Bharati Vidyapeeth 2007 B.E Biotechnology maths - Question Paper

Visvesvaraya Technological University
Model ques. Paper – I
B.E. exam
ENGINEERING MATHEMATICS-IV (06MAT41)
(Common to all Branches other than BT)
Time: 03 hours Max. Marks: 100
Note: ans any 5 ques. choosing atleast 2 from every part
PART-A
1. (a) Employ Taylor’s series method to obtain an approximate solution accurate to 4th decimal
place for the subsequent initial value issue at x = 0.1 and 0.2
2 three ; (0) 0 x dy
y e y
dx
= + =.
(b) Apply Runge-Kutta method to obtain an approximate value of y for x = 0.2 in steps of
x = 0.1, provided that two ; with (0) 1
dy
x y y
dx
=+ =.
(c) Using Milne’s predictor-corrector method obtain y when x = 0.8, provided two ;
dy
x y
dx
= -
(0) 0, (0.2) 0.02, (0.4) 0.0795, (0.6) 0.1762 y y y y = = = = . Apply corrector formula
twice. (7+7+6)marks
2. (a) describe analytic function. Derive Cauchy-Riemann equations in polar form.
(b) obtain an analytic function two 2 () , where =( )( four ). f z u iv u v x y x xy y =+ - - + +
(c) obtain the Bilinear transformation which maps the points 0, 1, z= ¥into the points
5, 1,3 w =- - respectively. elaborate the invariant points under this transformation?
(7+7+6) marks
3. (a) State and prove Cauchy’s integral formula.
(b) Expand
1
( )
( 1)(2 )
f z
z z
=
- -
as a Laurent’s series valid for () one ( )1 two i z ii z < < < .
(c)Using Cauchy’s residue theorem evaluate
( )3
cos
,
2 C
z z
dz
z p - ò where C : one 1 z- = .
(7+7+6) marks
4. (a) find the series solution of the equation:
2
2 nine (1 ) 12 four 0
dy dy
xx y
dx dx
- - + = .
(b) find the series solution of Bessel’s differential formula in the form
() ( ) n n y AJ x BJ x - = + .
(c) Prove that two 1
() {( 1) }
2 !
n
n
n n n
d
Px x
n dx
= - , where n is a positive integer.
(7 + seven + 6)marks
PART-B
5. (a) obtain a 2nd degree polynomial in the lowest square sense that fits to the subsequent data
x 0 one two three 4
y one 1.8 1.3 2.5 6.3
(b) obtain the correlation coefficient and regression lines of y on x and x on y for subsequent
data :
x one two three four 5
y two five three eight 7
(c) In a factory, machines A, B, C produce respectively 25%, 35%, and 40% of the total
production. Of which 5%, 4% and 2% are defective. An item is drawn at random was obtained
to be defective. obtain the probability that it was manufactured by A. (7 + seven + 6)marks
6.(a) describe random variable .A random variable(X=x) has the subsequent probability

Earning:   Approval pending.