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Let $'a'$ be an element in a group such that $ord(a) = 2020.$ Find the order of $43?$
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Let $G$ be a group with order $45$, and $H$ a non-abelian subgroup of $G$. Assuming $H \neq G$,what is the smallest possible order of $H$?
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Let $G=\langle{a\rangle}$. Find the smallest subgroup of $G$ containing $a^{2020}$ and $a^{1719}$ is? A.$a^{4}$ B.$a^{1719}$ C.$a^{2020}$ D.$a^{1}$
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GO 2021 Discrete Mathematics 3: 4
The number of complements of element $'g'$ is $\_\_\_\_\_$ \begin{tikzpicture} \tikzstyle{every node}=[rectangle,draw,scale=1,color=black!70!red,transform shape] \node (a) at (0,0) {a}; \node (b) at (0,3) {b}; \node (c) at (-2,1.5) {c}; \node (d) at (-1.5,1.5) { ... draw[-] (b) -- (h); \draw[-] (b) -- (i); \draw[-] (b) -- (j); \draw[-] (b) -- (k); \end{tikzpicture}
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GO 2021 Discrete Mathematics 3: 5
Which of the following is not correct? A.$(\mathbb{R^{\ast}},\times),$ the set of real numbers excluding $0$, under multiplication operation is a group. B.$(\mathbb{Q},\times),$ ... $\{0, 1, \ldots, n-1\}$, under addition modulo $n$ operation is a group.
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GO 2021 Discrete Mathematics 3: 6
What is the solution to the recurrence relation $a_{n} = a_{n-1} + n$ with $a_{0} = 0$? A.$a_{n} = \dfrac{n(n-1)}{2} $ B.$a_{n} = \dfrac{(n-1)(n+1)}{2} $ C.$a_{n} = \dfrac{(n-1)(n+1)}{4} $ D.$a_{n} = \dfrac{n(n+1)}{2} $
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GO 2021 Discrete Mathematics 3: 7
Consider a prime number $p$ and a positive integer $a$ such that $p$ divides $a$. The remainder when $a^{p-1}$ is divided by $p$ is A. $1$ B. $0$ C. $p-1$ D. Cannot be determined
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GO 2021 Discrete Mathematics 3: 8
Which of the following is/are correct? ("/" is the "divides" relation) $S1:\left[\{1,2,3,6,9,18,36,72\},\:/\right]$ is a lattice. $S2:\left[\{1,3,9,27,81,243\},\:/\right]$ is a total order. A.Only $S1$ B.Only $S2$ C.Both $S1$ and $S2$ D.None of the above
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GO 2021 Discrete Mathematics 3: 9
Which of the following is/are correct? A relation that is reflexive, anti-symmetric, and transitive is called a partial order. When every two elements in a set are comparable, the relation is called a total ordering (linear ordering). The poset $(Z,\leq)$ is not totally ordered ... of a set. A.I and II only B.I and III only C.I, II and III only D.I, II and IV only
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GO 2021 Discrete Mathematics 3: 10
Let $(G,\ast)$ be a group of order $P$, where $P$ is a prime number, then the number of subgroups of $G$ is? A.$0$ B.$P^{2}$ C.$P^{2}-2$ D.$2$
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GO 2021 Discrete Mathematics 3: 11
The value of $x_5$ for the recurrence relation defined by $x_n = 5x_{n-1} + 3$ with initial condition $x_1 = 3$ is $\_\_\_\_\_\_\_\_\_$
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GO 2021 Discrete Mathematics 3: 12
The following is the Hasse diagram of the poset $\left[\{a,b,c,d,e,f,g,h,i,j\},\preceq\right]$ %tikz image \begin{tikzpicture} \tikzstyle{every node}=[rectangle,draw,scale=1,color=blue!80!black,transform shape] \node ... and distributive lattice. C.not a complemented lattice and not a distributive lattice. D.a distributive lattice but not a Boolean algebra.
