QIESTION BANK ENGINEERING MATHEMATICS-III

SHORT QUESTIONS (EACH 2 MARKS)

UNIT-1
1. Show that f(z) = |z|2 is differentiable only at origin.

2. If u = x2 y2, find the corresponding analytic function.

3. Evaluate ( )n dz, where c is the circle | z a | = r.
4. An Electrostatic field in the X-Y plane is given by the potential function = 3 x2y y3, find the stream
function.
5. Define Harmonic Function and show that if f (z) = u + iv is an analytic function in some region then u and v
both are Harmonic.
6. If f(z) is a regular function of z, show that:
(2/x2 + 2/y2) |R f (z) |2 = 2 | f ' (z) |2
7. Expand the function f(z) = tan-1(z) in powers of z.
8. Expand the function f(z) = 1/z about z = 2 in Taylors Series and find its radius of convergence.
9. Determine the poles and residue at each pole of the function f(z) = cot(z).
10. Evaluate [() / z6] dz, where c is the circle | z | = 2.
UNIT-2
1. What do you mean by Integral Transform? Write down the Kernels in Laplace, Fourier and Mellins
Transform.
2. State and prove Change of Scale Property for Fourier Transform.
3. Find the Fourier transform of: f(x) = 1 for | x | < 1
0 for | x | > 1.
4. Derive the relation between Fourier and Laplace Transforms.
5. Define Z-Transforms and Explain Linearity Property.
6. State and prove Shifting Property for Fourier Transform.
7. Find the Fourier sine transform of [e-ax / x].
8. State and prove Modulation Theorem for Fourier Transform.
9. What do you mean by a Difference Equation?

10. Prove that: Z (np) = -z Z (np-1), p being a +ve integer.

UNIT-3
1. Explain Kurtosis.
2. Explain Skewness.
3. If on an average one ship in every ten is wrecked, find the probability that out of 5 ships expected to arrive,
4 at least will arrive safely.
4. Using Poisson distribution, find the probability that the ace of spades will be drawn from a pack of wellshuffled cards at least once in 104 consecutive trials.
5. A continuous random variable has probability density function f(x) = ke-x/5 , x 0
0
, elsewhere
Then find the value of k?
6. Ten percent of screws produced in a certain factory turns out to be defective. Find the probability that in a
sample of 10 screws chosen at random, exactly two will be defective.
Page 1 of 8

7. The first three moments of distribution, about the value 2 of the variable are 1, 16 and 40. Show that the
mean is 3, variance is 15 and 3 = - 86.
8. A car hire firm has two cars which it hires day by day. The number of demands for a car on each day is
distributed as a Poisson distribution with mean 1.5. Calculate the number of days in a year on which neither car
is on demand.
9. If there are 3 misprints in a book of 1000 pages. Find the probability that a given page will contain (i) no
misprint, (ii) more than 2 misprints.
10. Find the mode from the following data:
0-6
6-12
12-18
18-24
24-30
30-36
36-42
Age:
6
11
25
35
18
12
6
Frequency:
UNIT-4
1. Perform three iterations of the bisection method to obtain the smallest positive root of the equation
x3 5x + 1 = 0.
2. Define Central Difference Operator and Averaging Operator and establish relation between them.
3. Using the following table evaluate f (7.5)1
2
3
4
5
6
7
8
x:
1
8
27
64
125
216
343
512
y:
4. What do you mean by Rate of Convergence? Write down and compare the rate of convergence of Bisection,
False Position and Newton-Raphsons Methods.
5. Prove that: hD = log(1 + ) = - log(1 - ) = sinh-1(), where symbols have their usual meanings.
6. Prove that: [2 / E] ex. [E ex / 2 ex ] = ex, the interval of differencing being h.
7.
Estimate the missing term in the following table using Shift Operator directly0
1
2
3
4
x:
1
3
9
?
81
y:
8. Prove that: = 2/2 [ 1 + 2/4]1/2, where symbols have their usual meanings.
9. Evaluate: [ eax.log(bx)].
10. Which of the following numerical methods among Bisection, False Position and Newton-Raphsons
Methods can be used to solve the equation f(x) = (x 1)2 .(x 5)2 = 0. Explain with proper reason.
UNIT-5
1. Find the expression for the first order derivative using Forward Difference Formula when p = 0.
2. Perform two approximations (other than initial) of Jacobis Method for the system of Linear Equations:
4x + y + 3z = 17, x + 5y + z = 14, 2x y + 8z = 12.
3. Perform two approximations of Gauss-Seidel Method for the system of Linear Equations:
54 x + y + z = 110, 2x + 15y + 6z = 72, -x + 6y + 27z = 85.
4. Find the first order derivative of the function tabulated below, at the point x = 1.5
1.5
2.0
2.5
3.0
3.5
4.0
x:
3.375
7.000
13.625
24.000
38.875
59.000
y:
5. Find the first order derivative of the function tabulated below, at the point x = 0.4
0.1
0.2
0.3
x:
1.10517
1.22140
1.34986
y:

