B.E./B.Tech. DEGREE EXAMINATIONS, MAY/JUNE 2012

First Semester

MA2111 – MATHEMATICS – I

Common to all branches

(Regulations 2008)

Time : Three hours Maximum : 100 marks

Answer ALL questions

PART A – (10 x 2 = 20 marks)

1. If 3 and 6 are two eigen values of

1 1 3

1 5 1

3 1 1

A

====

, write down all the eigen values of1A−−−−

.

2. Write down the quadratic form corresponding to the matrix

0 5 1

5 1 6

1 6 2

A

−−−− ==== −−−−

.

3. Find the equation of the tangent plane to the sphere 2 2 2 2 4 6 6 0x y z x y z+ + + + − − =+ + + + − − =+ + + + − − =+ + + + − − = at

(((( ))))1, 2, 3 .

4. Write down the equation of the right circular cone whose vertex is at the origin, semi vertical angel is

αααα and axis is along z-axis.

5. For the catenary coshx

y cc

==== , find the curvature.

6. Find the envelope of the family of circles (((( ))))2 2 2x y rαααα− + =− + =− + =− + = , αααα being the parameter.

7. If x y z

uy z x

= + += + += + += + + , show that 0u u u

x y zx y z

∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂+ + =+ + =+ + =+ + =∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂

.

8. If

2 2 2

, 2 2y x y

u vx x

++++= == == == = , find(((( ))))(((( ))))

,

,

u v

x y

∂∂∂∂∂∂∂∂

.

9. Evaluate (((( ))))21

2 2

0 0

x

x y dydx++++∫ ∫∫ ∫∫ ∫∫ ∫ .

10. Change the order of integration in

0

( , ) a a

x

f x y dydx∫ ∫∫ ∫∫ ∫∫ ∫ .

PART B – (5 x 16 = 80 marks)

11. (a) (i) If iλλλλ for ( 1,2, ..., )i n==== are the non-zero eigen values of A , then prove that (1) ikλλλλ are the

eigen values of kA , where k being a non-zero scalar; (2) 1

iλλλλare the eigen values of

1A−−−−.

(ii) Verify Cayley- Hamilton theorem for the matrix

2 0 1

0 2 0

1 0 2

−−−− −−−−

and hence find1A−−−−

and4A .

Or

(b) Reduce the quadratic form 2 2 2 2 2 2x y z xy yz zx+ + − − −+ + − − −+ + − − −+ + − − − to canonical form through an

orthogonal transformation. Write down the transformation.

12. (a) (i) Find the equation of the sphere having the circle x y z y z+ + + − − =+ + + − − =+ + + − − =+ + + − − =2 2 2 10 4 8 0 ,

x y z+ + =+ + =+ + =+ + = 3 as the greatest circle.

(ii) Find the equation of a right circular cone generated when the straight line 2 3 6,y z+ =+ =+ =+ =

0x ==== revolves about z – axis.

Or

(b) (i) Find the two tangent planes to the sphere 2 2 2 4 2 6 5 0x y z x y z+ + − − − + =+ + − − − + =+ + − − − + =+ + − − − + = , which are

parallel to the plane 4 8 0x y z+ + =+ + =+ + =+ + = . Find their point of contact. (10)

(ii) Find the equation of the right circular cylinder of radius 3 and axis 1 3 5

2 2 1x y z− − −− − −− − −− − −= == == == =

−−−−.

13. (a) (i) Find the radius of curvature at any point of the cycloid ( sin ), (1 cos )x a y aθ θ θθ θ θθ θ θθ θ θ= + = −= + = −= + = −= + = − .

(ii) Find the circle of curvature ata a

,4 4

on x y a+ =+ =+ =+ = .

Or

(b) (i) Find the evolute of the parabola 2 4y ax==== .

(ii) Find the envelope of the system of ellipses

2 2

2 2 1x ya b

+ =+ =+ =+ = , where the parameters a and b are

connected by the relation 4ab ==== .

14. (a) (i) Transform the equation 2 0xx xy yyz z z+ + =+ + =+ + =+ + = by changing the independent variables using

u x y= −= −= −= − and v x y= += += += + .

(ii) Expand 2 3 2x y y+ −+ −+ −+ − in powers of ( 1)x −−−− and ( 2)y ++++ upto 3

rd degree terms.

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 32cc. Find the dimensions of the

box, that requires the least material for its construction.

15. (a) (i) Change the order of integration 2

1 2

0

x

x

xy dydx−−−−

∫ ∫∫ ∫∫ ∫∫ ∫ and hence evaluate.

(ii) Transform the integral into polar coordinates and hence evaluate

2 2

2 2

0 0

a a x

x y dydx−−−−

++++∫ ∫∫ ∫∫ ∫∫ ∫ .

Or

(b) (i) Find, by double integration, the area between the two parabolas 23 25y x==== and

25 9x y==== .

(ii) Find the volume of the portion of the cylinder 2 2 1x y+ =+ =+ =+ = intercepted between the plane

0x ==== and the paraboloid 2 2 4x y z+ = −+ = −+ = −+ = − .