GATE 2019 Paper with Answer Keys for Engineering Mathematics (XE-A)
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 absolute 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