0.4
1.49182

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6. Evaluate x dx, by Simpsons one-third rule using data e = 2.72, e2 = 7.39, e3 = 20.09, e4 = 54.60.
7. Using the Newton-Cotes Quadrature Formula:

() = nh [ y0 + y0 + ..] :where a = x0 and b = x0 + nh

Derive Trapezoidal Rule for Numerical Integration.
8. The velocity v (km. /min.) of a moped which starts from rest, is given at fixed intervals of time t (min.) as follows:
2
4
6
8
10
12
14
16
18
20
t:
10
18
25
29
32
20
11
5
2
0
v:
Estimate approximate distance covered in 20 min. by the moped using Simpsons 1/3 rule.
9. Using Picards method, find solution of dy/dx = (1 + xy) up to third approximation, when x0 = 0, y0 = 0.
10. A river is 80 feet wide. The depth d (in feet) of the river at a distance x from one bank is given by the following table:
0
10
20
30
40
50
60
70
80
x:
0
4
7
9
12
15
14
8
3
d:
Find approximately the area of the cross-section of the river using Simpsons 1/3 rule.

LONG QUESTIONS (EACH 15 MARKS)

UNIT-1

1. Evaluate:
2. Evaluate:

dx, using Contour Integration.

()
++

dx, using Contour Integration.

3. (5 + 5 + 5)
(a) If f (z) =
Prove that

()

+
()()

when z 0 and f (z) = 0 when z = 0

along any radius vector but not as in any manner.

(b) If u = 3x2y y3 find the analytic function f (z) = u + iv.

(c) Expand f (z) = ()() in Laurent Series valid for (i) | z 1 | > (ii) 0 < | z 2 | < 1.
4. (5 + 5 + 5)
(a) An Electrostatic field in the X-Y plane is given by the potential function = 3 x2y y3, find the stream
function and hence find complex potential.
(b) Show that the function f (z) =

(+)
+

when z 0 and f (z) = 0 when z = 0

is not analytic at the origin even though it satisfies Cauchy-Riemann equations at the origin.
(c) Evaluate the following integral:

() (+)

dz, Where C is the circle | z | = 2.

5. (5 + 10)
(a) Show that ex (x cos y y sin y) is a harmonic function. Find the analytic function for which ex (x cos y y
sin y) is imaginary part.
(b) If (u + v) = 2 sin (2x) / [ e2y + e-2y 2 cos (2x)] and f(z) = u + iv is an analytic function of z = x + iy, find
f(z) in terms of z.
6. (5 + 10)
(a) Derive Cauchy-Riemann Equations in Polar form and hence deduce:

2u/r2 + u/r + (1/r2) 2u/2 = 0

(b) Define Residues. State and prove Residue Theorem and hence evaluate:

[{ 2 + cos z2} / {(z 1)2. (z 2)}] dz where c is | z | = 3.

