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PAPER CODE NO.
MATH 013
THE UNIVERSITY
of LIVERPOOL
JANUARY 2009 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 represents the angle measured in radians, what is the value of measured in degrees? The formula for states that
.
Using this formula or otherwise find the exact value for , without using tables or a calculator. (Show all your working.) Hence determine the two angles in the range [ ] radians that satisfy the equation. 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) , for. ii) , for radians.
[9 marks]
3. Solve the following exponential equation and find x to 4 decimal places.
. [6 marks]
4. You are given the values of and , correct to six decimal places. Obtain the values of the following
, , ,
without using tables or a calculator , correct to four decimal places. (Show all your working.) [6 marks]
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SECTION B
9. i) Find two values of in the range radians satisfying the equation
. [7 marks]
ii) Using the identity rewrite in the form , where and is an angle between 0 and radians. Hence find all the solutions for the angle x in the range radians which satisfy the following equation
[8 marks]
10. (i) On separate diagrams sketch the curves for real x , and for. [4 marks] (ii) Solve the following equations:
,. [4 marks]
(iii) Climatologists have recently being trying to estimate the rise in the average global temperature in the past few decades from the year 1960 onwards. Their best calculations predict that the average global temperature is rising according to the formula
degrees Centigrade
w here is the time in years after the start date 1960 (so 1960 is equivalent to , and k is a constant. What was the average global temperature in 1960? In 2008 the climatologists measured the average global temperature to be
. Calculate the value of k , and the predicted average global temperature in the year 2020. [7 marks]
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11. (i) If are the roots of the equation , find the values of a) , b) , c) and d) , without determining the values of and individually. [8 marks]
(ii) Plot a table of the values of the following cubic polynomial
for. Sketch the curve of the polynomial, and find all the roots of. [7 marks]
12. (i) A complex number has modulus one and negative argument radians. Express each of the following complex numbers in the form (where a and b are real):
,
and plot them on the Argand diagram. [10 marks]
(ii) If calculate the values of a). b). c) in the form , where is as defined in part i). [5 marks]
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