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PAPER CODE NO.
MATH 013
THE UNIVERSITY
of LIVERPOOL
JANUARY 2008 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.
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SECTION A
1. If α represents the angle 7800 measured in degrees, what is the value of α measured in radians?
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 the two angles θ , in the range [ 0 0 , 3600 ] that satisfy the
equation tan ( )θ = 4sin^2 ( α ) 3. Your answers can be expressed in degrees or
radians. [7 marks]
2. Determine numerically all the values of the angle x which satisfy the following equations. You may express your answers in degrees or radians.
i) cos( ) x = 0. 75 , for 00 ≤ x ≤ 4500.
ii) sin ( 2 x ) = 0. 2 , for 0 ≤ x ≤ π radians.
[9 marks]
3. Solve the following logarithmic equation and find x to 4 decimal places.
Log e ( 4 x ) + Log e ( x + 1 ) −Log e ( ) x = 2
[6 marks]
4. You are given the values of log e ( 18 ) = 2. 890372 and log e ( 3 ) = 1. 098612 ,
correct to six decimal places. Obtain the values of the following
log e ( 54 ), log e ( 6 ), log e ( 324 ),
without using tables or a calculator , correct to four decimal places. (Show all your working.) [6 marks]
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SECTION B
9. Find two values of θ between 0 and π 2 radians satisfying the equation
5 − 5 cos (θ ) − 4 sin^2 ( )θ = 0.
[7 marks]
Using the identity cos( A + B ) =cos( A ) cos( B ) −sin( A )sin ( B )rewrite
2 3 cos ( ) x − 2 sin( ) x in the form R cos( x +φ), where R > 0 and φ is an angle
between 0 and π 2 radians.
Hence find all the solutions for the angle x in the range 0 ≤ x ≤ 2 π which
satisfy the following equation
2 3 cos ( x ) − 2 sin( ) x = 1.
[8 marks]
10. (i) On separate diagrams sketch the curves y = ex 3 for real x , and
⎟ ⎠
log
x y (^) e for x > 0.
[4 marks] (ii) Solve the following equations:
log 6 ( 216 ) = x , 2
log (^) ⎟=− ⎠
y.
[4 marks]
(iii) Jonny recently invested his savings in shares in the Eastern Rock Building Society. The value of his savings S (in pounds) satisfies the following equation:
S = (^3) + e kt
pounds,
w here t is the time in months after his original investment, and k is a constant. How much money did Jonny invest initially at t = 0? After four months he checks Eastern Rock’s share price and works out his savings are now worth £4000. Calculate the value of k , and how much money Jonny will have after 12 months if he keeps his shares. [7 marks]
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11. (i) If α and β are the roots of the equation 6 x^2 + 3 x + 1 = 0 , find the values of
a) αβ , b) α + β, c) α 2 + β^2 and d) (α − β) 2 , without determining the values
of α and β individually.
[8 marks]
(ii) Plot a table of the values of the following cubic polynomial
p ( x )= 3 x^3 − x^2 − 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 5 π / 6. 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) If w =− 2 − 2 i calculate the values of a) w.
b) Arg ( w ).
c) w
in the form z = a + b i.
[5 marks]
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