7. Evaluate: / (a + b cos )2 where a > b > 0 using Residue Theorem.

8. (7.5 + 7.5)
Page 3 of 8

(a) If f (z) is a regular function of z, show that:

{ |()|} + { |()|} = | f ' (z) |2

(b) Expand f (z) = sin {c ( z + 1/z) }.
9. (5 + 10)

(a) Show that the function u = log(x2 + y2) is harmonic. Find its harmonic conjugate.

(b) Verify Cauchys Theorem for the function f (z) = 3 z2 + iz 4 along the perimeter of square with vertices
(1 i) and (- 1 i).
10. (7.5 + 7.5)
(a) Verify Cauchys Theorem for the function f (z) = eiz along the boundary of the triangle with vertices at the
points (1 + i), (-1 + i) and (-1 i).

(b) Prove that:

d = , where a2 < 1.
UNIT-2

1. (7.5 + 7.5)
(a) State and prove convolution theorem for Fourier Transform.

(b) Find the Fourier cosine transform of f (x) = + . Hence derive Fourier sine transform of (x) = +.
2. (5 + 5 + 5)
(a) Find the Fourier Transform of f (x) = eiwx for a < x < b and f (x) = 0 otherwise.

(b) Find the Fourier Transform of f (x) = , where a > 0.

(c) Find the Fourier cosine transform of f (x) = e-2x + 4e-3x.
3. (7.5 + 7.5)

(a) Find the Fourier sine transform of e - | x | and hence evaluate

(b) Write down Parsevals Identity for sine and cosine transform and hence prove that:

( + )( + )

= (+)

4. (10 + 5)
(a) Solve by Z-transform the difference equation: yk+2 + 6yk+1 + 9yk = 2k ; y0 = y1= 0.
(b) Obtain Fourier cosine transform of f (x) = x,
for 0 < x < 1
2 x,
for 1 < x < 2
0,
for x > 2
5. (10 + 5)
(a) Use Z-transform to solve the difference equation: yn+2 2yn+1 + yn = 3n + 5 ; y(0) = y(1) = 0
(b) Taking the function f (x) = 1 for 0 < x < and f (x) = 0 for x > , show that:

] () = /2, 0 < x <

0, x >

6. If Fc(s) = tan-1 ( ), then find f (x).

7. Using the Z-transform, solve the following difference equation:

yk + () yk-2 = {()k cos (k/2)}; k 0

UNIT-3
1. (10 + 5)
(a) Calculate 1, 2 , 3 , 4 for the frequency distribution of heights of 100 students given in the following table
and hence find the coefficient of skewness and kurtosis:
Page 4 of 8

Height(cm.)
Class Interval
Frequency

144.5149.5
2

149.5154.5
4

154.5159.5
13

159.5164.5
31

(b) The first three moments about the origin are given by '1 =

164.5169.5
32

169.5174.5
15

(+)(+)

(+)

, '2 =

, '3 =

174.5179.5
3

Examine the skewness of the data.

2. The first four moments of a distribution about the value 4 of the variable are -1.5, 17, -30 and 108. Find the
moments about mean, 1 and 2.
Find also the moments about (i) the origin and (ii) the point x = 2.
3. (10 + 5)
(a) Using method of least square, derive the normal equations to fit the curve y = ax2 + bx. Hence fit this curve
to the following data:
1
2
3
4
5
6
7
8
x
1
1.2
1.8
2.5
3.6
4.7
6.6
9.1
y
(b) Find the least squares fit of the form y = a + bx2 to the following data:
-1
0
x
2
5
y

