LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – APRIL 2008
MT 1500 - ALG.,ANAL.GEOMET. CAL. & TRIGN. - I
Date : 07/05/2008
Time : 1:00 - 4:00
Dept. No.
XZ 1
Max. : 100 Marks
PART – A
Answer ALL the questions.
(10 x 2 = 20 marks)
1. Find the nth derivative of eax.
2. Prove that the subtangent to the curve y  x 2 is of constant length.
1 1
3. Find the coordinates of the center of curvature of the curve y  x 2 at  ,  .
2 4
4. What is the curvature of a (i) circle (ii) straight line.
5. Determine the quadratic equation having (3-2i) as a root.
6. If  ,  ,  are the roots of the equatim x 3  px 2  qx  r  0 . Show that

2
  3r  pq .
7. Prove that Cosh 2 x  Sinh 2 x  Cosh2 x .
1
 1 x 
8. Prove that tanh 1 x  log 
.
2
 1 x 
9. Find the pole of the line x  2ay  0 with respect to the parabola y2=4ax.
10. If e1 and e2 are the eccentricities of a hyperbola and its conjugate,
prove that e12  e2 2  1 .
PART – B
Answer any FIVE questions.
(5 x 8 = 40 marks)
11. At which point is the tangent to the curve x 2  y 2  5 parallel to the line 2 x  y  6  0 ?
12. Final the angle at which the radius vector cuts the curve
r
 1  e Cos .
13. Prove that the radius of curvature at any point of the cycloid x  a (  sin  )

and y  a (1  cos  ) is 4a Cos .
2
14. Show that if the roots of the equation x3  px 2  qx  r  0 are in arithmetic progression then
2 p3  9 pq  27r  0 .
1
15. If tan
x
x
 tan h , show that Cos x .Cos h  1 .
2
2
16. If sin( A  iB )  x  iy , prove that
i)
x2
y2

1
cos h 2 B sin h 2 B
x2
y2
ii)

1
sin 2 A cos 2 A
17. Find the locus of poles of chords of the parabola which subtend a right angle at the focus.
18. Find the equation of a rectangular hyperbola referred to its asymptotes as axes.
PART – C
Answer any TWO questions.
(2 x 20 = 40 marks)
19. a) If y  sin( m sin 1 x) prove that (1  x 2 ) y2  xy1  m2 y  0 and
(1  x 2 ) yn  2  (2n  1) xyn 1  (m 2  n 2 ) yn  0 .
b) Find the (p,r) –equation of the curve
2a
 1  cos and hence show that the radius of
r
curvature at any point varies as the cube of the focal distance.
20. a) Find the equation of the evolute of the parabola y 2  4ax .
b) Solve 3x 6  x 5  27 x 4  27 x 2  x  3  0 .
21. a) Find the real root of x 3  24 x  50  0 to two places of decimals using Horner’s method.
tan x  sin x
.
x 0
sin 3 x
b) Evaluate Lt
22. a) Prove that cos 8  128 sin 8   256 sin 6   160 sin 4   32 sin 2   1 .
b) Derive the equation of the tangent at the point whose rectorial angle is  on the conic

 1   cos  .
r
----------
2