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Recent questions and answers in Discrete Mathematics
+5
votes
5
answers
1
GATE201938
Let $G$ be any connected, weighted, undirected graph. $G$ has a unique minimum spanning tree, if no two edges of $G$ have the same weight. $G$ has a unique minimum spanning tree, if, for every cut of $G$, there is a unique minimumweight edge crossing the cut. Which of the following statements is/are TRUE? I only II only Both I and II Neither I nor II
answered
17 hours
ago
in
Graph Theory
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V0id
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15
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3k
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gate2019
engineeringmathematics
discretemathematics
graphtheory
graphconnectivity
+54
votes
9
answers
2
GATE2015139
Consider the operations $\textit{f (X, Y, Z) = X'YZ + XY' + Y'Z'}$ and $\textit{g (X, Y, Z) = X'YZ + X'YZ' + XY}$ Which one of the following is correct? Both $\left\{\textit{f} \right\}$ and $\left\{ \textit{g}\right\}$ are ... Only $\left\{ \textit{g}\right\}$ is functionally complete Neither $\left\{ \textit{f}\right\}$ nor $\left\{\textit{g}\right\}$ is functionally complete
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1 day
ago
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Set Theory & Algebra
by
Kuldeep Pal
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1.5k
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7.6k
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gate20151
settheory&algebra
functions
difficult
+23
votes
7
answers
3
GATE200628
A logical binary relation $\odot$ ... $(\sim A\odot B)$ $\sim(A \odot \sim B)$ $\sim(\sim A\odot\sim B)$ $\sim(\sim A\odot B)$
answered
1 day
ago
in
Set Theory & Algebra
by
haralk10
(
23
points)

1.5k
views
gate2006
settheory&algebra
binaryoperation
+6
votes
3
answers
4
ISRO200729
The set of all Equivalence Classes of a set A of Cardinality C is of cardinality $2^c$ have the same cardinality as A forms a partition of A is of cardinality $C^2$
answered
2 days
ago
in
Set Theory & Algebra
by
JashanArora
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1.8k
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2.4k
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isro2007
settheory&algebra
equivalenceclasses
+8
votes
5
answers
5
TIFR2018A6
What is the minimum number of students needed in a class to guarantee that there are at least $6$ students whose birthdays fall in the same month ? $6$ $23$ $61$ $72$ $91$
answered
3 days
ago
in
Combinatory
by
`JEET
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13.1k
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419
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tifr2018
pigeonholeprinciple
permutationandcombination
+1
vote
2
answers
6
#set theory #groups
Consider the set H of all 3 × 3 matrices of the type: $\begin{bmatrix} a&f&e\\ 0&b&d\\ 0&0&c\\ \end{bmatrix}$ where a, b, c, d, e and f are real numbers and $abc ≠ 0$. Under the matrix multiplication operation, the set H is: (a) a group (b) a monoid but not a group (c) a semigroup but not a monoid (d) neither a group nor a semigroup
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3 days
ago
in
Set Theory & Algebra
by
smsubham
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9.7k
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75
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settheory&algebra
groups
matrices
+34
votes
9
answers
7
GATE19941.6, ISRO200829
The number of distinct simple graphs with up to three nodes is $15$ $10$ $7$ $9$
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3 days
ago
in
Graph Theory
by
JashanArora
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1.8k
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9.7k
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gate1994
graphtheory
permutationandcombination
normal
isro2008
counting
+24
votes
13
answers
8
GATE201846
The number of possible minheaps containing each value from $\{1,2,3,4,5,6,7\}$ exactly once is _______
answered
4 days
ago
in
Combinatory
by
Praveenk99
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49
points)

8.4k
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gate2018
permutationandcombination
numericalanswers
+37
votes
11
answers
9
GATE2016126
The coefficient of $x^{12}$ in $\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}$ is ___________.
answered
4 days
ago
in
Combinatory
by
pritishc
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669
points)

9k
views
gate20161
permutationandcombination
generatingfunctions
normal
numericalanswers
+28
votes
5
answers
10
GATE20006
Let $S$ be a set of $n$ elements $\left\{1, 2,....., n\right\}$ and $G$ a graph with 2$^{n}$ vertices, each vertex corresponding to a distinct subset of $S$. Two vertices are adjacent iff the symmetric difference of the corresponding sets has exactly $2$ ... $G$ has the same degree. What is the degree of a vertex in $G$? How many connected components does $G$ have?
answered
4 days
ago
in
Set Theory & Algebra
by
Vimal Patel
(
137
points)

