Veer Narmad South Gujarat University 2010 B.E EnggMathematics & statistical Method - 2 ( Sem3 )( Civil ) - Question Paper
RN-6149
B. E. II (Sem. Ill) (Civil) Examination May / June - 2010 Engg. Mathematics & Statistical Method
Time : 3 Hours]
[Total Marks : 100
Instructions :
Seat No.:
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Fillup strictly the details of signs on your answer book.
Name of the Examination :
B. E. 2 (Sem. 3) (Civil)
Name of the Subject:
Engg. Mathematics & Statistical Method
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-Subject Code No. |
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-Section No. (1,2,.....) |
Student's Signature
1&2
(2) All questions are compulsory.
(3) Write each section in separate answerbooks.
SECTION - I
1 (a) Do as directed :
10
(1) Define divergence of a vector function.
(2) Give the relation between cartesian and spherical co-ordinates.
a jay
0 0
(4) Define beta function and show relation between beta and gamma function.
axx2
2 a
(5) Convert into polar form J J x2dydx
0 0
(b) Any three :
12
4 a 2 /ax
(1) Evalaute J J dydx by changing the order of
0 x2/
integration.
a x x+y
(2) Evaluate J J J* ex+y+zdz dy dx . 0 0 0
(3) Evaluate 11 , where R is the triangle
R
with vertices (0,0), (10, l), (l,l).
(4) Find the area enclosed by the curve r- sin 30.
2 (a) Attempt any one : 6
(1) Verify Green's theorem for f{x2 + y2j/-2xyj and C
is the rectangle in xy-plane bounded by y=0, y=b, x=0, x=a.
(2) Verify stokes theorem for F = yi + zj + xk, where S is
OOO
the upper half of the sphere x +y +z = 1 and C is its boundary.
(b) Attempt any two : 8
(1) Find the directional derivative of ty = 3e2x~y+z at
A( 1,1,-l) in the direction where B is the point (-3,5,6).
(2) A vector field is given by F = {x2 + xy2i + {y2 + x2yj
show that p is irrotational and find its scalar potential.
(3) Prove that divlgrad rn\ = n(n + \)rn~2.
3 (a) State and derive duplication formula for beta and 4 gamma function.
(b) Attempt any two : 10
\ dx J x
(1) Evaluate
V 7t/
dQ
(2) Evaluate j / j -7sin0 .
o 'Jsm6 0
r dx
(3) Evaluate J
o V l-x4
RN-6149] 2 [Contd...
(a) Do as directed : 10
(1) Find one solution of pz = qz = z2 + (x + y)2.
(2) Write an IBVP for wave equation.
(3) Write the properties for a probability function for a continuous random variable.
x2
(4) Check whether = x = 0,1,2,3,4 is a
probability function.
(5) Find P{-\<z<\).
(b) Explain and solve one dimensional wave equation. 8
(b) Determine the solution of one-dimensional heat 8
equation = c , where the boundary conditions are
u(0,t) = 0, u(l,t) = 0 (t>0) and u(x,0) = x, I being the
length of the bar.
(a) Solve any two : 6
(1) xp + yq = 3z
(2) (y-z)p + (x-y)q = z-x
(3) ptdLnx + qtmy = tanz
(b) Attempt any two : 10
(1) Six dice are thrown 729 times. How many times do you expect at least three dice to show a five or six ?
(2) Assume that the probability of an individual coalminer being killed in a mine accident during a
1
year is tttt. Use Poisson's distribution to calculate J 2400
the probability that in a mine employing 200 miners there will be at least one fatal accident in a year.
(3) A sample of 100 dry battery cells tested to find the length of life produced the following results x=12 hours, a = 3 hours.
Assuming the data to be normally distributed, what percentage of battery cells are expected to have life more than 15 hours. (S.N.V.Z. area between 0 and
1 is 0.3413)
2
application of X test.
(b) Attempt any two : 12
(1) The following are measurements of the air velocity (x cm/sec) and evaporation coefficient of burning fuel droplets in an impulse engine :
|
X |
20 |
60 |
100 |
140 |
180 |
220 |
260 |
300 |
340 |
380 |
|
y |
.18 |
.37 |
.35 |
.78 |
.56 |
.75 |
1.18 |
1.36 |
1.17 |
1.65 |
Fit a straight line to these data.
(2) The following table gives the number of accidents that took place in an idustry during various days of the week. Test if the accidents are uniformly distributed over the week.
|
Day |
Mon |
Tue |
Wed |
Thur |
Fri |
Sat |
|
No. of accidents |
14 |
18 |
12 |
11 |
15 |
14 |
(3) The heights and weights of five students are given below
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Find correlation coefficient. |
RN-6149] 4 [ 100 ]
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Attachment: |
| Earning: Approval pending. |
