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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.

Q34. Consider the differential equation

 with  and 

The numerical value of

 is

(A) -2

(B) -1

(C) 0

(D) 1

Solution

Let us Laplace transform to find  and later

The Laplace transform of function’s derivative [1] is

,  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 [2],

.

 

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

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

[2] Wiki entry on Laplace transform of function’s derivative [1]

[3] post on Q11 in GATE 2012 [2]

 

 


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URLs in this post:

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

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

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

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