```LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
FIRST SEMESTER – NOVEMBER 2012
MT 1503 - ANALYTICAL GEOMETRY OF 2D,TRIG. & MATRICES
Date : 10/11/2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
PART - A
(10 X 2 = 20)
1. Write down the expression of cos 4θ in terms of cosθ and sinθ.
2. Give the expansion of sinθin ascending powers of θ.
3. Express sin ix and cosix in terms of sin hx and coshx.
4. Find the value of log(1 + i).
 3 2
5. Find the characteristic equation of A = 
.
 2 3
6. If the characteristic equation of a matrix is 3  182  45  0 , what are its eigen values?
7. Find pole of lx + my + n = 0 with respect to the ellipse
x2 y2

 1.
a2 b2
8. Give the focus, vertex and axis of the parabola x 2  4ay.
9. Find the equation of the hyperbola with centre (6, 2), focus (4, 2) and e = 2.
10. What is the polar equation of a straight line?
PART – B
ANSWER ANY FIVE QUESTIONS.
(5 X 8 = 40)
11. Expandcos6θ in terms of sinθ .
12. If sinθ = 0.5033 show thatθ is approximately 30 13'6" .
13. Prove that sinh 1 x  log e ( x  x 2  1) .
14. If tany = tanα tanhβ ,tanz = cotα tanhβ, prove that tan (y+z) = sinh2βcosec2α.
 2 1
15. Verify Cayley Hamilton theorem for A = 
.
1 3 
16. Prove that the eccentric angles of the extremities of a pair of semi-conjugate diameters of an ellipse
differ by a right angle.
17. Find the locus of poles of all tangents to the parabola y 2  4ax with respect to y 2  4bx.
18. Prove that any two conjugate diameters of a rectangular hyperbola are equally inclined to the
asymptotes.
PART – C
ANSWER ANY TWO QUESTIONS:
(2 X 20 = 40)
19. (i) Prove that cos 8  1  32 sin 2   160 sin 4   256 sin 6   128 sin 8  .
(ii) Prove that 32 cos 6   cos 6  6 cos 4  15 cos 2  10 .
20. (i) Prove that
sin 2 cosh 2 x  cos 2 y
if tan   tanh x cot y, tan   tanh x tan y.

sin 2
cosh 2 x  cos y
(ii) Separate into real and imaginary parts tanh(x + iy).
  9 4 4
21. Diagonalise A =   8 3 4
 16 8 7
22. (i) Show that the locus of the point of intersection of the tangent at the extremities of a pair of
conjugate diameters of the ellipse
x2 y2
x2 y2
is
the
ellipse


1
.

 2.
a2 b2
a2 b2
(ii) Show that the locus of the perpendicular drawn from the pole to the tangent to the circle r = 2a
cosθ isr = a(1+cosθ).
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