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GO 2021 Discrete Mathematics 3: 13
The following is the Hasse diagram of the poset $\left[\{1,2,3,4,6,7,12,14,21,28,42,84\},\:\mid\right],$ where $\mid $ is the "divides" relation. \begin{tikzpicture} \tikzstyle{every node}=[rectangle,draw,scale=1,color=black!70!red,transform shape] ... $\gamma$ respectively, then the value of $2\alpha + 3 \beta + 4\gamma $ is $\_\_\_\_$
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GO 2021 Discrete Mathematics 3: 14
The following Hasse diagram describes a \begin{tikzpicture}[scale=1] \node (20) at (-3,2) {$s_{2,0}$}; \node (21) at (0,2) {$s_{2,1}$}; \node (22) at (3,2) {$s_{2,2}$}; \node (10) at (-3,0) {$s_{1,0}$ ... -- (22) -- (13) --(00); \draw (10) -- (21) -- (13); \end{tikzpicture} A.Join semi lattice B.Meet semi lattice C.Lattice D.Not a lattice
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GO 2021 Discrete Mathematics 3: 15
Which of the following is not correct regarding a bounded lattice? A.$[\:\{1,2,3,6,9,18\},\: \mid]$ is a bounded lattice. B.$[\:I,\:\leq]$ is not a bounded lattice, where $I$ is the set of integers. C.$[\:[0,1],\:\leq]$ is bounded lattice. D.$[\:(0,1),\:\leq]$ is bounded lattice.
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GO 2021 Discrete Mathematics 3: 16
The solution to the recurrence relation $a_{n}=a_{n-1}+2^{n}$ with $a_{0}=1$ is $\_\_\_\_\_$ A.$a_{n} = 2^{n} - 1 $ B.$a_{n} = 2^{n+1} - 2 $ C.$a_{n} = 2^{n+1} - 1 $ D.$a_{n} = 2^{n-1} - 1 $
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GO 2021 Discrete Mathematics 3: 17
Which of the following is NOT correct? A. A group $G$ has at least $2$ subgroups. B.Every subgroup of a cyclic group is cyclic. C.A cyclic group is not abelian. D.Any subgroup of an abelian group is also abelian.
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GO 2021 Discrete Mathematics 3: 18
The explicit formula for the sequence defined by the recurrence relation $x_n = 2x_{n-1} + 15x_{n-2} $ subject to the initial conditions, $x_1 = 2, x_2 = 4$ is. $(-3)^nC_1 + 5^nC_2$ $\frac{(-1)^{n+1}3^n + 5^n}{4}$ $\frac{(-1)^{n}3^n - 5^n}{4}$ $\frac{3^n -5^n}{4}$
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GO 2021 Discrete Mathematics 3: 19
Let $G$ be a finite group on $243$ elements. The size of the largest possible proper subgroup of $G$ is $\_\_\_\_\_$
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GO 2021 Discrete Mathematics 3: 20
The remainder when $99^{96}$ is divided by $97$ is
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GO 2021 Discrete Mathematics 3: 21
How many non-isomorphic abelian groups of order $32$ are there?
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GO 2021 Discrete Mathematics 3: 22
Let $G$ be a cyclic group of order $96$, then the number of subgroups of $G$ is $\_\_\_\_\_$
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GO 2021 Discrete Mathematics 3: 23
If G is cyclic group of order $48$, then how many element of order $8$ are in $G$?
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GO 2021 Discrete Mathematics 3: 24
How many ways can we fill the following table so that the binary operation $\ast$ is commutative? \begin{center} \begin{tabular}{| c | c| c | c | c |} \hline \textbf{$\ast$} & \textbf{a} &\textbf{b} &\textbf{c} & \textbf{d} \hline \textbf{a ... d & c \hline \textbf{d} & & b & & d \hline \end{tabular} \end{center} A.2 B.3 C.1 D.4
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GO 2021 Discrete Mathematics 3: 25
The remainder when $60^{61}$ is divided by $61$ is
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GO 2021 Discrete Mathematics 3: 26
If $G=\{1,2,3,4,5,6\}$ is a group with respect to $\otimes _{7}$ (multiplication modulo 'm'), which of the following is not true? A.The inverse of $1$ is $1$ B.The inverse of $6$ is $3$ C.The inverse of $2$ is $4$ D.The inverse of $3$ is $5$
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GO 2021 Discrete Mathematics 3: 27
Suppose $G$ is a group that has exactly $48$ distinct elements of order $5$. How many distinct subgroups of order $5$ does $G$ have? A.$12$ B.$6$ C.$1$ D.$4$
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What is the solution to the recurrence relation $f_{n} = f_{n-1} + f_{n-2}$ with $f_{0} = 0,f_{1} = 1$? A.$f_{n} =\dfrac{1}{\sqrt{5}}\bigg(\dfrac{(1 + \sqrt{5})}{2}\bigg)^{n} - \dfrac{-1}{\sqrt{5}}\bigg(\dfrac{(1 - \sqrt{5})}{2}\bigg)^{n}$ ...
