-
8/13/2019 M1 R08 MayJune 10.pdf
1/3
B.E./B.Tech. DEGREE EXAMINATION, May/JUNE 2010
Regulations2008
First Semester
Common to all branches
MA2111Mathematics I
Time : Three Hours Maximum : 100 Marks
Answer ALL Questions
PART A(10 x 2 = 20 Marks)
1. If 1 and 2 are the eienvalues of a 2 X 2 matrix A, what are
the eigenvalues of A2and A-1?2. State CayleyHamilton theorem.3.
Find the centre and radius of the sphere 2 2 22 6 6 8 9 0x y z x y
z .4. Find the equation of the right circular cone whose vertex is
the origin, axis is the yaxis, and semi
vertical angle is 30.
5. Find the radius of curvature for xy e at the point where it
cuts the yaxis.6. Find the envelope of the family of straight lines
1y mx
m
, where m is a parameter.
7. Given 2 1( , ) tan yu x y x x
, find the value of 2 22xx xy yy x u xyu y u .8. Write the
sufficient condition for ( , )f x y to have a maximum value at
(a,b).9. * Evaluate
R
dxdy , where R is the shaded region in the figure.
10.Change the order of integration for the double integral 10
0
( , )
x
f x y dxdy .
-
8/13/2019 M1 R08 MayJune 10.pdf
2/3
PART B(5 x 16 = 80 marks)
11. (a) (i) Find the eigen values and eigen vectors of the
matrix2 2 1
1 3 1
1 2 2
A
.
(ii) Using CayleyHamilton theorem, find1
A
when
2 1 2
1 2 1
1 1 2
A
.
OR
(b) Reduce the quadratic form2 2 2
2 5 3 4x y z xy to the Canonical form by orthogonalreduction and
state its nature.
12. (a) (i) Find the centre and radius of the circle given by 2
2 2 2 2 4 19 0x y z x y z and2 2 7 0x y z .(ii) Find the equation
of the cone whose vertex is the point 1,1,0 and whose base is
the
curve 2 20, 4y x z .
OR
(b) (i) Find the equation to the sphere passing through the
circle2 2 2
9,x y z 1x y z and cuts orthogonally the sphere
2 2 2
2 4 16 17 0x y z x y z .(ii) Find the equation of the right
circular cylinder of radius 3 and axis is the line
1 3 5
2 2 1
x y z .
13. (a) (i) Find the radius of curvature at any point of the
cycloid sinx a , 1 cosy a .(ii) Find the circle of curvature at /
4, / 4a a on x y a .
OR
(b) (i) Show that the evolute of the parabola2
4y ax is the curve 3227 4 2ay x a .(ii) Find the envelope of the
straight line 1
x y
a b , where a and bare connected by the
relation 2
ab c , c is a constant.
-
8/13/2019 M1 R08 MayJune 10.pdf
3/3
14. (a) (i) If ( , )u f x y where cos , sinx r y r , prove
that22 2 2
2
1u u u u
x y r r .
(ii) Expand by Taylors series the function2 2 2 2
( , ) 2 3f x y x y x y xy in powers of 2xand 1y upto the third
powers.
OR
(b) (i) Find the maximum and minimum values of2 2
2x xy y x y .(ii) A rectangular box open at the top, is to have
a volume of 32 cc. Find the dimensions of the
box, that requires the least material for its construction.
15. (a) (i) Change the order of integration in the interval2
2
0 /
a a x
x a
xy dydx and hence evaluate it.
(ii) By Transforming into polar coordinates, evaluate
2 2
2 2
x ydxdy
x y
over annular regionbetween the circles
2 216x y and 2 2 4x y .OR
(b) (i) Find the value of xyz dxdydz through the positive
spherical octant for which2 2 2 2
x y z a .
(ii) Find the volume of the tetrahedran bounded by the plane 1x
y z
a b c and the coordinate
plane 0, 0, 0x y z .
