Recent questions and answers in Engineering Mathematics - GATE Overflow

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1 answer

1

K H Rosen Ex
15. a) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces are chosen? b) How many cards must be chosen from a standard deck of 52 cards to guarantee that at least two of the four aces and ... cards must be chosen from a standard deck of 52 cards to guarantee that there are at least two cards of each of two different kinds?

answered 1 day ago in Combinatory by NedIsakoff | 42 views

+27 votes

6 answers

2

GATE2009-22
For the composition table of a cyclic group shown below: ... $a,b$ are generators $b,c$ are generators $c,d$ are generators $d,a$ are generators

answered 3 days ago in Set Theory & Algebra by Kushagra गुप्ता | 2.6k views

+7 votes

5 answers

3

ISI2017-MMA-29
Suppose the rank of the matrix $\begin{pmatrix}1&1&2&2\\1&1&1&3\\a&b&b&1\end{pmatrix}$ is $2$ for some real numbers $a$ and $b$. Then $b$ equals $1$ $3$ $1/2$ $1/3$

answered 4 days ago in Linear Algebra by Shankhajit Roy | 673 views

+19 votes

5 answers

4

GATE2009-24
The binary operation $\Box$ ... of the following is equivalent to $P \vee Q$? $\neg Q \Box \neg P$ $P\Box \neg Q$ $\neg P\Box Q$ $\neg P\Box \neg Q$

answered 4 days ago in Mathematical Logic by abichandani | 2.6k views

+63 votes

6 answers

5

GATE2015-1-39
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

answered 4 days ago in Set Theory & Algebra by Kushagra गुप्ता | 9.6k views

+52 votes

6 answers

6

GATE2006-25
Let $S = \{1, 2, 3,\ldots, m\}, m >3.$ Let $X_1,\ldots,X_n$ be subsets of $S$ each of size $3.$ Define a function $f$ from $S$ to the set of natural numbers as, $f(i)$ is the number of sets $X_j$ that contain the element $i.$ That is $f(i)=\left | \left\{j \mid i\in X_j \right\} \right|$ then $ \sum_{i=1}^{m} f(i)$ is: $3m$ $3n$ $2m+1$ $2n+1$

answered 5 days ago in Set Theory & Algebra by Kushagra गुप्ता | 3.6k views

+4 votes

2 answers

7

TIFR2017-A-2
For vectors $x, \: y$ in $\mathbb{R}^n$, define the inner product $\langle x, y \rangle = \Sigma^n_{i=1} x_iy_i$, and the length of $x$ to be $\| x \| = \sqrt{\langle x, x \rangle}$. Let $a, \: b$ be two vectors in $\mathbb{R} ^n$ ... $a, \: b$? Choose from the following options. ii only i and ii iii only iv only iv and v

answered 5 days ago in Linear Algebra by arks | 340 views

0 votes

2 answers

8

GATE1992-15.b
Let $S$ be the set of all integers and let $n > 1$ be a fixed integer. Define for $a,b \in S, a R b$ iff $a-b$ is a multiple of $n$. Show that $R$ is an equivalence relation and find its equivalence classes for $n = 5$.

answered 5 days ago in Set Theory & Algebra by Kushagra गुप्ता | 590 views

+14 votes

4 answers

9

GATE2002-3
Let $A$ be a set of $n(>0)$ elements. Let $N_r$ be the number of binary relations on $A$ and let $N_f$ be the number of functions from $A$ to $A$ Give the expression for $N_r,$ in terms of $n.$ Give the expression for $N_f,$ terms of $n.$ Which is larger for all possible $n,N_r$ or $N_f$

answered 5 days ago in Set Theory & Algebra by Kushagra गुप्ता | 1.1k views

+16 votes

5 answers

10

GATE1995-1.19
Let $R$ be a symmetric and transitive relation on a set $A$. Then $R$ is reflexive and hence an equivalence relation $R$ is reflexive and hence a partial order $R$ is reflexive and hence not an equivalence relation None of the above

answered 5 days ago in Set Theory & Algebra by Kushagra गुप्ता | 3k views

+27 votes

4 answers

11

GATE1991-15,b
Consider the following first order formula: ... Does it have finite models? Is it satisfiable? If so, give a countable model for it.