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GO 2021 Discrete Mathematics 3: 29
Calculate $a_{100} + a_{99}$ where $a_n$ is given by the recurrence relation $a_{n} = 2a_{n-1} - a_{n-2}$ with $a_{0} = 5,a_{1} = 4$?
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recurrence %-189 \textbf{solution:} it is linear homogeneous recurrence
and its characteristic equation is given by $t^{2} - 2t + 1 = 0$ $\implies t = \dfrac{2\pm \sqrt{4-4}}{2} = 1.$ the characteristic equation is having only one root $r = 1
$ and so as per the "single root theorem"
the sequence satisfies the explicit formula $a_n = c \times r^n + d\times n \times r^n
$ $\implies a_n = c + nd$ $a_0 = c = 5$ $a_1 = c+d = 4 \implies d = -1.$ so
$a_n = 5 - n .$ so
$a_{100} + a_{99} = 5-100 + 5 - 99 = 10 - 199 = -189$
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30
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GO 2021 Discrete Mathematics 3: 30
Number of maximal and minimal elements for the poset $[\:\: \{2,4,6,9,12,18,27,36,48,60,72\},\: \mid\:\: ],$ where $\mid$ is the "divides" relation is A.2,1 B.4,1 C.3,2 D.4,2
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GO 2021 Discrete Mathematics 2: 1
The minimum number of people that must be in a room to ensure that at least three were born on the same day of the week is $\_\_\_\_\_$
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GO 2021 Discrete Mathematics 2: 2
Consider a set $A$ with $6$ elements. Let $N_1$ denote the number of bijective functions from $A$ to $A$ and let $N_2$ denote the number of onto (surjective) functions from $A$ to $A.$ $N_2 - N_1 = \_\_\_\_\_$
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GO 2021 Discrete Mathematics 2: 3
Number of bit strings of length $10$ that do not end in “$111$” is $\_\_\_\_$
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GO 2021 Discrete Mathematics 2: 4
Let $G(x)= \frac{1}{(1-x)} =\sum_{0}^{\infty}f(i)x^i$ where $|x|< 1$. $f(i)$ is $1$ $i$ $2i$ $i+1$
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GO 2021 Discrete Mathematics 2: 5
The number of bit strings of length $6$ that do not contain “$1111$” as a substring is $\_\_\_\_$
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GO 2021 Discrete Mathematics 2: 6
The number of positive integers less than $1000$ which are divisible by $7$ but not by $11$ is $\_\_\_\_$
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GO 2021 Discrete Mathematics 2: 7
How many $10$ digit numbers greater than $19876543210$ have no two digits same? $8 \times 9!$ $9 \times 9!$ $7 \times 9!$ None of the above
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GO 2021 Discrete Mathematics 2: 8
How many bit strings of length $7$ contain more $0$’s than $1$’s? 93 84 64 none of these
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15 students took a quiz. The sum of their scores came out to be 100. Now consider the following two statements $S_1:$ All the scores are different. $S_2:$ There are two students having the same score. Which of the above statements is/are TRUE? Only Statement $S_1$ is true Only Statement $S_2$ is true Both $S_1$ and $S_2$ can be true Both $S_1$ and $S_2$ are false
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Let $S_1 = \{1,2,3,\ldots,100\}$ and $S_2 = \{a,b,c,\ldots,z\}.$ Number of one-one (injective) functions from $S_2 \to S_1$ when $f(m) = 13$ is $\_\_\_$ $^{100}P_{25}$ $^{99}C_{25}$ $^{99}P_{25}$ $1$
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