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

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

'2'

2    C 36089

n r* 2    , d²y_ k¹ - ab

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)y_n+₂ + (2n + 1) xy_n+1 + (n² - m²) y_n = 0 . (5 marks)

11.    Find c of the mean value theorem where f (x) = x - 3x² + 2x in

12.    Obtain the expansion of log (1 + sin x) in the form

x    + + *² +

^X~~2 6 ~ 12 24 ⁺'

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.

, -n/i-Jy , , 3u du _n

17.    If u = sin    , show that x + y - = 0 .

+ -Jy    ox ay

(4 marks)

1

(u,v) 3(x,y) 0(!/,u)

2

(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: calicut-university-2004-b-sc-computer-science-31735-1.jpg calicut-university-2004-b-sc-computer-science-31735-2.jpg calicut-university-2004-b-sc-computer-science-31735-3.jpg

Earning:   Approval pending.