Complex Numbers

GATE-CH-2001-1-2-math-1mark

2001-1-2-math

The value of \((1+i)^8\), where \(i= \sqrt {-1}\), is

\(8+4i\)
\(8-4i\)
16
8

GATE-CH-2002-1-1-math-1mark

2002-1-1-math

In the complex plane, the angle between lines \(1+i\) and \(-1+i\) (where \(i=\sqrt {-1}\)) is

\(\pi /4\)
\(\pi /2\)
\(3\pi /4\)
\(\pi \)

GATE-CH-2004-3-math-1mark

2004-3-math

The complex number \(2(1+i)\) can be represented in polar form as

\(2\sqrt {2}e^{i\pi /4}\)
\(\displaystyle \frac {\pi }{4}e^{i2\sqrt {2}}\)
\(\sqrt {2}e^{i\pi /4}\)
\(\displaystyle \frac {\pi }{4}e^{i\sqrt {2}}\)

GATE-CH-2005-3-math-1mark

2005-3-math

Given \(i = \sqrt {-1}\), the ratio \(\displaystyle \frac {(1+2i)}{(i-2)}\) is given by

1
\(-1\)
\(i\)
\(-i\)

GATE-CH-2007-1-math-1mark

2007-1-math

Given \(i=\sqrt {-1}\), the ratio \(\dfrac {(i+3)}{(i+1)}\) is given by

\(i\)
\(-2\)
\(-i+2\)
\(i+1\)

GATE-CH-2009-2-math-1mark

2009-2-math

The modulus of the complex number \(\dfrac {1+i}{\sqrt {2}}\) is

\(\dfrac {1}{2}\)
\(\dfrac {1}{\sqrt {2}}\)
1
\(\sqrt {2}\)

GATE-CH-2010-14-math-1mark

2010-14-math

Given that \(i = \sqrt {-1}\), \(\imath ^i\) is equal to

\(\pi /2\)
\(-1\)
\(i\ln i\)
\(e^{-\pi /2}\)

GATE-CH-2014-3-math-1mark

2014-3-math

If \(f^*(x)\) is the complex conjugate of \(\ f(x)=\cos (x)+i \sin (x)\), then for real \(a\) and \(b\), \(\ \displaystyle \int _a^b f^*(x)f(x)dx\ \) is ALWAYS

positive
negative
real
imaginary

GATE-CH-2015-4-math-1mark

2015-4-math

A complex-valued function, \(f(z)\), given below is analytic in domain \(D\): \[ f(z) = u(x,y) + i v(x,y) \qquad z = x + i y\] Which of the following is NOT correct?

\(\displaystyle \frac {df}{dz} = \frac {\partial v}{\partial y} + i \frac {\partial u}{\partial y} \)
\(\displaystyle \frac {df}{dz} = \frac {\partial u}{\partial x} + i \frac {\partial v}{\partial x} \)
\(\displaystyle \frac {df}{dz} = \frac {\partial v}{\partial y} - i \frac {\partial u}{\partial y} \)
\(\displaystyle \frac {df}{dz} = \frac {\partial v}{\partial y} + i \frac {\partial v}{\partial x} \)

GATE-CH-2016-3-math-1mark

2016-3-math

What are the modulus \((r)\) and argument \((\theta )\) of the complex number \(3+4i\) ?

\(\displaystyle r=\sqrt {7}, \ \ \theta =\tan ^{-1}\left (\frac {4}{3}\right )\)
\(\displaystyle r=\sqrt {7}, \ \ \theta =\tan ^{-1}\left (\frac {3}{4}\right )\)
\(\displaystyle r=5, \ \ \theta =\tan ^{-1}\left (\frac {3}{4}\right )\)
\(\displaystyle r=5, \ \ \theta =\tan ^{-1}\left (\frac {4}{3}\right )\)

GATE-CH-2017-2-math-1mark

2017-2-math

The real part of \(6e^{i\pi /3}\) is ____________

Answer

GATE-CH-2000-2-3-math-2mark

2000-2-3-math

The complex conjugate of \(\dfrac {1}{1+i}\) is

\(\dfrac {1}{1-i}\)
\((1-i)\)
\(0.5(1-i)\)
in the first quadrant of the complex plane

GATE-CH-2003-35-math-2mark

2003-35-math

The most general complex analytical function \(f(z) = u(x,y) + i v(x,y)\) for \(u=x^2-y^2\) is

\(z\)
\(z^2\)
\(z^3\)
\(1/z^2\)

GATE-CH-2005-35-math-2mark

2005-35-math

If \(z=x+iy\) is a complex number, where \(i=\sqrt {-1}\), then which of the following is an analytic function of \(z\)?

\(x^2+y^2\)
\(2ixy\)
\(x^2+y^2-2ixy\)
\(x^2-y^2+2ixy\)

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