1
3

2
0

4. (7.5 + 7.5)
(a) From the given data find the equation of lines of regression of x on y and y on x. Also calculate the
correlation coefficient.
2
4
6
8
10
x
5
7
9
8
11
y
(b) Calculate the coefficient of correlation between the marks obtained by 8 students in mathematics and
statistics.
A
B
C
D
E
F
G
H
Students
25
30
32
35
37
40
42
45
Mathematics
08
10
15
17
20
23
24
25
Statistics
5. (5 + 10)
(a) The probability that a pen manufactured by a company will be defective is 1/10. If 12 such pens are
manufactured, find the probability that
(i) exactly two will be defective,
(ii) at least two will be defective,
(iii) none will be defective.
(b) Fit a Poisson distribution to the following data which gives the number of yeast cells per square for 400
squares.
0
1
2
3
4
5
6
7
8
9
10
No. of
Total
cells per
square
(x)
103
143
98
42
8
4
2
0
0
0
0
400
No. of
squares(f)
It is given that e-1.32 = 0.2674

Page 5 of 8

6. Out of 8000 graduates in a town, 800 are females, out of 1600 graduate employees, 120 are females, 2 - test
to determine if any distinction is made in appointment on the basis of sex. The value of 2 for 1 degree of
freedom at 5% level is 3.841.
7. To test the effectiveness of inoculation against cholera, the following table was obtained:
Attacked
30
Inoculated
140
Not inoculated
170
Total
(The figure represents the number of persons)

Not attacked
160
460
620

Total
190
600
790

Use 2 -test to defend or refute the statement. The inoculation prevents attack from cholera.
8. In a sample of 1000 cases, the mean of a certain test is 14 and standard deviation is 2.5. Assuming the
distribution to be normal, find
(i) How many students score between 12 and 15?
(ii) How many score above 18?
(iii) How many score below 8?
(iv) How many score 16?
9. Fit a normal curve to the following data:
8.60
8.59
8.58
Length of
line (cm.)
2
3
4
Frequency

8.57

8.56

8.55

8.54

8.53

8.52

10

10. Calculate the coefficient of correlation and obtain the lines of the regression for the following data:
1
2
3
4
5
6
7
8
x
9
8
10
12
11
13
14
16
y

9
15

UNIT-4
1. (10 + 5)
(a) Find the real root of the equation x.log10 (x) = 4.77 correct to 4 decimal places using Newton-Raphson
Method.
(b) Find the real root of the equation x sin(x) + cos(x) = 0 between (2, 3) using Bisection Method.
2. (10 + 5)
(a) Derive Newton-Raphsons Formula and find the rate of convergence of Newton-Raphson Method.
(b) By using Newton-Raphson Method, find the root of x4 x 10 = 0, which is near to x = 2 correct to 3
decimal places.
3. (10 + 5)
(a) Find the root of the equation x3 5 x - 7 = 0 which lies between 2 and 3 by the method of False Position.
(b) Apply Newton-Raphson method to solve 3x cos(x) 1 = 0.
4. (10 + 5)
(a) Find the cube root of 48 correct to 3 decimal places using Newton-Raphson Method.
(b) Find an approximate value of the root of the equation x3 + x 1 = 0 near x = 1, using the method of false
position two times.
5. (10 + 5)
(a) Estimate from the following table the number of students who obtained marks between 40 and 45:
30-40
40-50
50-60
60-70
70-80
Marks
31
42
51
35
31
No. of students
Page 6 of 8

(b) Using Newtons forward interpolation formula find the cubic polynomial which takes on the following
values:
0
1
2
3
x:
1
2
1
10
y:
6. (7.5 + 7.5)
(a) Apply Newtons forward interpolation formula, compute the value of (5.5)1/2 given that (5)1/2= 2.236, (6)1/2=
2.449, (7)1/2= 2.646 and (8)1/2= 2.828, correct upto three places of decimals.
(b) From the following table, evaluate f (3.8) using Newton backward interpolation formula:
0
1
2
3
4
x:
1.00
1.50
2.20
3.10
4.60
f (x):
7. Given the values:
4
5
7
x:
48
100
294
f (x):
Evaluate f (8) using Lagranges Interpolation Formula.

