Elections are held throughout the country and the party that wins the most votes (among $545545$ voters forms the government.</p>
<p>Which of the following is true?</p>
<ol style="list-style-type: lower-alpha;">
<li>If the party forms the govt. by election TYPE C winning at least two-third of the constituencies, then it will also forms the govt. by election TYPE P.</li>
<li>If a party forms govt. by election TYPE C, then it will also form the govt. by election TYPE P.</li>
<li>If a party forms govt. by election TYPE P, then it will also form the govt. by election TYPE C.</li>
<li>All of the above</li>
<li>None of the above.</li>
</ol>
Numerical Abilityhttps://gateoverflow.in/25383/tifr2013-a-2Wed, 04 Nov 2015 12:52:09 +0000TIFR2013-A-1
https://gateoverflow.in/25382/tifr2013-a-1
<p>An infinite two-dimensional pattern is indicated below.</p>
<p><img alt="" height="225" src="https://gateoverflow.in/?qa=blob&qa_blobid=10928851139002106137" width="543"></p>
<p>The smallest closed figure made by the lines is called a unit triangle. Within every unit triangle, there is a mouse. At every vertex there is a laddoo. What is the average number of laddoos per mouse?</p>
<ol style="list-style-type:lower-alpha">
<li>3</li>
<li>2</li>
<li>1</li>
<li>$\frac{1}{2}$</li>
<li>$\frac{1}{3}$</li>
</ol>Numerical Abilityhttps://gateoverflow.in/25382/tifr2013-a-1Wed, 04 Nov 2015 12:36:17 +0000