1.8k
views
gate2000
settheory&algebra
normal
descriptive
sets
+1
vote
3
answers
11
GATE20014
Consider the function $h: N \times N \rightarrow N$ so that $h(a,b) = (2a +1)2^b  1$, where $N=\{0,1,2,3,\dots\}$ is the set of natural numbers. Prove that the function $h$ is an injection (oneone). Prove that it is also a Surjection (onto)
answered
4 days
ago
in
Set Theory & Algebra
by
Verma Ashish
Boss
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11.8k
points)

455
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gate2001
functions
settheory&algebra
normal
descriptive
+1
vote
1
answer
12
TIFR2019A6
A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is said to be $\textit{convex}$ if for all $x,y \in \mathbb{R}$ and $\lambda$ such that $0 \leq \lambda \leq1,$ $f(\lambda x+ (1\lambda)y) \leq \lambda f (x) + (1\lambda) f(y)$. Let $f:$\mathbb{R}$ $→$ $\mathbb ... $p,q$ and $r$ must be convex? Only $p$ Only $q$ Only $r$ Only $p$ and $r$ Only $q$ and $r$
answered
5 days
ago
in
Set Theory & Algebra
by
Satbir
Boss
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21.5k
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344
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tifr2019
engineeringmathematics
discretemathematics
settheory&algebra
functions
convexsetsfunctions
nongate
+7
votes
7
answers
13
GATE201935
Consider the first order predicate formula $\varphi$: $\forall x [ ( \forall z \: z \mid x \Rightarrow (( z=x) \vee (z=1))) \rightarrow \exists w ( w > x) \wedge (\forall z \: z \mid w \Rightarrow ((w=z) \vee (z=1)))]$ Here $a \mid b$ ... Set of all positive integers $S3:$ Set of all integers Which of the above sets satisfy $\varphi$? S1 and S2 S1 and S3 S2 and S3 S1, S2 and S3
answered
5 days
ago
in
Mathematical Logic
by
JashanArora
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1.8k
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5.1k
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gate2019
engineeringmathematics
discretemathematics
mathematicallogic
firstorderlogic
+5
votes
9
answers
14
GATE201912
Let $G$ be an undirected complete graph on $n$ vertices, where $n > 2$. Then, the number of different Hamiltonian cycles in $G$ is equal to $n!$ $(n1)!$ $1$ $\frac{(n1)!}{2}$
answered
6 days
ago
in
Graph Theory
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JashanArora
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3.3k
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gate2019
engineeringmathematics
discretemathematics
graphtheory
graphconnectivity
0
votes
2
answers
15
self doubt
What is the general formula for number of simple graph having n unlabelled vertices ??
answered
Nov 30
in
Graph Theory
by
Muneendra1337
(
11
points)

60
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simplegraph
0
votes
1
answer
16
group
if (G,*) is a cyclic group of order 97 , then number of generator of G is equal to ___
answered
Nov 29
in
Set Theory & Algebra
by
GoalSet1
(
203
points)

123
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groups
discretemathematics
settheory&algebra
+1
vote
2
answers
17
ISI2014DCG71
Five letters $A, B, C, D$ and $E$ are arranged so that $A$ and $C$ are always adjacent to each other and $B$ and $E$ are never adjacent to each other. The total number of such arrangements is $24$ $16$ $12$ $32$
answered
Nov 28
in
Combinatory
by
noob_coder
Junior
(
657
points)

26
views
isi2014dcg
permutationandcombination
arrangements
circularpermutation
+2
votes
1
answer
18
Kenneth Rosen Edition 6th Exercise 6.4 Question 13 (Page No. 440)
Use Generating function to determine,the number of different ways $10$ identical balloons can be given to four children if each child receives atleast $2$ ballons? Ans given $(x^{2}+x^{3}+.........................)^{4}$ But as there is a upper ... Which one is correct? plz confirm
answered
Nov 26
in
Combinatory
by
Kushagra गुप्ता
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1.9k
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198
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kennethrosen
discretemathematics
generatingfunctions
+36
votes
7
answers
19
GATE2008IT21
Which of the following first order formulae is logically valid? Here $\alpha(x)$ is a first order formula with $x$ as a free variable, and $\beta$ ... $[(\forall x, \alpha(x)) \rightarrow \beta] \rightarrow [\forall x, \alpha(x) \rightarrow \beta]$
answered
Nov 26
in
Mathematical Logic
by
JashanArora
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4.2k
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gate2008it
firstorderlogic
normal
+28
votes
3
answers
20
GATE20021.25, ISRO200830, ISRO20166
The maximum number of edges in a nnode undirected graph without self loops is $n^2$ $\frac{n(n1)}{2}$ $n1$ $\frac{(n+1)(n)}{2}$
answered
Nov 23
in
Graph Theory
by
Shailendra_
Junior
(
555
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5.5k
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gate2002
graphtheory
easy
isro2008
isro2016
graphconnectivity
+1
vote
2
answers
21
ISI2016PCBA3
A bit string is called legitimate if it contains no consecutive zeros $, e.g., 0101110$ is legitimate, where as $10100111$ is not. Let $a_n$ denote the number of legitimate bit strings of length $n$. Define $a_0=1$. Derive a recurrence relation for $a_n ( i.e.,$ express $a_n$ in terms of the preceding $a_i's).$
answered
Nov 22
in
Combinatory
by
chirudeepnamini
Active
(
3.1k
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29
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isi2016pcba
permutationandcombination
recurrencerelations
nongate
descriptive
+2
votes
3
answers
22
ISI2014DCG18
$^nC_0+2^nC_1+3^nC_2+\cdots+(n+1)^nC_n$ equals $2^n+n2^{n1}$ $2^nn2^{n1}$ $2^n$ none of these
answered
Nov 20
in
Combinatory
by
chirudeepnamini
Active
(
3.1k
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43
views
isi2014dcg
permutationandcombination
binomialtheorem
+1
vote
1
answer
23
ISI2018DCG14
In a room there are $8$ men, numbered $1,2, \dots ,8$. These men have to be divided into $4$ teams in such a way that every team has exactly $2$ ... of such $4$team combinations is $\frac{8!}{2^4}$ $\frac{8!}{2^4(4!)}$ $\frac{8!}{4!}$ $\frac{8!}{(4!)^2}$
answered
Nov 19
in
Combinatory
by
Yash4444
Junior
(
731
points)