answered 6 days ago in Mathematical Logic by Deepakk Poonia (Dee) | 2.2k views

+45 votes

5 answers

12

GATE2016-2-27
Which one of the following well-formed formulae in predicate calculus is NOT valid ? $(\forall _{x} p(x) \implies \forall _{x} q(x)) \implies (\exists _{x} \neg p(x) \vee \forall _{x} q(x))$ $(\exists x p(x) \vee \exists x q (x)) \implies \exists x (p(x) \vee q (x))$ ... $\forall x (p(x) \vee q(x)) \implies (\forall x p(x) \vee \forall x q(x))$

answered 6 days ago in Mathematical Logic by Kushagra गुप्ता | 7k views

+20 votes

3 answers

13

GATE1997-4.2
Let $A=(a_{ij})$ be an $n$-rowed square matrix and $I_{12}$ be the matrix obtained by interchanging the first and second rows of the $n$-rowed Identity matrix. Then $AI_{12}$ is such that its first Row is the same as its second row Row is the same as the second row of $A$ Column is the same as the second column of $A$ Row is all zero

answered Jul 2 in Linear Algebra by arks | 1.4k views

+48 votes

5 answers

14

GATE2017-1-31
Let $A$ be $n\times n$ real valued square symmetric matrix of rank 2 with $\sum_{i=1}^{n}\sum_{j=1}^{n}A^{2}_{ij} =$ 50. Consider the following statements. One eigenvalue must be in $\left [ -5,5 \right ]$ The eigenvalue with the largest ... greater than 5 Which of the above statements about eigenvalues of $A$ is/are necessarily CORRECT? Both I and II I only II only Neither I nor II

answered Jul 2 in Linear Algebra by arks | 8.3k views

+29 votes

9 answers

15

GATE2014-1-53
Which one of the following propositional logic formulas is TRUE when exactly two of $p,q$ and $r$ are TRUE? $(( p \leftrightarrow q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $( \sim (p \leftrightarrow q) \wedge r)\vee (p \wedge q \wedge \sim r)$ $( (p \to q) \wedge r) \vee (p \wedge q \wedge \sim r)$ $(\sim (p \leftrightarrow q) \wedge r) \wedge (p \wedge q \wedge \sim r) $

answered Jul 2 in Mathematical Logic by Kushagra गुप्ता | 4.1k views

0 votes

1 answer

16

ESE 2018
In a sample of 100 students, the mean of the marks (only integers) obtained by them in a test is 14 with its standard deviation of 2.5(marks obtained can be fitted with a normal distribution ).the percentage of students scoring 16 marks is a)36 b)23 c)12 d)10 (Area under standard normal curve between z=0 and z=0.6 is 0.2257 ; and between z=0 and z=1.0 is 0.3413)

answered Jul 2 in Probability by mohan123 | 442 views

+14 votes

4 answers

17

GATE2008-24
Let $P =\sum_{\substack{1\le i \le 2k \\ i\;odd}} i$ and $Q = \sum_{\substack{1 \le i \le 2k \\ i\;even}} i$, where $k$ is a positive integer. Then $P = Q - k$ $P = Q + k$ $P = Q$ $P = Q + 2k$

answered Jul 2 in Combinatory by vivekgatecs2020 | 1.4k views

+24 votes

3 answers

18

GATE2008-IT-25
In how many ways can $b$ blue balls and $r$ red balls be distributed in $n$ distinct boxes? $\frac{(n+b-1)!\,(n+r-1)!}{(n-1)!\,b!\,(n-1)!\,r!}$ $\frac{(n+(b+r)-1)!}{(n-1)!\,(n-1)!\,(b+r)!}$ $\frac{n!}{b!\,r!}$ $\frac{(n + (b + r) - 1)!} {n!\,(b + r - 1)}$

