AE56 Solution paper AMIETE-ET

8/18/2019 AE56 Solution paper AMIETE-ET

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AE56/AC56/AT56 ENGINEERING MATHEMATICS-II DEC 2014

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Q.2a. Show that the function ( ) xyzf = is not analytic at the origin even though

Cauchy-Riemann equations are satisfied thereof.


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b.Find the bilinear transformation which maps the points i z z == 21 ,2 and 23 −= z into

the points iww == 21 ,1 and 13 −=w .

Q.3a. Find the Taylor’s series expansion of a function of the complex variable

( )( )( )31

1

−−=

z z z f about the point, z = 4. Find its region of convergence.


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b. Determine the poles of the function ( )( ) ( )21 2

2

+−=

z z

z z f and find residue of )( z f of at

each pole.


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Q.4a. Evaluate grad2r

e where2222

z y xr ++= .


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b. Prove that( ) ( ) f r

dr

d

r r

r r f div 2

2

1 =

⋅


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Q.5 a. Apply Green’s theorem to evaluate xdydx yC

322 +∫

Where C is the boundary of closed region bounded between y = x and2

x y =

b. Apply Stoke’s theorem to calculate ydz zdy ydxC

624 ++∫

where C is the curve of intersection of z z y x 6222 =++ and 3+= x z


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Q.6a. Using Langrange’s interpolation formula, find the value of y corresponding to x

= 10 from the following table:

x 5 6 9 11

y 12 13 14 16


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b. Approximate the integral, dxe I x∫=1

0

using Simpson’s1 / 3

rd Rule with n = 8 interval.

Round off the result at 4 digits.


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Q.7a. Apply Charpit method to solve the equation pqqy px =+


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b. Using the method of separation of variable, solve ut

u

x

u+

∂∂

=∂∂

2 , where ( ) xe xu 360, −=


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Q.8 a. An urn ‘A’ contains 2 white nd 4 black balls. Another urn ‘B’ contains 5 whiteand 7 black balls. A ball is transferred from the urn ‘A’ to the urn ‘B’, then a ball is

drawn from urn ‘B’. Find the probability that it is white.


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b. Assuming half the population of a town consumes chocolates and that 100

investigators each take 10 individuals to see whether they are consumers, how many

investigators would you expect to report that three people or less were consumers?

Q.9 a. If the variance of the Poisson distribution is 2, find the probabilities for 4,3,2,1=r

from the recurrence relation of the Poisson distribution. Also find ( )4≥r P .


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b. The life of army shoes is ‘normally’ distributed with mean 8 months and standard

deviation 2 months. If 5000 pairs are issued how many pairs would be expected to need

replacement after 12 months? [Given that ( ) 0228.02P =≥ z andσµ= -x z ]


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Text books

1. Higher Engineering Mathematics –Dr. B.S.Grewal, 40th Edition 2007, Khanna

Publishers, Delhi

2. A Text book of engineering Mathematics – N.P. Bali and Manish Goyal , 7th Edition

2007, Laxmi Publication(P) Ltd

AE56 Solution paper AMIETE-ET

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