17
views
isi2018dcg
permutationandcombination
+1
vote
1
answer
24
ISI2016DCG21
The value of the term independent of $x$ in the expansion of $(1x)^{2}(x+\frac{1}{x})^{7}$ is $70$ $70$ $35$ None of these
answered
Nov 18
in
Combinatory
by
Verma Ashish
Boss
(
11.8k
points)

18
views
isi2016dcg
permutationandcombination
binomialtheorem
+15
votes
9
answers
25
GATE201830
Let $G$ be a simple undirected graph. Let $T_D$ be a depth first search tree of $G$. Let $T_B$ be a breadth first search tree of $G$. Consider the following statements. No edge of $G$ is a cross edge with respect to $T_D$. (A cross edge in $G$ is between ... then $\mid ij \mid =1$. Which of the statements above must necessarily be true? I only II only Both I and II Neither I nor II
answered
Nov 17
in
Graph Theory
by
Shailendra_
Junior
(
555
points)

5.5k
views
gate2018
graphtheory
graphsearch
normal
+3
votes
1
answer
26
Mock DFS Q
Consider DFS over undirected graph with 4 vertices <A;B;C;D>. The discovery and finishing times of them in the order A to D are given. Select the option from following showing more than one connected components: 1) <(1,6), (2,5), (3,4), (8,10)> 2) <(6,7), (2,5), (3,4), (8,9)> 3) <(4,5), (2,8), (1,7), (3,6)> 4) <(7,8), (1,2), (5,6), (3,4)>
answered
Nov 15
in
Set Theory & Algebra
by
vk82942349
(
11
points)

135
views
+1
vote
1
answer
27
Difference between DAG and Multistage graph
I have trouble understanding the difference between DAG and Multistage graph. I know what each of them is But I think that a multistage graph is also a DAG. Are multistage graphs a special kind of DAG?
answered
Nov 15
in
Graph Theory
by
mohithjagalmohanan
(
19
points)

75
views
graphtheory
graphalgorithms
graphconnectivity
multistagegraph
directedacyclicgraph
dag
+6
votes
3
answers
28
GATE20195
Let $U = \{1, 2, \dots , n\}$ Let $A=\{(x, X) \mid x \in X, X \subseteq U \}$. Consider the following two statements on $\mid A \mid$. $\mid A \mid = n2^{n1}$ $\mid A \mid = \Sigma_{k=1}^{n} k \begin{pmatrix} n \\ k \end{pmatrix}$ Which of the above statements is/are TRUE? Only I Only II Both I and II Neither I nor II
answered
Nov 14
in
Combinatory
by
Optimus Prime
Junior
(
543
points)

2.7k
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gate2019
engineeringmathematics
discretemathematics
permutationandcombination
+1
vote
2
answers
29
ISI2018DCG7
You are given three sets $A,B,C$ in such a way that the set $B \cap C$ consists of $8$ elements, the set $A\cap B$ consists of $7$ elements, and the set $C\cap A$ consists of $7$ elements. The minimum number of elements in the set $A\cup B\cup C$ is $8$ $14$ $15$ $22$
answered
Nov 14
in
Set Theory & Algebra
by
chirudeepnamini
Active
(
3.1k
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34
views
isi2018dcg
settheory
sets
+2
votes
2
answers
30
CMI2018A4
Let $G=(V, E)$ be an undirected simple graph, and $s$ be a designated vertex in $G.$ For each $v\in V,$ let $d(v)$ be the length of a shortest path between $s$ and $v.$ For an edge $(u,v)$ in $G,$ what can not be the value of $d(u)d(v)?$ $2$ $1$ $0$ $1$
answered
Nov 12
in
Graph Theory
by
SPodder
(
11
points)

