LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

   B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2010

    MT 1500 – ALG.,ANAL.GEOMET. CAL. & TRIGN. – I

 

 

 

Date : 10-11-10                     Dept. No.                                                 Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL the questions.                                                                                                (10 x 2 = 20)

 

1. Find y_n when
2. Show that, in the parabola y²=4ax,  the subtangent at any point is double the abscissa.
3. Find the radius of curvature of xy=30 at the point (3,10).
4. Define evolutes.
5. Form equation given that 3+2c is a root.
6. If α,β,γ, are the roots of the equation x³+px²+qx+r=0 find the value of ∑α².
7. Evaluate
8. Prove that cosh =
9. Find the polar of the point (3,4) with respect the parabola y²=4ax.
10. Define conormal and concyclic points.

 

PART  –  B

 

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

 

11. Show that in the curve hy²=(x+a)³ the square of the subtangent varies as the subnormal.
12. Find the radius of curvature at the point ‘t’ of the curve

x=a(cost+tsint); y=a(sint-tcost).

13. Find the coordinates of the centre of curvature at given point on the curve y=x²;
14. Solve the equation x⁴+2x³-5x²+6x+2=0 given that 1+is a root of it.
15. Find the real root of the equation x³+6x-2=0 using Horner’s method.
16. Expand sin³θ cos⁴θ in terms of sines of multiples of θ.
17. If sin(θ+iφ) =tanα + isecα , prove that cos2 θ cosh2φ =3.
18. Show that the area of the triangle formed by the two asymptotes of the rectangular hyperbola xy=c² and the normal at (x₁,y₁) on the hyperbola is .

PART – C

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

 

19. a) Prove that if y=sin(msin^-1x), then (1-x²)y₂-xy₁₊m²y=0.
20. b) Show that the evolute of the cycloid x=a(θ – sinθ);y=a(1-cosθ) is another cycloid.

 

20. a) solve 2x⁶-9x⁵+10x⁴-3x³+10x²-9x+2=0.
21. b) If α is a root of the equation x³+x²-2x-1=0 show that α² -2 is also a root.

 

21. a) if u=log tan show that tanh   = tan  and θ = -i log tan
22. b) sum to infinity the series
23. a) Find the locus of mid points of normal chords to the ellipse
24. b) Find the polar of the point (x₁, y₁) with respect to the parabola y²=4ax.

 

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