answered Jul 1 in Combinatory by vivekgatecs2020 | 3k views

+30 votes

6 answers

19

GATE2007-84
Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at $(i,j)$ then it can move to either $(i + 1, j)$ or $(i,j + 1)$. How many distinct paths are there for the robot to reach the point $(10,10)$ starting from the initial position $(0,0)$? $^{20}\mathrm{C}_{10}$ $2^{20}$ $2^{10}$ None of the above

answered Jul 1 in Combinatory by vivekgatecs2020 | 4.4k views

+42 votes

6 answers

20

GATE2004-75
Mala has the colouring book in which each English letter is drawn two times. She wants to paint each of these $52$ prints with one of $k$ colours, such that the colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of $k$ that satisfies this requirement? $9$ $8$ $7$ $6$

answered Jul 1 in Combinatory by vivekgatecs2020 | 4.9k views

+2 votes

1 answer

21

Counting number of pairs whose sum is less than k
How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k > 0$. Solve by summation rules. Solve by combinatorial argument.

asked Jun 9 in Combinatory by dd | 201 views

0 votes

1 answer

22

Kenneth Rosen Edition 7th Exercise 8.3 Question 16 (Page No. 535)
Solve the recurrence relation for the number of rounds in the tournament described in question $14.$

asked May 10 in Combinatory by Lakshman Patel RJIT | 236 views

0 votes

1 answer

23

Kenneth Rosen Edition 7th Exercise 8.3 Question 15 (Page No. 535)
How many rounds are in the elimination tournament described in question $14$ when there are $32$ teams?

asked May 10 in Combinatory by Lakshman Patel RJIT | 86 views

0 votes

2 answers

24

Kenneth Rosen Edition 7th Exercise 8.3 Question 14 (Page No. 535)
Suppose that there are $n = 2^{k}$ teams in an elimination tournament, where there are $\frac{n}{2}$ games in the first round, with the $\frac{n}{2} = 2^{k-1}$ winners playing in the second round, and so on. Develop a recurrence relation for the number of rounds in the tournament.

asked May 10 in Combinatory by Lakshman Patel RJIT | 97 views

0 votes

0 answers

25

Kenneth Rosen Edition 7th Exercise 8.3 Question 13 (Page No. 535)
Give a big-O estimate for the function $f$ in question $12$ if $f$ is an increasing function.

asked May 10 in Combinatory by Lakshman Patel RJIT | 96 views

+1 vote

2 answers

26

Kenneth Rosen Edition 7th Exercise 8.3 Question 12 (Page No. 535)
Find $f (n)$ when $n = 3k,$ where $f$ satisfies the recurrence relation $f (n) = 2f (n/3) + 4 \:\text{with}\: f (1) = 1.$

asked May 10 in Combinatory by Lakshman Patel RJIT | 66 views

0 votes

0 answers

27

Kenneth Rosen Edition 7th Exercise 8.3 Question 11 (Page No. 535)
Give a big-O estimate for the function $f$ in question $10$ if $f$ is an increasing function.

asked May 10 in Combinatory by Lakshman Patel RJIT | 25 views

0 votes

1 answer

28

Kenneth Rosen Edition 7th Exercise 8.3 Question 10 (Page No. 535)
Find $f (n)$ when $n = 2^{k},$ where $f$ satisfies the recurrence relation $f (n) = f (n/2) + 1 \:\text{with}\: f (1) = 1.$

asked May 10 in Combinatory by Lakshman Patel RJIT | 19 views

0 votes

1 answer

29

Kenneth Rosen Edition 7th Exercise 8.3 Question 9 (Page No. 535)
Suppose that $f (n) = f (n/5) + 3n^{2}$ when $n$ is a positive integer divisible by $5, \:\text{and}\: f (1) = 4.$ Find $f (5)$ $f (125)$ $f (3125)$

asked May 10 in Combinatory by Lakshman Patel RJIT | 18 views

0 votes

1 answer

30

Kenneth Rosen Edition 7th Exercise 8.3 Question 8 (Page No. 535)
Suppose that $f (n) = 2f (n/2) + 3$ when $n$ is an even positive integer, and $f (1) = 5.$ Find $f (2)$ $f (8)$ $f (64)$ $(1024)$

