PreDegree Maths Final Exam Autumn 2007
Time: 2 hours. Answer any 6 of the 7 questions. Calculators and log tables are
allowed. The table of Laplace transforms and formula sheet is attached.
Q1.
a.)
If A is an invertible 2  2 matrix such that A 2  A then show that A  I ,
the identity matrix.
b.)
For what values (if any) of x and a is the following matrix A invertible?
 1
A   x a
 x a
c.)
0
x a
0
x 6  6ax 1
0
2
2
Use Cramer’s Rule to find z in terms of a, b and c, where a, b and c are
constants, given that:
x  by  acz  b
2 x  cy  az  a
x  3ay  4az  b

Q2.
Recall that L( f (t ))   e  st f (t )dt .
0
Q3.
1
.
s2
a.)
Prove that L(t ) 
b.)
Prove the First Shifting Theorem, i.e. prove that
L(e at f (t ))  L( f (t )) | s s a .
a.)
Use Laplace transforms to solve the differential equation
d2y
dy
 5  6 y  cos x , given that y (0)  0 and y ' (0)  1 .
2
dx
dx
b.)
Without using Laplace transforms, solve the differential equation
d2y
dy
 5  6 y  cos x , given that y (0)  0 and y ' (0)  1 .
2
dx
dx
Q4.
a.)
Let x 
dy
3 sin t
and y  4 ln( 3t ) cos(t )  t 3 . Use the chain rule to find
2 cos t
dx
and find its value when t 
Q5.
x 2  3y 2

2
.
2z 2z

.
x 2 y 2
b.)
Let z 
c.)
Find
a.)
Solve the differential equation
b.)
Find
x
x

. Find
1
3
 2x 2  x  2
1
x2 1
dx .
1 dy
 e x  0 , given that y(0)  0 .
x dx
dx
(without using the formula from the log tables which states that

Q6.
1
x2  a2
dx  ln
x  x2  a2
 C ).
a
a.)
Draw on an Argand diagram the set of complex numbers z , where
| z  2  i | 2 .
b.)
Let z  (2  i 5 ) . Write z in polar form, find z 10 and write your answer
1
3
in standard (Cartesian) form.
Q7.
z1
.
z2
c.)
Let z1  3  2i and let z 2  2  3i . Find
a.)
Prove the formula in the log tables which states that
d  1 x 
a
.
 tan
 2
dx 
a  a  x2
d
1
1
sinh(ln( x))   2 .
dx
2 2x
b.)
Prove that
c.)
Use part b.) to show that sinh(ln(x) ) 
x 1

 C , where C is a constant.
2 2x