10
900

11
1210

13
2028

8. Given the values of f (x) = sin (x) in the table below, use divided difference to obtain a third degree
polynomial to approximate f (x). Hence find f (0.5) upto fifth place of decimal:
0.1
0.3
0.6
0.8
x:
0.09983
0.29552
0.56464
0.71736
f (x):
9. A varying current in a circuit was found to have the values tabulated below:
0
2
4
6
Time t (sec.)
0
1.960
3.684
4.952
Current i (amp.)
Estimate i when (i) t = 1 sec. (ii) t = 10.5 sec.

8
5.574

10
5.403

10. (10 + 5)
(a) Find the real root of the equation x.log10 (x) = 1.2 by Regula falsi method correct to 4 decimal places.
(b) Using Newton-Raphson Method find an iterative scheme to compute the reciprocal of a positive number.

UNIT-5
1. (7.5 + 7.5)
(a) Solve the following equations using Gauss-Elimination Method:
x + 2y + 3z + u = 3, 4x 6y z u = 27, 3x 2y 3z + 2u = 13, x + y + z u = 3.
(b) Solve the following equations using Gauss-Jordans Method:
10x 7y + 3z + 5u = 6, - 6x + 8y z 4u = 5, 3x + y + 4z + 11u = 2, 5x 9y 2z + 4u = 7.
2. Apply Crouts Method (Factorization Method) to solve the equations:
3a + 2b + 7c = 4, 2a + 3b + c = 5 and 3a + 4b + c = 7.
3. Apply Choleskys Method (Triangularisation Method) to solve the equations:
2p + 3q + r = 9, p + 2q + 3r = 6 and 3p + q + 2r = 8.
4. (10 + 5)
(a) Solve by Jacobis Method:
4x + y + 3z = 17, x + 5y + z = 14, 2x y + 8z = 12.
(b) Find the missing values in the following table using shift operator directly:
Page 7 of 8

x:
y:

0
6

5
10

10
?

15
17

20
?

25
31

5. Solve by Gauss-Seidel Method:

10x 2y z u = 3, - 2x + 10y z u = 15, - x y + 10z 2u = 27, - x y 2z + 10u = - 9.
6. Solve by Gauss-Seidel Method:
83x + 11y 4z = 95, 7x + 52y + 13z = 104, 3x + 8y + 29z = 71.
7. (10 + 5)
(a) Find the first and second derivatives of the function f (x) = (x)1/2 at x = 15 from the following table:
15
17
19
21
23
25
x
3.873
4.123
4.359
4.583
4.796
5.000
(x)1/2
(b) The velocities of a car which starts initially from rest (running on a straight road) at intervals of 2 minutes
are given below:
2
4
6
8
10
12
Time(min.)
22
30
27
18
7
0
Velocity(Km/h)
Apply Simpsons 3/8 Rule to find the distance covered by the car.
8. (7.5 + 7.5)
/

(a) Integrate numerically d.

(b) A rocket is launched from the ground, its velocity is registered during the first 80 seconds and is given in the
table below:
0
10
20
30
40
50
60
70
80
Time(t)
31.63
33.44
35.47
37.75
40.33
43.25
46.69
50.67
Velocity(m/s) 30.00
Find the acceleration of the rocket at time t = 80 sec.

9. Using Runge-Kutta Method of order 4, find y for x = 0.1, 0.2 and 0.3. Given that = xy + y with y(0) = 1.
10. (7.5 + 7.5)
(a) Use Picards method to approximate the value of y when x = 0.1, given that y = 1 when x = 0 and

= 3x + y2 (three approximations).

(b) Apply Runge-Kutta Method (fourth order), to find an approximate value of y when x = 0.2, given that

= x + y2 and y = 1when x = 0 taking h = 0.1 in two steps.

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