Summer 2008

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PreDegree Maths Final Exam Summer 2008
Time: 2 hours. Answer any 5 of the 6 questions. Calculators and log tables are
allowed. The table of Laplace transforms and formula sheet is attached.
Q1.
a.)
Find an example of a 2  2 matrix A such that A 2  A .
b.)
Use Cramer’s Rule to find z in terms of a, b and c, where a, b and c are
constants, given that:
ax  ay  2 z  5
bx  by  cz  2
cx  2cy  4cz  c

Q2.
Recall that L( f (t ))   e  st f (t )dt .
0
a.)
Prove that L(sin( wt )) 
w
, where w is a constant, without using the
s  w2
2
formula sheet.
b.)
Prove the Formula 9 without using Formula 14
(i.e. prove that L(te at ) 
1
without using the First Shifting
( s  a) 2
Theorem).
Q3.
a.)
Use Laplace transforms to solve the differential equation
d2y
dy
 6  5 y  sin( 2 x) , given that y(0)  1 and y' (0)  0 .
2
dx
dx
b.)
Without using Laplace transforms, solve the differential equation
d2y
dy
 6  5 y  sin( 2 x) , given that y(0)  1 and y' (0)  0 .
2
dx
dx
Q4.
Q5.
 2 w w
y

and evaluate it at the
 3 cos(xyz) . Find
xy z
x
point ( x, y, z, w)  (1,0,0,3) .
a.)
Let w  f ( x, y, z ) 
b.)
Find  
a.)
Solve the separable differential equation
b.)
Show that the derivative of cos 1 x is
1


 x 2  1 dx
2
 x  2x  3

d5y
0.
dx5
1
1 x2
(without using the formula in the log tables).
Q6.
a.)
Draw on an Argand diagram the set of complex numbers z , where
| 2  i  z | 3 .
b.)
Let z  1  i . Write z in polar form, find z 25 and write your answer in
standard (Cartesian) form.
c.)
Let z1    3i and let z 2  2  ei . Find
z1
.
z2
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