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 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
PART – A
Answer all questions: (10 x 2 = 20)
- Write down the expression of cos 4θ in terms of cosθ and sinθ.
- Give the expansion of sinθin ascending powers of θ.
- Express sin ix and cosix in terms of sin hx and coshx.
- Find the value of log(1 + i).
- Find the characteristic equation of A = .
- If the characteristic equation of a matrix is , what are its eigen values?
- Find pole of lx + my + n = 0 with respect to the ellipse
- Give the focus, vertex and axis of the parabola
- Find the equation of the hyperbola with centre (6, 2), focus (4, 2) and e = 2.
- What is the polar equation of a straight line?
PART – B
Answer any five questions. (5 X 8 = 40)
- Expandcos6θ in terms of sinθ .
- If sinθ = 0.5033 show thatθ is approximately .
- Prove that .
- If tany = tanα tanhβ ,tanz = cotα tanhβ, prove that tan (y+z) = sinh2βcosec2α.
- Verify Cayley Hamilton theorem for A =
- Prove that the eccentric angles of the extremities of a pair of semi-conjugate diameters of an ellipse differ by a right angle.
- Find the locus of poles of all tangents to the parabola with respect to
- 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)
- (i) Prove that .
(ii) Prove that .
- (i) Prove that if
(ii) Separate into real and imaginary parts tanh(x + iy).
- Diagonalise A =
- (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 is the ellipse
(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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