Download MAR ATHANASIUS COLLEGE KOTHAMANGALAM Fourth

MAR ATHANASIUS COLLEGE KOTHAMANGALAM
Fourth Semester Bsc.Mathematics
Part A
(Answer all questions)
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1.If α,β,γ are the roots of f(x)=0 write the equation whose roots are 𝛼 , 𝛽 , 𝛾
2.Give an example of an equation for which α=2,β=3 are three multiple roots.
3.Defien the flux of a three dimensional vetor field F.
Give an example of a standard reciprocal equation.
5.Find the gradient field of f(x,y,z)=xyz.
6 .Find the unit tangent vector to the curve , r(t)=6t3 i - 2t3 j- 3t3 k., 0≤t≤2.
7.Find the direction in which the function f(x,y)= x2/2 + y2/2 is increasing rapidly.
8.Find the equation of the plane through (0,2,-1) normal to 3i-2j-k.
9.What is the geometrical interpretation of the method of false position?
Part B
(Answer any six.Each question has two marks)
10.Evaluate ∫𝐶 𝑥 − 𝑦 + 𝑧 − 2 𝑑𝑠 where C is the straight line segment x=t ,y= 1-t,z=1.from
(0,1,1) to (1,0,1)
11.Find the circulation of the field F = (x-y)i+x j around the circle r(t)= cost i+sint j ; 0≤t≤2π.
12.Find the equation whose roots is greater by unity than the roots of the equation
x3-5x2+6x-3=0.
13.Solve the equation x4-12x3+49x2-78x+40 =0 by removing its second term.
14.Solve the equation x3-5x2-16x+80=0,given that the sum of the roots is zero.
15.Show that the curvature of a circle with radius ‘a’ is 1/a.
16.Find the distance from S(1,1,3) to the plane 3x+2y+6z = 6.
17.Find the derivative of f(x,y) = 2xy – 3y2 at (5,5) in the direction of A = 4 i +3j.
Part C
(Answer any 4.Each question carry 8 marks)
18.Solve the equation x3-7x2+36=0,given that the difference of the roots is 5.
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19. If α,β,γ are the roots of x3+px+q=0,prove that 5 ∑ 𝛼 5 = 6 ∑ 𝛼 3 . ∑ 𝛼 2
20.Show that F=(excosy+yz)i+(xz-exsiny)j+(xy+z)k is conservative and find the potential
function for it.
21.Find the curvature for the helix r(t) = a cost i + b sint j +bt k , where a ,b ≥ 0, a2+b2≠0.
22.Find a real root of the equation x sinx +cos x =0 correct to 3 decimal places using iteration
method.
22.The surfaces x2+y2 -2 =0 and x+z-4 =0 meets in an ellipse E..Find the parametric equations
for the line tangent to E at the point P(1,1,3)
Part D
(Answer any one.Each question carry 15 marks.)
24.Solve using Ferrari’s method x4-4x3+7x2-6x+2=0.
25.Use Newton-Raphson method to obtain a root for the equation 4(x-sinx) =1, correct to three
decimal places.