# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034 ```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – April 2009
ZA 01
MT 1500 - ALG.,ANAL.GEOMET. CAL. &amp; TRIGN. - I
Date &amp; Time: 20/04/2009 / 1:00 - 4:00
Dept. No.
Max. : 100 Marks
SECTION – A
10 x 2 = 20
1. Find the nth derivative of sin 3 2x .
2. Define subtangent of a parabola.
3. Write down the formula for radius curvature in terms polar co-ordinates.
l
4. Find the slope of the straight line  cos(   )  e cos  .
r
5. If  ,  ,  are the roots of the equation x3  px 2  qx  r  0 . Find the value of
6. Form the equation, one of whose roots is
1

2
3 5.
7. Prove that sin h 1 x  log e ( x  x 2  1) .
tan 2 x  2 tan x
.
x 0
x3
8. Evaluate lim
9. Find the pole of the line 2x  y with respect to the parabola y 2  2 x .
10. Define polar of a point P with respect to an ellipse.
SECTION – B
5 x 8 = 40
11. Show that the sub-normal at any point of a parabola is constant and equal to half the
latus rectum.
12. Find the co-ordinates of the centre of curvature of the curve
x  a(t  sin t ); y  a (1  cos t ) at ‘t’.
x2 y 2

 1 with pole at its centre.
a 2 b2
3
2
14. Solve the equation x  19 x  114 x  216  0 given that the roots are in GP.
3
15. If  ,  ,  are the roots of the equation x  px  q  0 find the value of
13. Find the pedal equation of the curve
 1
1  1
1  1
1 
 2
 2
 2

r   
r   r   .

1
.
16. Show that the eccentric angles of the extremities of a pair of conjugate diameters of an
ellipse differ by a right angle.
17. Prove that 32sin 4 q cos 2 q = cos 6q - 2 cos 4q - cos 2q + 2 .
18. Find the condition that the lines lx  my  n  0 and l x  m  n  0 may be conjugate
with respect to the hyperbola
x2 y 2

1.
a 2 b2
SECTION – C
2 x 20 = 40
19. a) If y  ( x  1  x 2 )m then prove that (1  x 2 ) yn  2  (2n  1) xyn 1  (n 2  m 2 ) yn  0 .
b) Find the nth derivative of e x log x .
(12+8)
20. a) Find the radius of curvature of
x
y

 1 at the points where it touches the coa
b
ordinate axes.
b) Solve 6 x5  x 4  43x3  43x 2  x  6  0 .
(10+10)
21. a) Calculate to two places of decimals the positive root of the equation
x3  24 x  50  0,
by Horner’s method.
b) Separate into real and imaginary parts tan h ( x  iy ) .
(12+8)
x(a  b cos x)  (sin x)
 1.
22. a) Determine a, b, c such that lim
x 0
x5
b) Prove that the area of the parallelogram formed by the tangents at the extremities of
two conjugate diameters of an ellipse is constant and equal to the product of the exes.
(10+10)
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