- DSP log - http://www.dsplog.com -

GATE-2012 ECE Q25 (math)

Posted By Krishna Sankar On October 9, 2012 @ 6:20 am In GATE | No Comments

Question 25 on math from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

## Q25. If $\Large{x=\sqrt{-1}}$, then the value of $\Large{x^x}$ is,

(a) $e^{-\pi/2}$

(b) $e^{\pi/2}$

(c) $x$

(d) 1

## Solution

Going by the definition of imaginary unit [1]$j^2 =-1$ and

From Euler’s formula [2]$e^{jy}=\cos(y) + j\sin(y)$.

Putting $y=\pi/2$,

$e^{j\pi/2}=\cos(\pi/2) + j\sin(\pi/2) = 0 + 1j = j$.

Substituting,

$\Large{x^x=j^j=e^{$$j\pi/2$$^j}=e^{-\pi/2}}$.

## References

[1] GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf [3]

[2] Imaginary unit [1]

[3] Euler’s formula [2]

URL to article: http://www.dsplog.com/2012/10/09/gate-2012-ece-q25-math/

URLs in this post:

[1] imaginary unit: http://en.wikipedia.org/wiki/Imaginary_unit

[2] Euler’s formula: http://en.wikipedia.org/wiki/Euler%27s_formula

[3] http://gate.iitm.ac.in/gateqps/2012/ec.pdf: http://gate.iitm.ac.in/gateqps/2012/ec.pdf