# LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – APRIL 2012
MT 1500 - ALGEBRA, ANALY. GEO., CALCULUS &amp; TRIGONOMETRY
Date : 28-04-2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
PART - A
Answer ALL the questions:
(10 X 2 = 20 Marks)
1
1. Find the nth derivative of
.
ax  b 2
2. Find the slope of the straight line
l
 cos     e cos .
r
3. Write the formula for the radius of curvature in Cartesian form.
4. Define Cartesian equation of the circle of the curvature.
2
5. If  ,  ,  are the roots of the equation x3+px2+qx+r=0. Find the value of  .
6. Diminish the roots x4+x3-3x2+2x-4 =0 by 2.
tan 2 x  2 tan x
.
x 0
x3
7. Evaluate Lt
8. Prove that cosh x 
1  tanh 2 x
.
1  tanh 2 x
9. Define Pole and Polar of a ellipse.
10. In the hyperbola 16x2-9y2 = 144, find the equation of the diameter conjugate to the diameter x =2y.
PART - B
Answer any FIVE questions:
(5 X 8 = 40 Marks)
2x 1
11. Find the nth derivative of
.
2 x  12 x  3
12. Find the angle between the radius vector and tangent for the curve r  a1 cos  at


2
.
13. Solve the equation x3-4x2-3x+18=0 given that two of its roots are equal.
14. Solve the equation x4-5x3+4x2+8x-8=0 given that 1- is a root.
15. Expand cos 6 in terms sin  .
16. Separate real and imaginary parts tan 1 x  iy  .
17. P and Q are extremities of two conjugate diameters of the ellipse
PQ 2  SP  SQ   2b 2 .
x2 y2

 1 and S is a focus. Prove that
a2 b2
2
18. The asymptotes of a hyperbola are parallel to 2x+3y=0 and 3x-2y =0 . Its centre is at (1,2) and it passes
through the point (5,3). Find its equation and its conjugate.
PART - C
Answer any TWO questions:

19. (a) If y  x  1 x 2

m
(2 x 20=40 Marks)


, show that 1  x 2 y2  xy1  m 2 y  0.
(b) Prove that the sub-tangent at any point on y  be
x
a
is constant ant the subnormal is
y2
.
a
(10 +10)
20. (a) Find the radius of curvature at any point on the curve r 2  a 2 cos 2 .
(b) Show that the evolute of the cycloid x  a  sin  , y  a1  cos  is another
cycloid .
(10+10)
21. (a) Solve 6x5+11x4-33x3-33x2+11x+6=0.
(b) Find by Horner’s method, the roots of the equation x3  3x  1  0, which lies between 1 and 2
correct to two decimal places.
(10+10)
22. (a) Prove that 32 cos 6   cos 6  6 cos 4  15 cos 2  10.
(b) Prove that the Product of the perpendicular drawn from any point on a hyperbola to its
asymptotes is constant.
(10+10)
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