Calicut University 2006 B.Sc Mathematics CH-DIFFERENTIAL
SECOND YEAR B.Sc. DEGREE EXAMINATION, MARCH / APRIL 2006
Part III - Mathematics (Main)
PAPER II-DIFFERENTIAL AND INTEGRAL CALCULUS
15853 (Pages : 2) Name......................................
Reg. No..................................SECOND YEAR B.Sc. DEGREE EXAMINATION, MARCH/APRIL 2006
Part IIIMathematics
Paper IIDIFFERENTIAL AND INTEGRAL CALCULUS
mo : Three Hours Maximum : 60 Marks
Maximum marks that can be earned from Unit I is 15, Unit II is 15,
Unit 111 is 10 and Unit IV is 20.
Unit I
/
Find the radius if curvaturc of the curve y2 = 1 Ojc - 6 at ( 1 , 2 ) (4 marks )
2. Find the radius if curvature of the curve x = a Cos Q , y = b Sin 6 (4 marks )
3 Show that the evolute of the hyperbola x = a sec 0 , y = b tan 6
1 I , -.2
is (ax)i -{by)) = [a2 +b2)* (5 marks)
v_,4.' Prove that the asymptotes of x2y2 = c2(x2 + y2) are the sides of a square
(3 marks)
Find the position and nature of the double point on the curve .v3 + y* - 3axy = 0 (4 marks )
Draw the curve y x'\x - i) (5 marks )
Unit II ''I
2
Find the reduction formula for Jsin xdx where n is a positive integer7.
sj
and hence evaluate Jsin4 xdx
(4 marks)
(4 marks ) (4 marks )
(4 marks) (4 marks ) (5 marks)
(4 marks ) (4 marks)
Turn over .ft
0
8. Find the area of the ellipse
9 Prove that the perimeter of the cardioid r = a ( 1 + cos Q ) is 8 a
10. Find the volume of the solid obtained by evolving the cardioid r = a ( 1 + cos 9 ) about the initial line
11. Find the area of the surface generated by revolving the curve
y = 2 %/x , 1 < x < 2 about the x - axis
12. Find the Moment of Inertia of the area bounded by the curve r2 = a2 cos2d about its axis
Unit III
x
13. Using Maclaurins theorem find
cosx
3 3 5 3 5 7
14. Sum the infinite series 1 + + +-+
4 4 8 4.8.12
U
i-.
(4 marks) (5 marks)
(4 marks) (5 marks )
(5 marks)
(4 marks ) (5 marks)
(5 marks )
V5 iaulklj )
-i
18. If u = sin
o'u
dy
-1>
dz'
' i/> < 2 15 Sum the series T" o/J + 3 n\
16. Show that log, x - -- + - + ---L +---
x + 1 2(* + 1 Y 3(x + l )3
Unit IV
17. If z = ( y2 - x') log ( x - y ) prove that y + x - x2 - y2
dx dy
20. Ifz = x2y + y3,x = log t , y = e 1, find - as a function of t
di . ,
21. Find the minimum value of the flinction u = xJ+xy+y2+3x + 4
22. Find the maximum value of-' y -- y 2 subject to the condition 2 x + y - 6 0
19. If u is a homogeneous polynomial of degree n in x and y prove that
->I -n2 -,2
v u on au ,
dtj ck: I
-== prove that x--vy - = tan u
Jx + yjy dx dy ?
1 I-X
23. -Evaluate, Jl_j" (x2 +_y: \kcty
T + yXT + zT~i' = n(n
x -f v
dx'
*--<> y-.-0
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Attachment: |
| Earning: Approval pending. |