asked May 10 in Combinatory by Lakshman Patel RJIT | 18 views

0 votes

1 answer

31

Kenneth Rosen Edition 7th Exercise 8.3 Question 7 (Page No. 535)
Suppose that $f (n) = f (n/3) + 1$ when $n$ is a positive integer divisible by $3,$ and $f (1) = 1.$ Find $f (3)$ $f (27)$ $f (729)$

asked May 10 in Combinatory by Lakshman Patel RJIT | 26 views

0 votes

0 answers

32

Kenneth Rosen Edition 7th Exercise 8.3 Question 6 (Page No. 535)
How many operations are needed to multiply two $32 \times 32$ matrices using the algorithm referred to in Example $5?$

asked May 10 in Combinatory by Lakshman Patel RJIT | 18 views

0 votes

0 answers

33

Kenneth Rosen Edition 7th Exercise 8.3 Question 5 (Page No. 535)
Determine a value for the constant C in Example $4$ and use it to estimate the number of bit operations needed to multiply two $64$-bit integers using the fast multiplication algorithm.

asked May 10 in Combinatory by Lakshman Patel RJIT | 17 views

0 votes

1 answer

34

Kenneth Rosen Edition 7th Exercise 8.3 Question 4 (Page No. 535)
Express the fast multiplication algorithm in pseudocode.

asked May 10 in Combinatory by Lakshman Patel RJIT | 29 views

0 votes

0 answers

35

Kenneth Rosen Edition 7th Exercise 8.3 Question 3 (Page No. 535)
Multiply $(1110)_{2} \:\text{and}\: (1010)_{2}$ using the fast multiplication algorithm.

asked May 10 in Combinatory by Lakshman Patel RJIT | 21 views

0 votes

0 answers

36

Kenneth Rosen Edition 7th Exercise 8.3 Question 2 (Page No. 535)
How many comparisons are needed to locate the maximum and minimum elements in a sequence with $128$ elements using the algorithm in Example $2$?

asked May 10 in Combinatory by Lakshman Patel RJIT | 16 views

0 votes

1 answer

37

Kenneth Rosen Edition 7th Exercise 8.3 Question 1 (Page No. 535)
How many comparisons are needed for a binary search in a set of $64$ elements?

asked May 10 in Combinatory by Lakshman Patel RJIT | 28 views

+1 vote

0 answers

38

Kenneth Rosen Edition 7th Exercise 8.2 Question 52 (Page No. 527)
Prove Theorem $6:$Suppose that $\{a_{n}\}$ satisfies the liner nonhomogeneous recurrence relation $a_{n} = c_{1}a_{n-1} + c_{2}a_{n-2} + \dots + c_{k}a_{n-k} + F(n),$ where $c_{1}.c_{2},\dots,c_{k}$ ... solution of the form $n^{m}(p_{t}n^{t} + p_{t-1}n^{t-1} + \dots + p_{1}n + p_{0})s^{n}.$

asked May 6 in Combinatory by Lakshman Patel RJIT | 32 views

0 votes

0 answers

39

Kenneth Rosen Edition 7th Exercise 8.2 Question 51 (Page No. 527)
Prove Theorem $4:$ Let $c_{1},c_{2},\dots,c_{k}$ be real numbers. Suppose that the characteristic equation $r^{k}-c_{1}r^{k-1}-\dots c_{k} = 0$ has $t$ distinct roots $r_{1},r_{2},\dots,r_{t}$ ... $\alpha_{i,j}$ are constants for $1 \leq i \leq t\:\text{and}\: 0 \leq j \leq m_{i} - 1.$

asked May 6 in Combinatory by Lakshman Patel RJIT | 19 views

0 votes

0 answers

40

Kenneth Rosen Edition 7th Exercise 8.2 Question 53 (Page No. 527)
Solve the recurrence relation $T (n) = nT^{2}(n/2)$ with initial condition $T (1) = 6$ when $n = 2^{k}$ for some integer $k.$ [Hint: Let $n = 2^{k}$ and then make the substitution $a_{k} = \log T (2^{k})$ to obtain a linear nonhomogeneous recurrence relation.]

asked May 6 in Combinatory by Lakshman Patel RJIT | 18 views

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