LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034
B.Sc., DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – NOVEMBER 2004
MT - 1500/MAT 500 - ALGEBRA, ANAL. GEOMETRY,
CALCULUS & TRIGONOMETRY
01.11.2004
Max:100 marks
1.00 - 4.00 p.m.
SECTION - A
Answer ALL Questions.
1.
2.
3.
4.
5.
(10 x 2 = 20 marks)
If y = sin (ax + b), find yn.
Show that in the parabola y2 = 4ax, the subnormal is constant.
Prove that cos h2x = cos h2x + sin h2x.
Write the formula for the radius of curvature in polar co-ordinates.
Find the centre of the curvature xy = c2 at (c, c).
lim
6. Prove that θ  0
sin a θ a
 .
sin b θ b
7. Form a rational cubic equation which shall have for roots 1, 3 
2 .
8. Solve the equation 2x3 - 7x2 + 4x + 3 = 0 given 1+ 2 is a root.
9. What is the equation of the chord of the parabola y2 = 4ax having (x, y) as mid - point?
10. Define conjugate diameters.
SECTION - B
Answer any FIVE Questions.
(5 x 8 = 40 marks)
11. Find the nth derivative of cosx cos2x cos3x.
12. In the curve xm yn = am+n , show that the subtangent at any point varies as the abscissa of
the point.
13. Prove that the radius of curvature at any point of the cycloid
θ
x = a ( + sin ) and y = a (1  cos ) is 4 a cos .
2
m
m
14. Find the p-r equation of the curve r = a sin m .
lim θ (a  b sin θ)  c sin θ
1 .
θ5
15. Find the value of a,b,c such that θ  0
16. Solve the equation
6x6  35x5 + 56x4  56x2 + 35x  6 = 0.
17. If the sum of two roots of the equation x4 + px3 + qx2 + rx + s = 0 equals the sum of the
other two, prove that p3 + 8r = 4pq.
18. Show that in a conic, the semi latus rectum is the harmonic mean between the segments
of a focal chord.
1
SECTION -C
Answer any TWO Questions.
(2 x 20 = 40 marks)
1
19. a) If y = e a sin x , prove that
(1  x2) y2  xy1  a2y = 0.
Hence show that (1  x2) yn+2  (2n +1) xyn+1  (m2 + a2) yn = 0.
(10)
b) Find the angle of intersection of the cardioid r = a (1 + cos ) and r = b (1  cos ).
(10)
20. a) Prove that
cos 7θ
= 64 cos6   112 cos4 + 56 cos2  7.
cos θ
(12)
1  tan hx
 cos h 2 x  sin h 2 x.
1  tan hx
21. a) If a + b + c + d = 0, show that
b) Show that
(8)
a 5  b5  c5  d 5 a 2  b 2  c 2  d 2 a 3  b3  c3  d 3

.
.
5
2
3
(12)
b) Show that the roots of the equation x3 + px2 + qx + r = 0 are in Arithmetical
progression if 2 p3  9pq + 27r = 0.
(8)
22. a) Prove that the tangent to a rectangular hyperbola terminated by its asymptotes is
bisected at the point of contact and encloses a triangle of constant area.
(8)
b) P and Q are extremities of two conjugate diameters of the ellipse
a focus. Prove that PQ2  (SP  SQ)2 = 2b2.
x2
a2

y2
b2
1 and S is
(12)
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