**Q. 1 – Q. 7 carry one mark each & Q. 8 – Q. 11 carry two marks each.**

Q.1- Let ???? be the Poisson random variable with parameter ???? = 1. Then, the probability ????(2 ≤ ???? ≤ 4) equals

(A) \frac{19}{24e}

(B) \frac{17}{24e}

(C) \frac{13}{24e}

(D) \frac{11}{24e}

Ans: (B) \frac{17}{24e}

Q.2- For the series {\textstyle\sum_{n=1}^\infty}\frac{\left(x+1\right)^n}{n\;2^n},-8<x<\infty, which of the following statements is **NOT** correct?

(A) The series converges at ???? = −3

(B) The series converges at ???? = −1

(C) The series converges at ???? = 0

(D) The series converges at ???? = 1

Ans: (D) The series converges at ???? = 1

Q.3- Let ????(????) = ????̅???? ^{−|????|2} , where ????̅ is the complex conjugate of ????. Then, it is differentiable on

(A) |????| > 1

(B) |????| < 1

(C) |????| = 1

(D) the entire complex

plane ℂ

Ans: (C) |????| = 1

Q.4- If the transformation ????(????, ????) = ???? ^{????} ????(????, ????) reduces the partial differential equation

− 2 ???????? − ???????? + ???? = 9 to the equation ???????? − ???????? 2 = 9 ????(????), then ????(????) equals

(A) -e^{-x}

(B) e^{-x}

(C) -2e^{-x}

(D) 2e^{-x}

Ans: (A) -e^{-x}

Q.5- The value of ???? for which the system of equations

???? − ???? − 3???? = 3

2???? + ???? = 0

−2???? − 7???? = ????

has a solution is ** _** .

Ans: 6 to 6

Q.6- The value of the line integral \frac2\pi ∮????(−???? ^{3} ???????? + ???? ^{3} ????????), where ???? is the circle ????^{ 2} + ???? ^{2} = 1 oriented counter clockwise, is ** _** .

Ans: 3 to 3

Q.7- Let ????_{1} (????) and ????_{2} (????) be two linearly independent solutions of the differential equation x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}-4y=0,x>0. If ????_{1} (????) = ???? ^{2 }, then \lim_{x\rightarrow\infty}\;y_2\left(x\right) is ** _** .

Ans: 0 to 0

Q.8- If Q=\begin{pmatrix}3&2&4\\2&0&2\\4&2&3\end{pmatrix} and ???? = (????_{1} ????_{2} ????_{3} ) is the matrix where ????_{1 }, ????_{2} and ????_{3} are linearly independent eigenvectors of the matrix ????, then the sum of the **absolut**e values of all the elements of the matrix ????^{−1} ???????? is

(A) 6

(B) 10

(C) 14

(D) 22

Ans: (B) 10

Q.9- If ????(????) = ???????? ^{3 }+ ???????? ^{2 }+ ???????? + ???? is the polynomial obtained by Lagrange interpolation satisfying ????(0) = −8, ????(1) = −7, ????(2) = −6 and ????(4) = 20, then the value of ???? − ???? + ???? is

(A) 1

(B) 3

(C) 5

(D) 7

Ans: (D) 7

Q.10- The number of critical points of the function ????(????, ????) = ???? ^{3} + 3???????? ^{2} − 15???? − 12???? at which there is neither maximum nor minimum is ** _** .

Ans: 2 to 2

Q.11- Let I=\frac{10^5i}{2\pi}\phi_y\frac{dz}{\left(z-4\right)\left(z^7-1\right)} , where i=\sqrt{-1} and ???? is the circle |????| = 2 oriented counter clockwise. Then, the value of ???? rounded off to one decimal place is ** _** .

**END OF THE QUESTION PAPER**

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