Kerala University 2005 B.Sc Mathematics group 1 - Question Paper
Name........................................................
FIRST YEAR B.Sc. DEGREE EXAMINATION, MARCH/APRIL 2005
Part IIIGroup IMathematics Paper ICALCULUS, ANALYTIC GEOMETRY AND TRIGONOMETRY
Time : Three Hours Maximum : 65 Marks
Maximum marks tha& can be earned from each unit is 13.
Each question carries 5 marks.
1. Ify = (x2 - 1)", show that (x2 - 1) yM + 2 + 2 xyn +1 - n (n + 1) yn 0.
2. State Rolle's theorem. Verify the theorem for the {traction f(x) = jc (x + 3) e~*12.
3 5
<-1 XX
3. Using Maclaurin's series, prove that tan x = x -
3 5
du du
4. State Euler's theorem on homogeneous functions and apply it to show that * dx *y dy
where u = sin-1
x + y
Unit II
(1 1) X 2 y
5. Find the radius of curvature of the curve x + y* = xy at the point
2 2 Y y
6. Find the evolute of the ellipse = 1.
2 *2 a o
7. Show that the points of inflexion of the curve y2 = (x- m)2 (x - n) lies on the line 3 x + m = 4 n.
8. Find the envelope of the family of straight lines x cos a + y sin a = I sin a cos a, the parameter being a.
Unit in
9. Define polar of a point with respect to a conic. Show that the locus of poles of normal chords of the parabola y2'' = 4 ax is (* + 2 a) y2 + 4 a = 0.
2 2
10. Show that the locus of middle points of chords of the ellipse r- + = 1 touching the circle
2 b
( 2 2 fL + 2,2 a b
11. Prove that the tangents at the ends of a pair of conjugate diameter, of an ellipse form a parallelogram of constant area.
12. The asymptotes of a hyperbola are parallel to the lines x + 2y 7 = 0 and 3 ae + 2 ;y - 1 = 0 and the centre is at (1, 1). If the hyperbola passes through (2, 5), find its equation. Find also the equation to the conjugate hyperbola.
Unit IV
13. Show that the general second degree equation ax2 + 2hry + by2 + 2gx + 2fy + c=0 represents a parabola if ab h2 = 0.
14. Show that 5 x2 - 6 xy + 5 y2 + 22 * 26 y + 29 = 0 represents an ellipse and find its centre. Obtain the equation to the ellipse referred to its centre as new origin and axes parallel to the original axes.
15. Find the polar equation of tangent at any point of the conic - = 1 + e cos Q.
16. Find the polar equation of a circle whose centre is (y,, 0j) and radius q. Deduce the equation of the circle passing through the point.
UnifcV
17. Expand cos6 0 sin2 0 in a series of cosines of multiples of 0.
18. Resolve into real factors x1 + 1.
1 13
19. Find the sum to infinity of the series sin a + sin (a + J3) + sin (a + 2p) + . - .
2 2.4
20. If tan (A + i B) = x + iy, prove that x2 + y3 + 2x cot 2A = 1.
|
Attachment: |
| Earning: Approval pending. |
