Jadavpur University 2007 B.E Mechanical Engineering MATHEMATICS-IIIM - Question Paper
Wednesday, 23 January 2013 09:10Web
Ex/ME/Math/T/121/75/2006
FIRST MECHANICAL ENGG. second SEMESTER exam 2006
MATHEMATICS-IIIM
Time : 3 hours Full Marks: 100
( 50 marks for every group )
Use a separate answer-script for every group.
Group—A
ans any 3 ques..
Two marks for general proficiency.
1. (i) Show that a monotonically increasing
sequence which is bounded above is convergent to its
supremum. 5
(ii) Test the convergence and obtain the limit of
the sequence
8
(iii) obtain
3
2. (i) If
and
then show that { an } is monotone increasing and { bn } is
monotone decreasing and both have the identical limit. 7
[ Turn over
( two )
(ii) Examine the convergence of the subsequent series:
3+3+3
3. (i) Show that the series
is convergent for þ > one and divergent for þ = one , 8
(ii) Show that the series
is convergent. 8
4. (a) obtain the Laplace transform of
(i) t(3 sin 2t?2 cos 2t)
(ii) sin v t
(iii) e–t(3 sinh 2t?5 cosh 2t). 3+4+3
(b) obtain the inverse Laplace transform of
6
5. Solve the subsequent differential equations using
Laplace transform
(i) (D+2)2y = 4e–2t, y(0) = –1, y’(0) = 4
(ii) [tD2+(l–2t)D-2]y=0, y(0)=l,
y’(())= 2. 8+8
( three )
Group—5
ans any 3 ques...
Two marks for general performance.
6.(a) Draw the histogram of the subsequent
frequency distribution and obtain the proportion of firms
with annual sales greater than Rs. 70,000 :
Annual Sales (Rs. ’000) : 0-20 20-50 50-100
No. of firms : 20 50 69
Annual Sales (Rs. ’000): 100-250 250-500 500-1000
No. of firms : 30 25 19
8
(b) You are provided the subsequent incomplete
frequency distribution. It is known that the total
frequency is 1,000 and that the median is 413.11.
Estimate by computation the missing frequencies.
Values : 300-325 325-350 350-375 375-400
Frequency : five 17 80 ?
400-425 425-450 450-475 475-500
326 ? 88 9
8
7. (a) From the subsequent cumulative frequency
distribution of marks found by 22 students compute
(i) Arithmetic mean and (ii) Mode. 8
Marks No. of students
beneath 10 3
” 20 8
” 30 17
” 40 20
” 50 22
[ Turn over
( four )
(b) calculate the standard deviation from the
subsequent distribution of marks found by 90 students:
Marks : 20-29 30-39 40-49 50-59
No. of
students : five 12 15 20
60-69 70-79 80-89 90-99
18 10 six 4
( Use short-cut method for calculation ). 8
8. (a) obtain the 2 lines of regression from the
subsequent data :
Age of husband (x) : 25 22 28 26 35
Age of spouse (y) : 18 15 20 17 22
20
14
22
16
40
21
20
15
18
14
8
(b) (i) Prove that correlation co-efficient does not
depend on the origin and scale of the observations.
(ii) Show that correlation co-efficient lies
ranging from –1 and +1. 8
9- (a) For any 3 events A, B, C of a random
experiment E, prove that
P(A+B+C) = P(A)+P(B)+P(C)–P(AB)-P(BC)
–P(CA) + P(ABC) 6
( five )
(b) Suppose 2 fair dice are thrown Let E
denote the event that the sum of the 2 upturned faces
is six and F denote the event that the upturned face of
the 1st dice is 4. Are the events E and F independent?
Justify your ans. 4
(c) There are 2 identical urns containing
respectively four white and three red balls and three white and
7 red balls. An urn is chosen at random and a ball is
drawn from it. obtain the prabability that the ball drawn
is white. If the ball drawn is white, what is the
probability that it is from the 1st urn? 6
10. (a) The distribution function of a random variable
is described as follows :
F(x) = A 8
=C 0 = x < 2
= D two = x < 8
where A, B, C, D are constants.
If P { X = 0 } =
6
1 and P {X > 1}=
3
2
then obtain the
values of A, B, C, D . 4
(b) obtain the value of k so that the subsequent
function is a probability density function of a continuous
random variable X.
( six )
f(x) = x 0 < x < 1
=k–x 1
compute P
? ? ?
? ? ?
= =
2
3
2
1 X 6
(c) Let X denote the number of white balls
drawn when five balls are drawn without replacement
from an urn containing four white and six black balls. obtain
tho probability mass function of X. 6
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| Earning: Approval pending. |
