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GATE-2012 ECE Q34 (signals)

Posted By Krishna Sankar On December 29, 2012 @ 5:23 pm In GATE | No Comments

Question 34 on signals from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.

Solution

Let us Laplace transform to find  and later

,  where with real numbers  and .

Using integration by parts,

.

Rearranging,

.

Extending this to find the Laplace Transform of the second derivative of the function,

.

Coming back to the problem,

Taking Laplace transform,

.

To find the inverse Laplace transform, let us revisit the Laplace transform for some simple functions.

For , the Laplace transform is,
.

From the discussion in the post on Q11 in GATE 2012 ,

.

Also from the earlier discussion in this post,

Applying the above equations to find the inverse Laplace transform

.

Taking the differential,

.

Plugging in  ,

Based on the above, the right choice is (D) 1

References

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

 Wiki entry on Laplace transform of function’s derivative 

 post on Q11 in GATE 2012 

URL to article: http://www.dsplog.com/2012/12/29/gate-2012-ece-q34-signals/

URLs in this post:

 Laplace transform of function’s derivative: http://en.wikipedia.org/wiki/Laplace_transform#Proof_of_the_Laplace_transform_of_a_function.27s_derivative

 discussion in the post on Q11 in GATE 2012: http://www.dsplog.com/2012/12/27/gate-2012-ece-q11-signals/

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