Vinayaka Missions University 2008 B.Sc Mathematics CALCULAS : t : Correspondence - Question Paper

COURSE CODE – 1030502
UG DEGREE exam – SEP 2008
B.SC (MATHS / MPC)
CALCULAS
(For Candidates Admitted from Calendar 2007 Onwards)
Time: three Hours Max Marks: 75
part – A
ans all the Questions: 15X1=15
1. Write the nth derivative of log ( ax + b ).
2. describe Limits?
3. Write down the formula for curvature?
4. Write down the formula for Leibnitz’s theorem?
5. State the cauchy’s mean value theorem?
6. elaborate successive differentiation?
7. Example for continuity?
8. Write down the L.Hospital rule?
9. provide an example for homogenous equations?
10. describe functions of 2 variables?
11. describe minima?
12. Write down the Bernoulli’s formula?
13. Write any 2 properties of definite integral?
14. Example for double integral?
15. Write down the formula for length of the curve from the point
( x, y ) to ( x2
, y2 )?
parts – B
ans any 5 Questions: five X six = 30
16. a) If y = a cos ( log x ) + b sin ( log x ). Show that
d2y dy
x2 ----- + x ---- + y = 0
dx2 dx
(Or)
(b). obtain the envelopes of the ellipse x2 y2
---- + ---- =1
a2 b2
17. (a). State and prove Taylor’s theorem.
(Or)
(b). obtain the nth derivative
i. sin3 2x
ii. 1
-----------
( ax + b )2
18. (a). obtain the maxima or minima of sin x sin y sin ( x + y )
(Or)
Tan- 1y tan -1x
(b). If u = x2 -- -- y2 ---- prove that
x y
?2u x2 - y2 ?2u
-------- = ---------- = ----------
?x ?y x2 + y2 ?y ?x
2a a a
19. (a). Prove that ? f(x)dx = ? f(x)dx + ? f(2a - x)dx
0 0 0
(Or)
a x x+y
(b). Evaluate ? ? ? e
x+y+z dz dy dx
0 0 0

p/2 v sin x
20. (a). Evaluate ? dx
0 v sin x + v cos x
(Or)
(b). obtain the length of the curve 3ay2 = x ( x – a ) 2
part – C
ans any 2 questions: two X 15 = 30
21. (a). obtain the envelope of the circle whose diameters are double
ordinates of a parabola.
(b). obtain the nth derivative of sin5 x cos4 x
22. If y = ( x v one + x2 ) m , prove that
( one + x2 ) yn+2 + ( 2n + one ) xy n +1 + (n2 – m2 ) yn = 0
23. State and prove Euler’s theorem p/2
24. (a). Reduction formula or ? xn sin x dx
0
(b). Write the formula for length of an are in cartesian
co ordinates and polar coordinates
25. (a). obtain the area of the Surface of the sphere of redius a
(b). Solve ? x three cos x/2 dx by using Bernouilli’s formula.

Earning:   Approval pending.