Other Engineering Answers

TASK H1: with each of the 4 marked statements immediately below, explain, in
plain English,
a) what the statement does and
b) its purpose in the program.

const int MAXSTACKSIZE = 5; - TASK H1.1 Explain this statement
const int BOTTOMOFSTACK = -1; - TASK H1.2 Explain this statement
typedef char StackElement; -TASK H1.3 Explain this statement



TASK H1.4 Explain this statement


typedef struct {
StackElement contents[MAXSTACKSIZE];
int top;
} Stack;

Find the equation of the line which passes through and makes an angle
30
0
with the line (3)
(b) Find the distance of the line obtained in part (a), from the origin by expressing it
in the normal form. Also find the intercepts made by this line on the coordinate
axes. (3)
4
(c) Obtain the equation of the plane passing through the line and
which is perpendicular to the plane (3)
(d) Find the vertices, eccentricity, foci and asymptotes of the hyperbola
Also trace it. Under what conditions on the line will be tangent
to this hyperbola? Explain geometrically.

a) Let a quadratic form have the expression x
2
+ y
2
+ 2z
2
+ 2xy + 3xz with respect to the
standard basis B
1 = f(1; 0; 0); (0; 1; 0); (0; 0; 1)g. Find its expression with respect to the
basis B
2 = f(1; 1; 1); (0; 1; 0); (0; 1; 1)g (3)
b) Consider the quadratic form
Q : 2x
2
4xy + y
2
+ 4xz + 3z
2
i) Find a symmetric matrix A such that Q = X
t
AX .
ii) Find the orthogonal canonical reduction of the quadratic form.
iii) Find the principal axes of the form.
iv) Find the rank and signature of the form. (5)

a) Check whether the forms 2x
2
+ 3y
2
+ 5z
2
4xz 6yz and 4x
2
+ 3y
2
+ z
2
6xy 2xz
are orthogonally equivalent. (3)
b) Use Gram-Schmidt orthogonalisation process to find an orthonormal basis for the
subspace of C
4
generated by the vectors (1; i; 0; i), ( i; 0; 1; 2) and (0; i; 1; 1). (5)
c) Which of the following matrices are Hermitian and which are Unitary? Justify your
answer. (4)
A =
2
4
1 i 0
i 1 1 i
0 1 + i 2
3
5
; B =
2
4
1
p
2
0
1
p
2
0 1 0
p
2i 0
p
2i
3
5

Let
A =
2
4
5 4 4
6 7 6
12 12 11
3
5
a) Find the adjoint of A. Find the inverse of A from the adjoint of A. (4)
b) Find the characteristic and minimal polynomials of A. Hence find its eigenvalues and
eigenvectors. (6)
c) Why is A diagonalisable? Find a matrix P such that P
1
AP is diagonal. (2)
d) Verify Cayley-Hamilton theorem for A. Hence, find the inverse of A

a) Which of the following functions are 1-1 and which are onto? Justify your answer.
i) f : R ! R
0
given by f (x) = x
2
where R
0
is the set fx 2 Rjx 0g.
ii) f : R ! R given by f (x) = x
2
+ x + 1. (3)
b) If we consider the expression
1
2 3x
as a function on R, what will be its domain and
range? Will it have an inverse? Justify your answer. (2)
c) Let a =
(
1
2
p
2
;
p
3
2
p
2
;
1
p
2
)
and b =
(
1
p
2
; 0;
1
p
2
)
.
i) Find the direct cosines of a and b.
ii) Find the angle between a and b. (2)
d) Check that the vectors u =
(
3
5
;
4
5
; 0
)
, v =
(
4
5
;
3
5
; 0
)
and w = (0; 0; 1) are orthonormal.
Further, write the vector a = (1; 1; 2) as a linear combination of the vectors.

Obtain the geometric, polar and exponential representationsof (i5 – 1)
–1
.

Obtain the solution set of the system x – 3y + 4z = 9, 4x +3y + 2z = 7, y – 2x = 5 – 10z by
elimination.

A stamping machine produces ‘can tops’ whose diameters are normally distributed with a standard deviation of 0.02 inch. At what nominal mean diameter should the machine be set, so that no more than 9 % of the ‘can tops’ produced have diameters exceeding 3.5 inches?

why steel has carbon in its structure? is there any simple explanation? it's for homework