25
views
cmi2018
graphtheory
undirectedgraph
simplegraph
shortestpath
+32
votes
7
answers
31
GATE20021.8
"If $X$ then $Y$ unless $Z$" is represented by which of the following formulas in prepositional logic? ("$\neg$" is negation, "$\land$" is conjunction, and "$\rightarrow$" is implication) $(X\land \neg Z) \rightarrow Y$ $(X \land Y) \rightarrow \neg Z$ $X \rightarrow(Y\land \neg Z)$ $(X \rightarrow Y)\land \neg Z$
answered
Nov 11
in
Mathematical Logic
by
Navneet Singh Tomar
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621
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4.4k
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gate2002
mathematicallogic
normal
propositionallogic
0
votes
2
answers
32
MadeEasy Full Length Test 2019: Combinatory  Permutations And Combinations
The number of ways 5 letter be put in 3 letter boxes A,B,C. If letter box A must contain at least 2 letters.
answered
Nov 11
in
Combinatory
by
SaurabhKatkar
(
151
points)

207
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discretemathematics
permutationandcombination
madeeasytestseries2019
madeeasytestseries
0
votes
2
answers
33
solution of rosen
can anyone has the soft copy of kenneth h rosen solutions?
answered
Nov 11
in
Mathematical Logic
by
Ram Swaroop
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+2
votes
2
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34
Discrete Mathematics Thegatebook
how many positive integers between 50 and 100, (a) divisible by 7 (b) divisible by 11 (c) divisible by 7 and 11?
answered
Nov 10
in
Combinatory
by
Ram Swaroop
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303
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inclusionexclusion
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3
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35
Rosen chapter 5.5 question 50
How many ways are there to distribute 5 distinguishable objects into three indistinguishable boxes?
answered
Nov 9
in
Mathematical Logic
by
SaurabhKatkar
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151
points)

242
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0
votes
2
answers
36
Rosen 7e Exercise8.1 Question no10 Page no511
Find a recurrence relation for the number of bit strings of length n that contain the string 01.
answered
Nov 9
in
Combinatory
by
Sandeep shahu
(
71
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59
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kennethrosen
discretemathematics
permutationandcombination
#recurrencerelations
recurrence
+1
vote
1
answer
37
CMI2018A5
How many paths are there in the plane from $(0,0)$ to $(m,n)\in \mathbb{N}\times \mathbb{N},$ if the possible steps from $(i,j)$ are either $(i+1,j)$ or $(i,j+1)?$ $\binom{2m}{n}$ $\binom{m}{n}$ $\binom{m+n}{n}$ $m^{n}$
answered
Nov 9
in
Combinatory
by
`JEET
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13.1k
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37
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cmi2018
permutationandcombination
paths
+1
vote
1
answer
38
CMI2016A9
ScamTel has won a state government contract to connect $17$ cities by highspeed fibre optic links. Each link will connect a pair of cities so that the entire network is connectedthere is a path from each city to every other city. The contract requires the network to remain ... $34$ $32$ $17$ $16$
answered
Nov 9
in
Graph Theory
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`JEET
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13.1k
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54
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cmi2016
graphtheory
graphconnectivity
path
+32
votes
5
answers
39
GATE200624
Given a set of elements N = {1, 2, ..., n} and two arbitrary subsets A⊆N and B⊆N, how many of the n! permutations $\pi$ from N to N satisfy min($\pi$(A)) = min($\pi$(B)), where min(S) is the smallest integer in the set of integers S, and $\pi$(S) is the set of integers obtained by applying ... $n! \frac{A ∩ B}{A ∪ B}$ $\dfrac{A ∩ B^2}{^n \mathrm{C}_{A ∪ B}}$
answered
Nov 7
in
Set Theory & Algebra
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Sourajit25
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gate2006
settheory&algebra
normal
sets
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1
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40
#ACE ACADEMY BOOKLET QUESTION
The solution of $\sqrt{a_n} – 2\sqrt{a_{n1}} + \sqrt{a_{n2}} = 0$ where $a_0 = 1$ and $a_1 = 2$ is ${\Big[\frac{2^{n+1} + (1)^n}{3}\Big]}^2$ $(n+1)^2$ $(n1)^3$ $(n1)^2$
answered
Nov 5
in
Combinatory
by
techbd123
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discretemathematics
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