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PAPER CODE NO.
MATH 013
THE UNIVERSITY
of LIVERPOOL
SEPTEMBER 2004 EXAMINATIONS
Bachelor of Engineering : Year 1 Bachelor of Science : Year 1
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. Determine the radian measure of the angle , expressed as a rational multiple of
α=− 4800
Using the formula for sin( A + B ), or otherwise, find the exact value for
sin ( α ), without using tables or a calculator.
Hence determine all the angles θ , in the range [− 3600 , 3600 ]satisfying
sin ( ) θ = sin( α ).
[7 marks]
2. Find all the solutions for θ in the range [ 0 , 1800 ], which satisfy
2 sec 2 (θ ) + 3 tan(θ ) = 1.
[6 marks]
3. Find (to 4 decimal places) the value of x which satisfies
log e ( 4 x ) + log e ( x^3 ) = 9.
[5marks]
4. Use logarithms to solve the equation
7 6 − x^ = 4 x. [5 marks]
5. Write down the first seven rows of Pascal’s triangle. Hence or otherwise find the coefficient of x^9 in the expansion of
3 6 x + 2. [6 marks]
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SECTION B
9. Assuming the Difference Formula for the cosine function:
sin^ ( x − y )^ =sin(^ x )cos^ (^ y )^ −cos(^ x )sin^ (^ y ),
show that sin( x − π 2 ) =−cos( x ), for all x. [2marks] Express 4 sin ( ) x − 3 cos( ) x in the form A sin( x − φ), where the phase angle φ is acute and A > 0. The angle should be expressed in radians. Hence solve the equation
4 sin x − 3 cos x = ,
where 0 ≤ x ≤ π. Comment on the case when the right hand side of this
equation is replaced by 5 3. [13 marks]
10. (i) On separate diagrams sketch the curves y = loge( x )for and
for all x.
x > 0 y = e − x
[4 marks] (ii) Solve the following equations:
log 2 ( x ) = 5 , (^) log (^) y ( 243 ) (^) = 5. [4 marks] (iii) A thermometer is used to measure the temperature of a house. Inside the house the temperature is 200 C. At time t = 0 the thermometer is moved to the outside of the house, where the air temperature is only. Three minutes later the reading of the thermometer has dropped to. Assuming the temperature of the thermometer, T , drops according to Newton’s Law of cooling, one can show that
50 C
100 C
T = 5 + Qe −^ kt.
Use the above information to calculate the two constants Q and k. How long after being put outside does it take the thermometer to register 70 C?
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11. (i) If α and β are the roots of the equation , write down the
values of a)
− 3 x^2 + 5 x − 3 = 0
αβ , b) α + β, c)^ α^2 +^ β^2 and d)(α^ − β)^2 ,^ without determining
the values of α and β individually.
[6 marks] (ii) Given the following cubic polynomial
p ( x ) = 2 x^3 + 6 x^2 − 4 x − 4 ,
calculate the values of p (− 4 ) , p ( − 3 ), p ( − 2 ), p ( − 1 ), p ( 0 ), p ( 1 ) and p ( 2 ). Hence find all the roots p ( x )= 0 , and sketch the curve. [9 marks]
12. (i) A complex number has modulus 1 and argument 5 π / 6. Express each of
the following complex numbers in the form a + b i:
z
z , z^2 , z^3 ,^1 ,
and plot them (separately) on the Argand diagram. [10 marks]
(ii) If ( x + i y ) 2 = a +i b show that x^2 − y^2 = a , 2 xy = b. Hence evaluate
8 + 6 i. [5 marks]
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