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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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Engineering Mathematics 2012
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 2
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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Engineering Mathematics 2012
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 3
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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Engineering Mathematics 2012
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 4
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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Engineering Mathematics 2012
Prepared by C.Ganesan, M.Sc., M.Phil., (Ph:9841168917) Page 5
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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