PART – A
(10 x 2 = 20)
Answer All the Questions
1. State any two properties of eigen values of
a matrix.
2. Use
Cayley-Hamilton theorem to find the inverse of the matrix 
3. Find the coefficient of xn in the expansion of 
4. Find the coefficient of xn in the expansion of
log (1 + x + x2
+ x3 + x4)
5. Find the radius of curvature for the curve
x = at2, y = 2at.
6. Write
the formula to find the center of the circle of curvature and equation to the
circle of curvature.
7. Write the tailor’s series expansion of f(x, y) about the point (a, b).
8. State
the conditions for f(x, y) to have a maximum
or a minimum value.
9. Solve (D2 + 9) y = sin 3x, where 
.
.
10. Write
the Euler’s homogenous linear differential equation of order n.
PART – B (5 x 12 = 60)
Answer All the Questions
11. (a) Find the Eigen values and Eigen vectors of the matrix 
(b) Diagonalise the matrix A given above by similarity
transformation.
(or)
12. (a) Find the inverse of the matrix 
by using Cay;ey-Hamilton
theorem.
by using Cay;ey-Hamilton
theorem.
(b) Obtain an orthogonal transformation,
which will transform the quadratic form 6x2
+ 3y2 + 3z2 – 4xy – 2yz + 4zx into a canonical form.
13. (a) If p-q is small compared to p or q, show that 
(b) Show that the coefficient of xr in the expansion of is 
. Hence, show that 
. Hence, show that
(or)
14. (a) Find
the sum of 
(b) Show
that 
15. (a) Prove that if the center of curvature of the ellipse 
at one end of the minor axis lies at the other end, then the
eccentricity of the ellipse is 
.
at one end of the minor axis lies at the other end, then the
eccentricity of the ellipse is .
(b) Obtain the equation of the evolute of the curve: x =
a(cosq + q sinq); y = a(sinq - q cosq).
(or)
16. (a) Find
the envelope of the family of straight lines
y
= x tan a +
2 sec a.
(b) Prove that the evolute of the tractrix
x =a(cos t + log(tan
)), y = a sint is a catenary.
)), y = a sint is a catenary.
17. (a) Find and classify the extreme values, if any, of the function f(x, y) = y2 + x2y + x4.
(b) A rectangular box open at the top is to have a capacity
of 108 cubic meters. Find the dimensions of the box requiring least material
for its transaction.
(or)
18. (a) Determine the points on the ellipse, defined by the
intersection of the surface x + y = 1
and x2 + 2y2 + z2
= 1 which are nearest to and farthest from the origin.
(b) Find the maximum and minimum distance from the origin to
the curve 5x2 + 6xy + 5y2
– 8 = 0.
19. (a) Solve 
(b) Solve
the simultaneous equations 
(or)
20. (a) Solve xy// -
2 (x + 1) y/+ (x + 2) y = (x – 2) e2x,by the method
of reduction of order.
(b) Solve by the method of variation of
parameters:
y// - 2y/ + 2y = ex tan x.
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