LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2011

MT 3501/ MT 3500 – ALGEBRA, CALCULUS AND VECTOR ANALYSIS

 

 

 

Date : 12-04-2011              Dept. No.                                                    Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL questions.                                                                                                 (10 ´ 2 = 20)

1. Evaluate
2. Find when u = x² – y²; v = x² + y²
3. Solve
4. Find the complete integral of z = px + qy +p²q²
5. Find grad f if f = xyz at (1, 1, 1)
6. Evaluate divergence of the vector point function
7. Find L[sin² 2t]
8. Find
9. Find the sum of all divisors of 360.
10. Find the remainder when 2¹⁰⁰⁰ divisible by 17.

 

PART – B

Answer any FIVE questions.                                                                                        (8 ´ 5 = 40)

11. Change the order of integration and evaluate
12. Express in terms of Gamma functions and evaluate
13. Solve p² + pq = z²
14. Solve xp + yq = x
15. Show that the vectoris irrotational.
16. Evaluate: (a) L[cos 4t sin 2t]                   (b) L[e^-3t sin²t]
17. Find
18. Show that 18! + 1 is divisible by 437.

 

PART – C

Answer any TWO questions.                                                                                       (2 ´ 20 = 40)

19. (a) Evaluate where the region V is bounded by x + y+ z = a (a > 0),
    x = 0; y = 0; z = 0.

(b)  Evaluate  where R is the region in the positive quadrant for which
x + y £ 1.

(c)  Show that

20. (a) Solve (x² + y² + yz)p + (x² + y² – xz)q = z(x+y)

(b)  Find the complete integral and singular integral of p³ + q³ = 8z

21. (a) Solve y¢¢ + 2y¢ – 3y = sin t given that y(0) = y¢(0) = 0

(b)  State and prove the Weirstrass inequality.

22. (a) State and prove Wilson’s theorem.

(b)  Verify Green’s theorem in the XY plane for where C is the closed curve in the region bounded by y = x; y = x².

 

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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)

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).
5. Find the characteristic equation of A = .
6. If the characteristic equation of a matrix is , what are its eigen values?
7. Find pole of lx + my + n = 0 with respect to the ellipse
8. Give the focus, vertex and axis of the parabola
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 .
13. Prove that .
14. If tany = tanα tanhβ ,tanz = cotα tanhβ, prove that tan (y+z) = sinh2βcosec2α.
15. Verify Cayley Hamilton theorem for A =
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 with respect to

 

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 .

 

(ii) Prove that .

20. (i) Prove that if

(ii) Separate into real and imaginary parts tanh(x + iy).

21. Diagonalise A =
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 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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