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)
- Find yn when
- Show that, in the parabola y2=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 x3+px2+qx+r=0 find the value of ∑α2.
- Evaluate
- Prove that cosh =
- Find the polar of the point (3,4) with respect the parabola y2=4ax.
- Define conormal and concyclic points.
PART – B
Answer any FIVE questions. (5 x 8 = 40)
- Show that in the curve hy2=(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=x2;
- Solve the equation x4+2x3-5x2+6x+2=0 given that 1+is a root of it.
- Find the real root of the equation x3+6x-2=0 using Horner’s method.
- Expand sin3θ cos4θ 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=c2 and the normal at (x1,y1) on the hyperbola is .
PART – C
Answer any TWO questions. (2 x 20 = 40)
- a) Prove that if y=sin(msin-1x), then (1-x2)y2-xy1+m2y=0.
- b) Show that the evolute of the cycloid x=a(θ – sinθ);y=a(1-cosθ) is another cycloid.
- a) solve 2x6-9x5+10x4-3x3+10x2-9x+2=0.
- b) If α is a root of the equation x3+x2-2x-1=0 show that α2 -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 (x1, y1) with respect to the parabola y2=4ax.
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