MA2111 Mathematics I ANNA UNIVERSITY QUESTION PAPER November/December 2010 ...

B.E./B.Tech.Degree Examinations, November/December 2010

Regulations 2008

First Semester

Common to all branches

MA2111 Mathematics I

Time: Three Hours Maximum: 100 Marks

Answer ALL Questions

Part A - (10 x 2 = 20 Marks)
1. If ¡1 i s an eigen value of the matrix A = µ 1 ¡2
¡3 2 ¶, ¯nd the eigen values of A4
using properties.
2. Use Cayley-Hamilton theorem to ¯nd A4 ¡ 8A3 ¡ 12A2 when A = · 5 3
1 3 ¸:
3. Find the centre and radius of the sphere 2(x2 + y2 + z2) + 6x ¡ 6y + 8z + 9 = 0.
4. State the conditions for the equation ax2 +by2 +cz2 +2fyz +2gzx+2hxy +2ux+
2vy + 2wz + d = 0 to represent a cone with vertex at the origin.
5. Find the radius of curvature of y = ex at x = 0.
6. Find the envelope of the family of straight lines y = mx +
1
m
, where m is a
parameter.
7. If u = f(x ¡ y; y ¡ z; z ¡ x), show that
@u
@x
+
@u
@y
+
@u
@z
= 0.
8. If u =
x + y
1 ¡ xy
and v = tan¡1 x + tan¡1 y, ¯nd
@(u; v)
@(x; y)
.
 132  132  132 
9. Evaluate
1 Z0
2 Z0
ex+y dx dy.
10. Evaluate
1 Z0
2 Z0
3 Z0
xyzdzdydx:
Part B - (5 x 16 = 80 Marks)
11. (a) (i) Using Cayley-Hamilton theorem, ¯nd the inverse of the matrix A = 24
¡1 0 3
8 1 7
¡3 0 8
35
.
(8)
(ii) Find the eigen values and eigen vectors of the matrix 24
2 2 1
1 3 1
1 2 2
35
.
(
8
)
OR
11. (b) Reduce the quadratic form
2x2
1 + x2
2 + x2
3 + 2x1x2 ¡ 2x1x3 ¡ 4x2x3
to canonical form by an orthogonal transformation. Also ¯nd the rank, index,
signature and nature of the quadratic form. (16)
12. (a) (i) Find the equation of the sphere having its centre on the plane 4x¡5y¡z =
3 and passing through the circle x2 + y2 + z2 ¡ 2x ¡ 3y + 4z + 8 = 0;
x ¡ 2y + z = 8. (8)
(ii) Find the equation of the right circular cone whose vertex i s the origin,
whose axis i s the line
x
1
=
y
2
=
z
3
and which has semi-vertical angle 30±.
(8)
OR
12. (b) (i) Find the equation of the sphere passing through the circle x2 + y2 + z2 +
x¡3y +2z ¡1 = 0, 2x+5y ¡z +7 = 0 and cuts orthogonally the sphere
x2 + y2 + z2 ¡ 3x + 5y ¡ 7z ¡ 6 = 0. (8)
(ii) Find the equation of the right circular cylinder whose axis is
x ¡ 1
2
=
y ¡ 2
1
=
z ¡ 3
2
and radius 2. (8)


13. (a) (i) Find the equation of the circle of curvature of the curve px + p y = pa
at ³a
4
;
a
4´. (8)
(ii) Prove that the radius of curvature of the curve xy2 = a3 ¡x3 at the point
(a, 0) is
3a
2
: (8)
OR
13. (b) (i) Show that the evolute of the hyperbola
x2
a2 ¡
y2
b2 = 1 i s (ax)2=3 ¡(by)2=3 =
(a2 + b2)2=3. (10)
(ii) Find the envelope of
x
a
+
y
b
= 1, where a and b are connected by the
relation a2 + b2 = c2; c being constant. (6)
14. (a) (i) Find the Taylor's series expansion of ex cos y in the neighborhood of the
point ³1;
¼
4 ´ upto third degree terms. (8)
(ii) If u = log(x2 + y2) + tan¡1 ³y
x´, prove that uxx + uyy = 0. (8)
OR
14. (b) (i) Discuss the maxima and minima of the function f(x; y) = x4 +y4 ¡2x2 +
4xy ¡ 2y2. (8)
(ii) Find the Jacobian of y1; y2; y3 with respect to x1; x2; x3 if y1 =
x2x3
x1
; y2 =
x3x1
x2< br class="c26" />; y3 =
x1x2
x3
. (8)
15. (a) (i) Evaluate
1 Z0
1 Zx
e¡y
y
dx dy by changing the order of integration. (8)
(ii) Evaluate
1 Z0
1 Z0
e¡(x2+y2) dx dy by converting to polar coordinates. Hence
deduce the value of
1 Z0
e¡x2
dx. (8)
OR
15. (b) (i) Using triple integration, ¯nd the volume of the sphere x2 + y2 + z2 = a2.
(8)
(ii) Find the area bounded by the parabolas y2 = 4 ¡ x and y2 = x by double
integration. (8)