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2 votes

1 answer

7641

ISI2011-PCB-A-3b
The numbers $1, 2, \dots , 10$ are arranged in a circle in some order. Show that it is always possible to find three adjacent numbers whose sum is at least $17$, irrespective of the ordering.

go_editor asked in Combinatory Jun 3, 2016

772 views

1 vote

0 answers

7642

ISI2011-PCB-A-3a
Consider an $m \times n$ integer lattice. A path from $(0, 0)$ to $(m, n)$ can use steps of $(1, 0)$, $(0, 1)$ or diagonal steps $(1, 1)$. Let $D_{m,n}$ be the number of such distinct paths. Prove that $D_{m,n} = \Sigma_k \begin{pmatrix} m \\ k \end{pmatrix} \begin{pmatrix} n+k \\ m \end{pmatrix}$

go_editor asked in Combinatory Jun 3, 2016

536 views

1 vote

2 answers

7643

ISI2011-PCB-A-2b
An $n \times n$ matrix is said to be tridiagonal if its entries $a_{ij}$ are zero except when $|i−j| \leq 1$ for $1 \leq i, \: j \leq n$. Note that only $3n − 2$ entries of a tridiagonal matrix are non-zero. Thus, an array $L$ of size ... matrix. Given $i, j$, write pseudo-code to store $a_{ij}$ in $L$, and get the value of $a_{ij}$ stored earlier in $L$.

go_editor asked in Linear Algebra Jun 3, 2016

779 views

1 vote

2 answers

7644

matrix
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define 4 $\times$ ... Then rank(A) equals 1 or 2 0 4 2 or 3

kvkumar asked in Linear Algebra Jun 2, 2016

734 views

1 vote

1 answer

7645

limits
Let $f(x,y) = \begin{cases} 1, & \quad if \: xy=0, \\ xy, & \quad xy \neq 0. \end{cases}$ Then $f$ is continuous at $(0,0)$ and $\frac{\partial f}{ \partial x} (0,0)$ exists $f$ is not continuous at $(0,0)$ ... $f$ is not continuous at $(0,0)$ and $\frac{\partial f}{ \partial x} (0,0)$ does not exist

kvkumar asked in Calculus Jun 2, 2016

429 views

0 votes

1 answer

7646

kvkumar asked in Probability Jun 2, 2016

287 views

0 votes

1 answer

7647

kvkumar asked in Set Theory & Algebra Jun 2, 2016

351 views

0 votes

1 answer

7648

kvkumar asked in Set Theory & Algebra Jun 2, 2016

197 views

1 vote

2 answers

7649

eigen vectors
Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix $ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & -\sin t & \cos t \end{pmatrix}$. If $\lambda_1+ \lambda_2+\lambda_3=\sqrt{2} +1$ ... $\{ - \frac{\pi}{4}, \frac{\pi}{4} \}$ $\{ - \frac{\pi}{3}, \frac{\pi}{3} \}$

kvkumar asked in Linear Algebra Jun 2, 2016

690 views

2 votes

1 answer

7650

calculus
Consider the function $f(x)=\begin{cases} \int_0^x \{5+ \mid 1-y \mid \} dy & \quad if \: x>2 \\ 5x+2 & \quad if \: x \leq 2 \end{cases}$ Then $f$ is not continuous at x=2 $f$ is continuous and differentiable everywhere $f$ is continuous everywhere but not differentiable at x=1 $f$ is continuous everywhere but not differentiable at x=2

kvkumar asked in Calculus Jun 2, 2016

336 views

0 votes

1 answer

7651

kvkumar asked in Probability Jun 2, 2016

219 views

0 votes

1 answer

7652

Kenneth Rosen Edition 6th Exercise 2.1 Question 21c (Page No. 120)
21 c) How many elements does the given set have where a and b are distinct elements? P(P(∅))

khushtak asked in Set Theory & Algebra Jun 2, 2016

553 views

1 vote

1 answer

7653

Linear Algebra \ determinant

Desert_Warrior asked in Linear Algebra Jun 2, 2016

by Desert_Warrior

465 views

2 votes

1 answer

7654

a graph with n vertices and n-1 edges that is not a tree is called

Anuanu asked in Graph Theory Jun 2, 2016

by Anuanu

12.3k views

1 vote

1 answer

7655

Find the Eigen Vector
Find the Eigenvector of $\begin{bmatrix} 1 & cos\theta\\ cos \theta & 1 \end{bmatrix}$

Anuanu asked in Linear Algebra Jun 2, 2016

by Anuanu

1.6k views

0 votes

1 answer

7656

first order logic

Sanjay Sharma asked in Mathematical Logic Jun 2, 2016

by Sanjay Sharma

421 views

1 vote

0 answers

7657

ISI2012-PCB-CS-4
A fan of order $n$ is a graph on the vertices $\{0, 1, \dots, n\}$ with $2n − 1$ edges defined as follows: vertex $0$ is connected by an edge to each of the other $n$ vertices, and vertex $i$ ... the number of spanning trees of the fan of order $n$. Calculate $f_4$. Write a recurrence for $f_n$. Solve for fn using ordinary generating functions.

go_editor asked in Graph Theory Jun 2, 2016

684 views

1 vote

0 answers

7658

Kenneth Rosen Edition 6th Exercise 1.4 Question 44 (Page No. 62)
Use quantifiers and logical connectives to express the fact that a quadratic polynomial with real number coefficients has at most two real roots. answer I think is ∀a∀b∀c[∃x ∃y ((ax2+ bx+c=0 )∨ (ay2+ by+c=0)) ∧ (∀w(aw2+ bw+c=0) $\rightarrow$ (w=x ∨ w=y)]

khushtak asked in Mathematical Logic Jun 1, 2016

1.6k views

1 vote

0 answers

7659

UGC NET CSE | September 2013 | Part 3 | Question: 67
The tracing model in Prolog describes program execution in terms of certain events. These events are call and exit call and fail call, exit and redo call, exit, redo and fail

Sanjay Sharma asked in Mathematical Logic Jun 1, 2016

by Sanjay Sharma

844 views

1 vote

1 answer

7660

UGC NET CSE | June 2013 | Part 2 | Question: 40
The truth value of the statements: $\exists ! xP(x) \rightarrow \exists xP(x) \text{ and } \exists ! x \rceil P(x) \rightarrow \rceil \forall xP(x)$, (where the notation $\exists ! x P(x)$ denotes the proposition “There exists a unique $x$ such that $P(x)$ is true”) are: True and False False and True False and False True and True

Sanjay Sharma asked in Mathematical Logic Jun 1, 2016

by Sanjay Sharma

1.6k views

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