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

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__PART – A__

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

- Find y
_{n}when - Show that, in the parabola y
^{2}=4ax, the subtangent at any point is double the abscissa. - Find the radius of curvature of xy=30 at the point (3,10).
- Define evolutes.
- Form equation given that 3+2c is a root.
- If α,β,γ, are the roots of the equation x
^{3}+px^{2}+qx+r=0 find the value of ∑α^{2}. - Evaluate
- Prove that cosh =
- Find the polar of the point (3,4) with respect the parabola y
^{2}=4ax. - Define conormal and concyclic points.

__PART – B__

__ __

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

- Show that in the curve hy
^{2}=(x+a)^{3}the square of the subtangent varies as the subnormal. - Find the radius of curvature at the point ‘t’ of the curve

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

- Find the coordinates of the centre of curvature at given point on the curve y=x
^{2}; - Solve the equation x
^{4}+2x^{3}-5x^{2}+6x+2=0 given that 1+is a root of it. - Find the real root of the equation x
^{3}+6x-2=0 using Horner’s method. - Expand sin
^{3}θ cos^{4}θ in terms of sines of multiples of θ. - If sin(θ+iφ) =tanα + isecα , prove that cos2 θ cosh2φ =3.
- Show that the area of the triangle formed by the two asymptotes of the rectangular hyperbola xy=c
^{2}and the normal at (x_{1},y_{1}) on the hyperbola is .

__PART – C__

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

- a) Prove that if y=sin(msin
^{-1}x), then (1-x^{2})y_{2}-xy_{1+}m^{2}y=0. - b) Show that the evolute of the cycloid x=a(θ – sinθ);y=a(1-cosθ) is another cycloid.

- a) solve 2x
^{6}-9x^{5}+10x^{4}-3x^{3}+10x^{2}-9x+2=0. - b) If α is a root of the equation x
^{3}+x^{2}-2x-1=0 show that α^{2}-2 is also a root.

- a) if u=log tan show that tanh = tan and θ = -i log tan
- b) sum to infinity the series
- a) Find the locus of mid points of normal chords to the ellipse
- b) Find the polar of the point (x
_{1}, y_{1}) with respect to the parabola y^{2}=4ax.

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