CSIR NET -2020

CSIR NET MATHEMATICS 2020-JUNE

HELD ON 26TH-NOVEMBER 

 

QUESTIONS AND SOLUTIONS

PART-A 

 

1.Find the value of $ f(c) $ if $ f(x+2)=(x+1)^{34}-(x+1)^{35}+5 $

 

(A) 5

(B) 7

(C) 6

(D) 72 

 

 

 Solution Click Here

 

 

2.the shortest distance between the parallel lines A and B in the following figure

(A) $\sqrt{2} $

(B) 2

(C) $2\sqrt{2} $

(D) $ 2\sqrt{3} $

 Solution 

 

PART-B

 

1.$ f:N\rightarrow N $ be bounded function.which the following statement is NOT true?

 

(A)  $ {\lim}_{x \to \infty }\sup f(n) \in  N  $

(B)  $ \lim_{x \to \infty }\inf f(n) \in  N $

(C) $ \lim_{x \to \infty }\inf( f(n)+n) \in  N $

(D) $ \lim_{x \to \infty }\sup( f(n)+n) \notin  N $

Solution Click Here 

 

2.Let A and B  be $ 2\times 2 $  matrices.then which of the following is true?

 

(A) det(A+B)+det(A-B)=detA+detB

(B)det(A+B)+det(A-B)=2detA-2detB

(C) det(A+B)+det(A-B)=2detA+2detB

(D)det(A+B)-det(A-B)=2detA-2detB

 

Solution Click Here 

 

 

 3.if $ A= \begin{bmatrix}
3& -2\\
2 & -1
\end{bmatrix} $, then $ A^{20} $ equals

 

 

(A) $  \begin{bmatrix} 41 & 40  \\ -40 & -39  \end{bmatrix} $

 

(B)$  \begin{bmatrix} 41 & -40  \\ 40 & -39  \end{bmatrix} $

 

(C) $  \begin{bmatrix} 41 & -40  \\ -40 & -39  \end{bmatrix} $

 

(D) $  \begin{bmatrix} 41 & 40  \\ 40 & -39  \end{bmatrix} $

 

 

Solution Click Here 

4.Let A be a  $ 2 \times 2 $  real matrix with $ \det A=1 $ and trace A=3.what is the value of trace $ A^2$

 

(A) 2

(B)10

(C) 9

(D) 7

Solution Click Here

5. Which of the following real quadratic forms on $R^2$ is positive definite 

 

(A) $Q(X,Y)=XY$

(B) $Q(X,Y)=(X^2+XY)$

(C)$Q(X,Y)=X^2+2XY+Y^2$

(D) $Q(X,Y)=X^2-XY+Y^2$   

 Solution Click Here

6.Let $\gamma$ be the positively  oriented circle in the complex plane given by $\lbrace Z \in C : \vert Z-1 \vert =1 \rbrace $ then $\frac{1}{2\pi i}\int_{\gamma}\frac{dz}{z^3-1}$ equals

 

(A) 3

(B) 1/3

(C)2 

(D) 1/2

 Solution Click Here

7.The maximum and minimum values of $5x+7y$,when $\vert x\vert +\vert y \vert \leq 1$

 

(A)  $5$and $-5$

(B) $5$and $-7$

(C)  $7$and $-5$ 

(D) $7$and $-7$

Solution Click Here

8.Let $\gamma$ be the positively  oriented circle in the complex plane given by $\lbrace Z \in C : \vert Z-1 \vert =1/2 \rbrace $ then $\int_{\gamma}\frac{ze^{1/z}}{z^2-1}dz$ equals 

(A)  $i\pi e$

(B)  $-i\pi e$

(C)  $\pi e$

(D)  $- \pi e$

 

 

PART-C

 

 

1.Let P be a square matrix such that $P^2=P$ which of the following statements are true?  

(A) Trace of P is an irrational number

(B)  Trace of P = rank of P

(C)  Trace of P is an integer

(D) Trace of P is an imaginary complex number