Mathematics:Teaching, Learning and Exploring.
You are not logged in.
Pages: 1
TENTATIVE DATE OF SINGLE MCQ EXAMINATION 18.12.2016.
For More Details check (www.csirhrdg.res.in.)
DATE OF START OF ON-LINE SUBMISSION OF APPLICATION FORM: 16.08.2016.
The National Board for Higher Mathematics (NBHM) invites applications
for the grant of scholarships to students for pursuing studies leading to a
masters degree in mathematics.
Selection will be made on the basis of a written test followed by an interview of shortlisted candidates. The written test will be held on Saturday,September 17, 2016. The test will be of two and a half hours’ duration and involve questions requiring short answers, primarily in the areas of analysis and algebra, but also in other areas of mathematics covered in undergraduate mathematics programmes. (Please visit the NBHM website or http://www.imsc.res.in (under Quick Links) for the question papers of previous years.)
The written test will be conducted at the following centres:
Zone 1: Chandigarh, New Delhi, Srinagar.
Zone 2: Allahabad, Indore, Jabalpur, Raipur.
Zone 3: Goa, Mumbai, Pune, Vallabh Vidyanagar.
Zone 4: Bhubaneswar, Guwahati, Kolkata, Ranchi, Shillong.
Zone 5: Bangalore, Chennai, Cochin, Hyderabad.
Important Dates
July 29, 2016 Last date for receiving applications.
August 29, 2016 Applicant should contact zonal coordinator
if hall ticket is not received by this date.
September 9, 2016 Applicant should contact centre-in-charge
suggested by zonal coordinator if hall ticket
is not received by this date.
September 17, 2016 Date of written test.
JAM 2017 will be held on February 12, 2017(Sunday) for admission to integrated Ph.D degree programmes at IISc Bangalore, and M.Sc (Two Year), Joint M.Sc +Ph.D Dual Degree and other post-bachelor degree programes at IITs. For the academic year 2017-2018. Candidates have to apply ONLINE only from September 05, 2016 to October 06, 2016. For more details please visit http://jam.iitd.ac.in or refer to the Employment News dated August27, 2016.
Given \(n \times n\) matrix B define \(e^B\) by $$e^B= \sum_{j=0}^{\infty} \frac{B^j}{j !} $$ Let p be the characteristic polynomial of B. Then the matrix \(e^{p(B)}\) is???
A)\( I_{n \times n}\)
B)\( 0_{n \times n}\)
C)\( e I_{n \times n} \)
D)\( \pi I_{n \times n} \)
First look at \(e^B\) , It gives \(e^B= \sum_{j=0}^{\infty} \frac{B^j}{j !} \)
\(e^B= \sum_{j=0}^{\infty} \frac{B^j}{j !}=I + \frac{B^1}{ 1!}+\frac{B^2+}{ 2!}+\frac{B^3}{ 3!}+\dots \)
Now p is characteristic polynomial of B \(\implies \) \(p(B)=0\) By cayleyHamilton theorem.
Now calculate \(e^{p(B)}\) .
\(e^{p(B)}=\sum_{j=0}^{\infty} \frac{p(B)^j}{j !}=I + \frac{p(B)^1}{ 1!}+\frac{p(B)^2+}{ 2!}+\frac{p(B)^3}{ 3!}+\dots \)
\(e^{p(B)}=\sum_{j=0}^{\infty} \frac{p(B)^j}{j !}=I + 0+0+0+\dots \)
\(e^{p(B)}=I_{n \times n} \)
The differential equation
$$ (\alpha xy^3 + y \cos x )dx+ (x^2y^2+ \beta \sin x )dy =0 $$
A) \( \alpha = \frac{3}{2},\beta = 1 \)
B) \( \alpha = \frac{2}{3},\beta = 1 \)
C) \( \alpha = 1,\beta = \frac{3}{2} \)
D) \( \alpha = 1,\beta = \frac{2}{3}\)
The sequence of function $$f_n(x)=x^n, \forall x \in [0,1], n=1,2,3,\dots $$ is
A) Pointwise convergent to a continuous on function on $[0,1]$
B) Not pointwise convergent on $[0,1]$
C) Pointwise convergent but not uniformly convergent on $[0,1]$
D) Uniform convergent on $[0,1]$
The Value of $$100\left[\frac{1}{(1)(2)}+\frac{1}{(2)(3)}+ \frac{1}{(3)(4)}+\ldots+\frac{1}{(99)(100)}\right]$$
A)is 99;
B)lies between 50 and 98;
C) is 100;
D)is different from valuse specified in the forgoing statments.
[Hint:use $$\frac{1}{n(n+1)}=\frac{1}{n} -\frac{1}{n+1}$$
Ans is (A) is 99.
Q(NET DEC 2014)
Which of the following are eigenvalues of the matrix
\[ \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0& 0 & 1 & 0 \\
0& 0 & 0 & 0 & 0 & 0 \\
1& 0 & 0 & 0 & 0 & 0 \\
0& 0 & 0 & 0 & 0 & 0 \\
0& 0 & 1 & 0 & 0 & 0 \end{pmatrix} \]
A) 1
B)\(-1\)
C) i
D)\(-i\)
Ans
Q.(NET Dec 2014)
Which of the following matrices have Jordan canonical form equal to \[ \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
A) \[ \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
B) \[ \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \]
C) \[ \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
D) \[ \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \]
Q.(NET Dec 2014)
The Determinant of \(n \times n \) permutation matrix is
\[ \begin{pmatrix} 0 & 0 & \cdots & 1 \\ 0 & 0 & 1 & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & \cdots & 0\end{pmatrix} \]
A) \( (-1)^n \)
B) \( (-1)^{\frac{n}{2}} \)
C) \( -1 \)
D) \( 1 \)
Ans .
Q.(SET Nov 2011)
The dimension of the vector spaces of \(n \times n\) matrices all of whose components are 0 expect possibly the diagonal components is
A) \(n^2\)
B) \(n-1\)
C) \(n^2-1\)
D) \(n \)
Ans D
The basis of \(n \times n\) matrices all of whose components are 0 expect possibly the diagonal components ( i.e we have only diagonal elements ) are \(e_{11}, e_{22},e_{33}, ....,e_{nn} \) so that dim is n
Let A be the matrices.
\[ A_{n,n} = \begin{pmatrix} a_{1,1} & 0 & \cdots & 0 \\ 0 & a_{2,2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{n,n} \end{pmatrix}
=a_{11} \begin{pmatrix} e_{11}\\ 0\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ 0 \end{pmatrix}+ a_{22} \begin{pmatrix} 0\\ e_{22}\\ 0\\ \vdots\\ \vdots\\ \vdots\\ 0 \end{pmatrix}+ \dots +a_{nn} \begin{pmatrix} 0\\ 0\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ e_{nn} \end{pmatrix}\]
Q.(Net June 2011) The number of elements in the set \(\{ m:1\leq m \leq 1000 \} \),where m and 1000 are relatively prime is
A) 100
B) 250
C) 300
D)400
Ans D
If n is any number then \(\phi (n) \)= number of relatively prime to n .
The number of elements in the \(\{ m:1\leq m \leq 1000 \} \) m and 1000 are relatively prime is
\(\phi(1000)=1000(1-\frac{1}{2})(1-\frac{1}{5})=400 \)
Pages: 1