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PAPER CODE NO.
MATH 013
THE UNIVERSITY
of LIVERPOOL
JANUARY 2006 EXAMINATIONS
Bachelor of Engineering : Foundation Year
Bachelor of Science : Foundation Year
MATHEMATICAL METHODS
TIME ALLOWED : Three Hours
INSTRUCTIONS TO CANDIDATES
You may attempt all questions. All answers to
Section A and the best THREE answers to Section B
will be taken into account.
Numerical answers should be given correct to
four places of decimals.
Page 1 of 5 Continued
of LIVERPOOL
SECTION A
1. Determine the radian measure of the angle α of , expressed as a
rational multiple of
0 − 120
The formula for sin ( A − B )states that
sin ( A − B ) =sin( A ) cos( B ) −cos( A ) sin( B ).
Using this formula or otherwise find the exact value for sin(^ α^ ), without using
tables or a calculator. (Show all your working.)
Hence determine all the angles θ , in the range [ ] satisfying
0 0 − 360 , 360
sin ( ) θ = sin( α ). Your answers can be expressed in degrees or radians.
[6 marks]
2. Sketch the graph of y = tan( x )in the range− π ≤ x ≤ π. Determine
numerically the solutions of tan ( x ) = 2. 5 and tan( x ) =− 2. 5 in the same range.
[9 marks]
3. Solve the equation
log ( 4 ) log ( ) 7
4 e x^ +^ ex =.
[7 marks]
4. You are given the values of log e ( 100 ) = 4. 605170 and log e ( 5 ) = 1. 609438 ,
correct to six decimal places. Obtain the values of the following
log e ( 500 ) , log e ( 20 ), log e ( 25 ),
without using tables or a calculator , correct to four decimal places. (Show all
your working.)
[6 marks]
5. Write down the first six rows of Pascal’s triangle. Hence or otherwise find the
coefficient of in the expansion of
4 x
5 1 − 4 x.
[6 marks]
PAPER CODE ……M013……… PAGE 2 OF 5 CONTINUED
of LIVERPOOL
SECTION B
9. Find two values of θ between satisfying the equation
0 0 0 and 180
6 sin (θ ) 4 cos(θ )
2 = +.
[7 marks]
Using the identity ( ) ( ) ⎟
cos 2
sin sin 2 sin
A B A B
A B or otherwise,
find the range of values of a for which the equation
( x + ) + ( x + ) = a
0 0 sin 225 sin 135 ,
has real solutions. For the case a^ =^1 /^2 , find all the solutions in the interval
.
0 0 0 ≤ x ≤ 360
[8 marks]
10. (i) On separate diagrams sketch the curves
x y e 3
= for real x , and
y = log e ( ) x + 2 for x > 0.
[4 marks]
(ii) Solve the following equations:
log 16 x = , log y ( 121 ) = 2.
[4 marks]
(iii) A body of mass falls under gravity through a resistive
medium which exerts a resistive force proportional to the body’s velocity v. It
turns out that after a time t (measured in seconds) v is given by the following
equation
m = 0. 02 kg
kt m e k
A
k
mg v
− = − ,
where g is the acceleration due to gravity (take ) and k and A are
constants. If the body falls initially from rest, show that
2 9 8 ms
− g =.
A = mg. As
the body’s velocity approaches. Calculate the value of k and hence
show that after 1 second the body is already falling at a speed of
approximately.
t → ∞ 1 10 ms
−
1 6 25 ms
− .
[7 marks]
PAPER CODE ……M013…… PAGE 4 OF 5 CONTINUED
of LIVERPOOL
11. (i) If α and β are the roots of the equation , find the values of
a)
2 x + x + =
αβ , b) α + β, c) and d)
2 2
2 α − β , without determining the values
of α and β individually.
[8 marks]
(ii) Plot a table of the values of the following cubic polynomial
3 2 p x = x − x − x +
for x =− 2 , − 1 ,0,1,2, 3 and 4. Sketch the curve of the polynomial, and find all
the roots of p ( x )= 0.
[7 marks]
12. (i) A complex number z has modulus one and argument π / 4. Express each of
the following complex numbers in the form a + b i(where a and b are real):
z
z z z
2 3 ,
and plot them on the Argand diagram.
[10 marks]
(ii) Find the modulus and argument of the complex number w , where
i
3 +i 1 +i w =.
[5 marks]
PAPER CODE ……M013…… PAGE 5 OF 5 END
