This document contains instructions and questions for an Applied Mathematics exam for a Mechanical Engineering course. It includes 12 multi-part questions covering various mathematical and statistical topics like solving equations numerically, calculus, series solutions, transforms, probability, and random variables. Students are instructed to solve 6 of the 12 questions and assume data if necessary. The use of calculators and tables is permitted.

1. TKN/KS/16/7371 1 P.T.O
B.E. (Mechanical Engineering) Fourth Semester (C.B.S.)
Applied Mathematics - IV Paper - I
P. Pages : 3 TKN/KS/16/7371
Time : Three Hours Max. Marks : 80
_____________________________________________________________________
Notes : 1. All questions carry marks as indicated.
2. Solve Question 1 OR Questions No. 2.
3. Solve Question 3 OR Questions No. 4.
4. Solve Question 5 OR Questions No. 6.
5. Solve Question 7 OR Questions No. 8.
6. Solve Question 9 OR Questions No. 10.
7. Solve Question 11 OR Questions No. 12.
8. Assume suitable data whenever necessary.
9. Use of non programmable calculator is permitted.
10. Use of Normal Distribution Table is Permitted.
1. a) Find a positive root of 2x-3 sin x = 5 correct to four decimal places using false position
method.
6
b) Apply Gauss-Seidal method to find the solution of the following system:
10x + y + z = 12
2x + 10y + z = 13
x + y + 5z = 7
6
OR
2. a) Find an iterative formula to find the reciprocal of a given number N and hence find the value
of 19
1 , by using Newton's Raphson Method.
6
b) By Crout's method, solve the system:
2x + 3y + z = -1
5x + y + z = 9
3x + 2y + 4z = 11
6
3. a) Apply the fourth order Runge-Kutta method to find y (0.2) given that 1y,yxy 0  ,
taking h = 0.1.
7
b) Find the largest Eigen value and corresponding Eigen vector for the matrix.











324
202
423
A
By iteration method.
7
OR
4. a) Use Modified Euler method to obtain y (0.2), given
    1.0h,10y,yxlog
dx
dy
 .
7
b) Using Milne's method find y (4.4) & y (4.5), if 02yxy5 21
 , given y (4) = 1, y (4.1) =
1.0049, y (4.2) = 1.0097, y (4.3) = 1.0143.
7
*0479*
NagpurStudents.Org

2. TKN/KS/16/7371 2
5. a) If Z [f (n)] = F (z), then prove that Z [n f (n)] =  zFz
dz
d , hence find  2
nZ . 6
b)
Find
   








2z21z3
2z91
Z .
6
OR
6. a) Find Z-transform of sin (2n+3) and cos (2n+3). 6
b) Solve by using Z-transform-
0nfor0yand1ythatgiven,uy2y3y n0nn1n2n   .
6
7. a) Solve in series by Frobenius method.
  0y412x1x9
dx
dy
2dx
y2d
 .
7
b) Prove that
i)     xJxxJx 1n
n
n
n
dx
d

ii)     xJxxJx 1n
n
n
n
dx
d



7
OR
8. a) Show that     





 



x
0
n
n
1n|n2
1
1n
n
xJxdxxJx . 7
b)
If  





 then
1x0,x
0x1,0
xf show that
        .....xPxPxPxP 433
3
216
5
12
1
04
1 
7
9. a)   x
3CxfLet  , x = 1, 2, 3, ..... is the probability function for a discrete random variable
X. Determine
i) The constant C
ii) The distribution function &
iii)  3xP  .
6
b) Let X & Y be continuous r.v. having density function.
 
 







Otherwise,0
2y0
2x0,yxc
y,xf
22
Determine
i) The constant C
ii)  41Y,41XP 
iii)  23x21P 
iv)  23YP  .
8
OR
NagpurStudents.Org

3. TKN/KS/16/7371 3 P.T.O
10. a) Let X and Y be random variables having joint density function
 
 


 


Otherwise,0
2y0,1x0,
y,xf 5
yxx3
Find E (x), E (y), Var (x), Var (y).
8
b) Let X be a random variable having density function
 
 


 

Otherwise,0
3x0,x9x
xf
2
81
4
Find its moments generating function and first four moments about origin.
6
11. a) If 3% of electric bulbs manufactured by a company are defective, find the probability that
in a sample of 100 bulbs
i) Exactly 2
ii) Between 1 & 3
iii) Atmost 2 bulbs will be defective
7
b) A continuous random variable X is uniformly distributed having probability density
function.
 


 
 
Otherwise0
bxa;
xf ab
1
Find its
i) Mean
ii) Variance and
iii) Moment generating function
7
OR
12. a) The mean grade on a final examination was 72 and the S.D. was 9. The top 10% of the
students are to received A's. What is the minimum grade a student must get in order to
receive an A?
7
b) The time required to repair a machine is exponentially distributed with parameter 21 .
What is the probability that the repair time exceeds 2 hrs ?
7
*********
NagpurStudents.Org

4. TKN/KS/16/7371 4
NagpurStudents.Org