Calicut University 2004 B.Sc Computer Science analytical geometry and calculus - Question Paper
c36089
C 30089 (Pages : 3)
Name. .Sl.Jii......Reg. No.....................
SECOND YEAR B.Sc. DEGREE EXAMINATION, MARCH/APRIL 2004
Part IIIMathematics (Subsidiary)
Paper IANALYTIC GEOMETRY AND CALCULUS (2001 AdmissionsImprovement)
Time : Three Hours Maximum : 65 Marks
Maximum marks that can be obtained from Unit I is 20, Unit II is 30 and Unit III is 15.
Unit I (Analytic Geometry)
1. Translate the equation x2 + y2 + z2 - 4z into cylindrical and spherical polar co-ordinates.
(4 marks)
2. Find the angle through which the axes must be turned so that the expression ax2 + 2hxy2 + by2 may become an expression in which there is no term involving XY.
(4 marks)
3. Find the foci and equations of the directrices of the hyperbola 4x2 - 9y2 ~ 8x - 18y -41 = 0-
(5 marks)
4. Prove that the foot of the perpendicular from the focus to any tangent to the parabola y2 = 4ajc lies on the tangent at the vertex.
(4 marks)
5. Show that the locus of the point of intersection of perpendicular tangents to the hyperbola
*2 y2
~Y ~ 7T ~ is the director circle x2 + y2 = a2 - b2 a b J
(5 marks) (4 marks) (4 marks)
6. Derive the equation of the conic in polar coordinates.
7. Discuss and sketch the locus of y2 + z2 = 4x-
Unit II (Differential Calculus)
8. Differentiate w.r.t. x, the following :
(a) cos'*1 (tanh x).
(b) 2 cosh 1 ( ) + sinh 1
2
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'2'
2 C 36089
9. If ax + 2hxy + by = 1, show that ~7~j ~
dx* {hx + by)2 ' (4 marks)
10. If y = j* + Vl + x2 j , prove that (1 + x2)yn+2 + (2n + 1) xyn+1 + (n2 - m2) yn = 0 . (5 marks)
11. Find c of the mean value theorem where f (x) = x - 3x2 + 2x in
(4 marks)
12. Obtain the expansion of log (1 + sin x) in the form
X~~2 6 ~ 12 24 +'
(5 marks) (4 marks)
13. Evaluate lim (sec 0 - tan 0). e /2
O A
14. Determine the constants a and 6 so that the curve y = x + ax + bx has an inflexion at the point (3, - 9).
(4 marks)
15. Find the co-ordinates ,of the centre of curvature at 0 on the curve x = a cos 0, y ~ b sin 6.
(5 marks)
16. Find all the asymptotes of the curve y3 - 6xy2 + llx2y - 6*3 + x + y = 0.
(5 marks) (5 marks)
, -n/i-Jy , , 3u du n
17. If u = sin , show that x + y - = 0 .
+ -Jy ox ay
1
2
1
2
Test the continuity at the point x ~
fix) = |
|
1 2 |
(4 marks)
(u,v) 3(x,y) 0(!/,u)
(5 marks)
19. Let f (x) be defined in [0, 1] as follows
Unit III (Integral Calculus)
2
1
20. Evaluate by Simpsons rule J dx, where h - -1, correct to four decimal places. (5 marks)
i *
21. Obtain a reduction formula for J sec*x dx. (5 marks)
22. Find the area enclosed by the cardiod r = a (1 + cos 0). (5 marks)
23. Find the perimeter of the curve r = a cos 0. (6 marks)
24. Show that the surface area of the solid generated by revolving the curve** = a cos3 8,
3 1*2 2 y - a sin 0 about the x-axis is to* .
5
(5 marks)
Attachment: |
Earning: Approval pending. |