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Engineering Mathematics 2012

Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 1

SUBJECT NAME : Engineering MathematicsI

SUBJECT CODE : 181101/MA 2111

MATERIAL NAME : PartA questions

MATERIAL CODE : JM08AM1013

Name of the Student: Branch:

UnitI (Matrices)

1.

Given :

1 0 0

2 3 0

1 4 2

A

. Find the eigen values of 2A .

2.

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

of 1

A

.

3.

Write down the quadratic form corresponding to the matrix

0 5 1

5 1 6

1 6 2

A

.

4.

The product of two eigenvalues of the matrix A

6 2 2

2 3 1

2 1 3

is 16. Find the third

eigenvalue ofA .

5.

For a given matrix A of order 3, 32A and two of its eigen values are 8 and 2.

6.

Check whether the matrix B is orthogonal? Justify.

cos sin 0

sin cos 0

0 0 1

B

.

7.

Can1 0

0 1A

be diagonalized? Why?

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8.

If the sum of two eigen values and trace of a 3 X 3 matrix A are equal, Find the value

of A .

9.

Use CayleyHamilton theorem to find

4 3 24 5 2A A A A I

when 1 24 3

A

.

10.

If 1 and 2 are the eienvalues of a 2 X 2 matrix A, what are the eigenvalues of A2and A

-1?

11.

State CayleyHamilton theorem.

12.

Find the nature of the Quadratic Form 2 2 2

1 2 3 1 2 2 32 2 2x x x x x x x

.

UnitII (Three Dimensional Analytical Geometry)

1.

Write the equation of the tangent plane at 1,5,7 to the sphere

2 2 22 3 4 14x y z .

2.

Find the equation of the tangent plane at

1,4,2 on the sphere x y z2 2 2

x y z 2 4 2 3 0 .

3.

Find the equation of the tangent plane to the sphere2 2 2

x y z 2 4x y

6 6 0z

at 1,2,3 .

4.

Find the equation of the sphere concentric with x y z x y z 2 2 2

4 6 8 4 0

and passing through the point

1,2,3 .

5.

Find the equation of the sphere having the points 2, 3,4 and 1,5,7 as the ends

of a diameter.

6.

Check whether the two spheres 2 2 2 6 2 8 0x y z y z

and

2 2 26 8 4 20 0x y z x y z are orthogonal.

7.

Find the centre and radius of the sphere

2 2 22 6 6 8 9 0x y z x y z .

8.

Find the equation of the right circular cone whose vertex is at the origin and axis is the

line1 2 3

x y z

having semi vertical angel of 45.

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9.

Find the equation of the cone whose vertex is the origin and guiding curve is

2 2 2

1, 14 9 1

x y zx y z .

10.

Find the equation of the right circular cone whose vertex is the origin, axis is the y

axis, and semivertical angle is 30.

11.

Write down the equation of the right circular cone whose vertex is at the origin, semi

vertical angel is and axis is along z-axis.

UnitIII (Differential Calculus)

1.

For the catenary coshx

y cc

, find the curvature.

2.

Find the radius of curvature for

x

y e at the point where it cuts the yaxis.

3.

Define the circle of curvature at a point1 1,p x y on the curve ( )y f x .

4.

Find the curvature of the curve 2 2

2 2 5 2 1 0x y x y .

5.

Write down the formula for Radius of curvature in terms of Parametric Coordinates

System.

6.

Find the envelope of the lines2 2 2

y mx a m b where m is the parameter.

7.

Find the envelope of family of straight linesa

y mxm

, m being the parameter.

8.

Find the envelope of the family of straight lines1

y mxm

, where m is a parameter.

9.

Write the properties of Evolutes.

10.

Find the envelope of the family of straight lines cos sinx y where

is the

parameter.

11.

Find the envelope of the family of circles

2 2 2x y r , being the parameter.

UnitIV (Functions of several variables)

1.

Using Eulers theorem, given ( , )u x y is a homogeneous function of degree n, prove

that2 22 ( 1)

xx xy yy x u xyu y u n n u

.

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2.

Using the definition of total derivative, find the value ofdu

dtgiven

24u y ax ;

2, 2x at y at .

3.

If 3 2 2 3u x y x y

where 2x at and 2y at then find dudt

?

4.

Finddu

dtif sin( / )u x y , where

2,

tx e y t

.

5.

If

2 2 2

, ,2 2

y x yu v

x x

find( , )

( , )

u v

x y

.

6.

If x u v2 2

andy uv2 , find the Jacobian of x and ywith respect to uand v .

7.

If2 2

2 , , cos , sinu xy v x y x r y r

then compute( , )

( , )

u v

r

?

8.

Write the sufficient condition for ( , )f x y to have a maximum value at (a,b).

9.

If cos , sinx r y r

find( , )

( , )

x y

r

.

10.

If y

u x , show thatu u

x y y x

2 2

.

11.

Given 2 1( , ) tan

yu x y x

x

, find the value of2 22

xx xy yy x u xyu y u

.

12.

Ifx y z

uy z x

, findu u u

x y zx y z

.

UnitV (Multiple Integral)

1.

Write down the double integral, to find the area between the circles 2sinr and

4sinr

2.

Evaluate

sin

0 0

r drd

.

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3.

Evaluate

1

0

x

x

xy x y dxdy

.

4.

Evaluate

21

2 2

0 0

x

x y dydx

.

5.

Evaluate

0 0

( )

a b

x y dxdy

.

6.

Evaluate2 2

C

x dy y dx

where C is the path y xfrom 0,0 to 1,1 .

7.

* Evaluate

R

dxdy

, where R is the shaded region in the figure.

8.

Change the order of integration in2

1 2

0

( , )

x

x

I f x y dxdy

.

9.

Change the order of integration for the double integral

1

0 0

( , )

x

f x y dxdy

.

10.

Change the order of integration in

0

( , )

a a

x

f x y dydx

.

11.

Change the order of integration

1 1

0 y

dxdy

.

12.

Express f x y dxdy

0 0

( , ) in polar co-ordinates.

13.

Evaluate

y x y

dxdydz

1

0 0 0

.

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