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B.E/B.Tech DEGREE EXAMINATION, Jan 2009
MA 2111 – MATHEMATICS -I
Time : 3 hours Maximum Marks:100
PART A (10 x 2 = 20 MARKS )
1. For a given matrix A of order 3, [A]=32 and two of its eigenvalues are 8 and. Find the sum of the eigenvalues.
2. Check whether the matrix B is orthogonal? Justify?
3.Write the equation of tangent plane at (1,5,7) to the sphere
(x-2)2 + (y-3) 2 + (z-4) 2 = 14
4. Find the equation of the right circular cone whose vertex is at the origin and axis is the line having semi vertical angle of 45”
5. Find the envelope of the lines y=mx where m is the parameter.
6. Define the circle of curvature at a point p() on the curve y = f(x)
7. Using Euler’s theorem, given u(x,y) is a homogeneous function of degree n, prove that
x2 uxx + 2xy uxy + y2uyy = n(n-1)u
8. Using the definition of total derivative, find the value of given u=y2-4ax X=at2, y=2at.
9. Write down the double integral, to find the area between the circles r= and r=4.
10. Define the circle of curvature at a point p() on the curve y = f(x)
PART B (5 x 16 = 80 MARKS )
11. a) (i) Find the characteristic equation of the matrix A given A=. Hence find A-1 and A4
ii) Find the eigen values and eigen vectors of A=
b) Reduce the given quadratic for Q to its canonical form using orthogonal transformation Q = x2 + 3y2 + 3z2 – 2yz.
12. a) Obtain the equation of the sphere having the circle x2 +y2+z2=9, x+y+z=3 as a great circle.
ii) Find the equation of the right circular cylinder whose axis is the line x=2y=-z and radius 4.
b) Find the equation of the right circular cone whose vertex is at the origin and axis is the line = = and which has semi vertical angle of 30o.
ii) Find the equation of the sphere described on the line joining the points (2,-1,4) and (-2,2,-2) as diameter. Fine the area of the circle in which this sphere is cut by the plane 2x+y –z =3.
13. (a) Fine the evolute of the hyperbola ==1 considering it as the envelope of its normal.
ii) Find the radius of curvature of the curve + = at
(b) Find the equation of circle of curvature of the parabola y2 =12x at the point (3,6).
(ii) Find the envelope of the family of lines + = 1 subject to the condition
14. a) If u=x2 show that uxxy = uxyz.
ii) If u = log +tan-1 (y / x) prove that uxx + uyy =0.
iii) Find Jacobian of the transformation x=r sin cos, and z=r cos .
b) (i) Find the the maximum value of subject to the condition x+y+z=a
ii) Find the Taylor series expansion ex sin y at the point up to 3rd degree terms.
15. a) Find the area inside the circle r=a but lying outside the cardioid
r = a (1-cos )
b) Change to spherical polar co-ordinate and hence evaluate where V is the volume of the sphere
ii) Change the order of integration and hence evaluate x2 dy dx