Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU) B-Tech 4th Semester (Fourth Semester) 2015-16 NAS 401EAS Engg Mathematics Iii Question Paper
(Following Paper ID and Roll No. to be ?lled in your
Answer Books)
M RollNoJll 1111
B. TECH.
Theory Examination (Semester-IV) 2015-16
ENGG MATHEMATICS-III
Time : 3 Hours Max. Marks : 100
Section-A
l. Attempt all questions of this section. Each question carry
equal marks. (2XI0 = 20)
(3) Write the cauchy?s Reimaun conditions in polar coor-
dinates system.
(b) Write the statement of generalized cauchy?s integral
formula for n111 derivative of an analytic function at the
point Z = 20.
(c) Find the z ? transform of U? = {an}
(d) Writethenormalequationsto ?tacm've y=ax2+b by
leastsquammetbod.
(1) P.T.O.
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(e) Ifcovariance between x and y variable is 10 and the
variance of x and y are respectively 16 and 9, ?nd the
coe?icient of correlation.
(f) The regression equations calculated ?om a given set of
observations for two random variable are
x = ?0.4y + 6.4 and y = ?0.6x + 4.6 calculate mean
values of x and y.
(g) Write the Newton?s Raphson iterative formula to ?nd
the value of J? .
(h) Find the missing data in the given table :
x 0 1 2 3
f(x) 580 556 ? 465
(i) If ?n) is given in following table :
x o 0.5 1
f(x) 1 0.8 0.5
then using trapezoidal rule, evaluate
[f(x)dx
(j) Find the third forward difference with the arguments
2, 4, 6, 8 of the function?x)= x3- 2x
(2)
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Section-B
2. Attempt any ?ve questions from this section.
(10X5 = 50)
(a) Find the Laurent series for the ?mction
722+9z?18
?2): 23?92
, Z is complex variable
valid for the regions
(i) O<|z|<3 (ii)|z|>3
(b) Using calculus of residue, evaluate the following
integral
f dx
(02 +112)2
(c) Find the inverse Fourier sine transform of 18?"
x
(d) Using least square method, ?t a second degree poly-
nomial from the following data :
x012345678
y 12.0 10.5 10.0 8.0 7.0 8.0 7.5 8.5 9.0
Also estimate y at x = 6.5
i (3)
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(e) For the following data, calculate the ?nite differences
and obtain the forward and backward difference poly-
nomials. Also interpolate at x = 0.25 and x = 0.35
x 0.1 0.2 0.3 0.4 0.5
f(x) 1.40 1.56 1.76 2.00 2.28
(1) Construct the divided diference table for the data.
x 0.5 1.5 3.0 5.0 6.5 8.0
m
f(x) 1.62 5.87 31.0 131.0 282.12 521.0
Hence ?nd the interpolating polynomial and an ap-
proximation to the value of?z).
(g) Solve the system of equations AX=B, where
2 l 1 ?2 ?10
4 0 2 1 8
A: B:
3 2 2 0 , 7
1 3 2 ?1 ?5
using the LU decdomposition method. Take all the
diagonal elements of L as 1.
(4)
(h) Solve the initial value problem
W 2
? = ?2 , 0 = l
h w.?)
with h = 0.1 on the interval [003]. Use the fourth order
Runge-Kutta method.
Section-C
Note: Attempt any two questions from this section. Each
question carry equal marks. (15X2=30)
3. (a) Show that for the function give as -
M ifz..0
f(z)= xz+yz
0 ifz=0
The C-R conditions are satis?ed at origin but derivative of
f(z) at origin does not exist.
(b) Verify that the function on 4(xy) = xy is harmonic and ?nd
its conjugate harmonic ?mction. Express wiv as an analytic
function?z).
u=f?f-y
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(c) Find the Fourier transform of Block function?t) of height 1
and duration a de?ned by
1 f 1 :3
f(t)= ?" ' 2
0 otherwise
4. (a) Using Z ? tranform, solve the difference equation
_ n
u"+2 ?4u"+1 + 314,, ? 5
with uo=u1=1
(b) The ?rst four moments of a distribution about x = 4 are
1, 4, 10, 45. Comment on the skewness and Kmtosis of
the distribution.
(c) For 210 observations on price (x) and supply 02) the
following data were obtained
21:: =130,2y = 220,282 = 2288
2x2 = 5506 and 29 = 3467
Obtain the two lines of regression.
5. (a) Find the root of the euqation xe" =3by regula talsi
method correct up to two decimal places in the interval (1 ,
1.5).
(6)
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(b) Prove the following identities :
A2 AIM
(i) [fj'ux ? Ell!
. A2 I E(e?)_e,
(11) ET
(0) The velocity v of a particle at distance 3 ?om a point on
its path is given by the following table :
s(m.) 0 10 20 30 40 50 60
v(m./s.) 47 58 64 65 61 52 38
Estimate the time taken to travel 60m. Using Simpson?s one-
third rule.
(7)
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This post was last modified on 29 January 2020