Complex variables 1-marks

 GATE 2019 Set-2 

1. An analytic function f) of complex variable iy may be written as

f) = u) + iv) . Then, u) and v) must satisfy

(A) \(\large\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} and \frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}\)

(B) \(\large\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} and \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\)

(C) \(\large\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y} and \frac{\partial u}{\partial y}=\frac{\partial v}{\partial x}\)

(D) \(\large\frac{\partial u}{\partial x}=-\frac{\partial v}{\partial y} and \frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\)

Answer

 GATE 2016 Set-1 

1. \(\large f(z)=u(x,y)+iv(x,y)\)is an analytic function of complex variable \(\large z=x+iy\) where \(\large i=\sqrt{-1}\). If \(\large u(x,y)=2xy\) then \(\large v(x,y)\) may be expressed as

(A) -x2+y2+constant

(B) x-y2+constant

(C) x2+y2+constant

(D) -(x2+y2)+constant

Answer

 GATE 2016 Set-2 

1. A function f of the complex variable \(\large z=x+iy\), is given as \(\large f(z)=u(x,y)+iv(x,y)\) where \(\large u(x,y)=2kxy\) and \(\large v(x,y)=x^2-y^2\). The value of k, for which the function is analytic, is _____

Answer

 GATE 2015 Set-1 

1. Given two complex numbers \(\large z_1=5+5\sqrt{3}i\) and \(\large z_2=\frac{2}{\sqrt{3}}+2i\), the argument of \(\large \frac{z_1}{z_2}\)in degrees is

(A) 0

(B) 30

(C) 60

(D) 90

Answer

 GATE 2014 Set-1 

1. The argument of the complex number \( \large \frac{1+i}{1-i}\), where\( i=\sqrt{-1}\), is

(A) \(\large -\pi\)

(B)\(\large -\frac{\pi}{2}\)

(C)  \(\large \frac{\pi}{2}\)

(D) \(\large \pi\)

Answer

 GATE 2013 

1. The eigenvalues of a symmetric matrix are all

(A) Complex with non-zero positive imaginary part.

(B) Complex with non-zero negative imaginary part.

(C) real

(D) pure imaginary

Answer

 GATE 2011 

1. The product of two complex numbers 1+i and 2-5i is

(A) 7-3i

(B) 3-4i

(C) -3-4i

(D) 7+3i

Answer

 GATE 2010 

1. The modulus of the complex number \(\large \left ( \frac{3+4i}{1-2i} \right )\) is

(A) 5

(B) √5

(C) 1/√5

(D) 1/5

Answer

 GATE 2008 

1. In the Taylor series expansion of ex about x=2 , the coefficient of (x-2)4 is

(A) 1/4!

(B) 24 /4!

(C) e2/4!

(D) e4/4!

Answer

 GATE 2007 

1. If φ (x,y) and Ψ (x, y) are functions with continuous second derivatives, then φ (x, y) + i Ψ (x, y) can be expressed as an analytic function of x + i y (i = √−1), when

(A) \(\large \frac{\partial \varphi}{\partial x}=-\frac{\partial \psi}{\partial x};\;\frac{\partial \varphi}{\partial y}=\frac{\partial \psi}{\partial y}\)

(B) \(\large \frac{\partial \varphi}{\partial y}=-\frac{\partial \psi}{\partial x};\;\frac{\partial \varphi}{\partial x}=\frac{\partial \psi}{\partial y}\)

(C) \(\large \frac{\partial \varphi^2}{\partial x^2}+\frac{\partial \varphi^2}{\partial y^2}=\frac{\partial \psi^2}{\partial x^2}+\frac{\partial \psi^2}{\partial y^2}=1\)

(D) \(\large \frac{\partial \varphi}{\partial x}+\frac{\partial \varphi}{\partial y}=\frac{\partial \psi}{\partial x}+\frac{\partial \psi}{\partial y}=0\)

Answer

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