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```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – NOVEMBER 2012
MT 1500 - ALGEBRA, ANALY. GEO., CALCULUS & TRIGONOMETRY
Date : 08/11/2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
PART – A
(10 x 2 = 20 marks)
1. Write the nth derivative of y  log (ax  b).
2. If y = a cos(log x)  b sin (log x) show that x 2 y2  xy1  y  O.
3. Define the evolute of a curve.
4. Find the p-r equation of the curve r = a sin .
5. Determine the quadratic equation having 3 – 2 i as a root.
6. Diminish the roots of x 4  5x 3  7 x 2  4 x  5  0 by 2.
7. Show that cosh 2 x  sinh 2 x  1.
8. Express sinh 1 x in locus of logarithmic function.
9. Define a rectangular hyporbola.
10. Write down the angle between the asymptotes of the hyperbola
x2 y2

 1.
a2 b2
PART – B
(5 x 8 = 40 marks)
11. Show that in the parabola y 2  4ax the subtangent at any point is double the abscissa and the
subnormal is a constant.
12. Find the radius of curvature at the point ‘O’ on x  a(cos    sin  ); y  a(sin    cos ) .
13. Show that if the roots of
x 3  px 2  qx  r  o are in arithmetic progresssi on then 2p 3  9 pq  27r  O.
14. Find the p-r equation of the curve
2a
 1  cos with respect to the focus as the pole.
r
15. Separate tan( x  iy ) into real and uniaguinary parts.

16. Find the sum of the series

1
sin n
.
2n
17. Find the locus of poles of ale Laugets to y 2  4ax with respect to y 2  4bx.
18. Derive the polar equation

1  e cos  of a comic.
r
PART – C
( 2 x 20 = 40 marks)

1

19. a) If y  ea sin x prove that 1  x 2 y n 2  (2n  1) xyn1  (n 2  a 2 ) y n  0.
b) Show that r = a sec2

2
and r = b cosec2

2
intersect at right angles.
20. a) Find the minimum value of x 2  5 y 2  6 x  10 y  12.
 3a 3a 
b) Find the radius of curvature of x 3  y 3  3axy at  ,  .
 2 2
21. a) Solve: x 3  19 x 2  114 x  216  0 given that the roots are in geometric progression.
b) Solve: 6 x 5  11x 4  33x 3  33x 2  11x  6  0 .
22. a) Express cos8 in locus of power of sin.
b) If e1 and e2 are the eccentricities of a hyperbola and its conjugate show that e12  e22  1 .
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