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C o l l e g e o f S c i e n c e , Te c h n o l o g y a n d A p p l i e d A r t s
o f Tr i n i d a d a n d To b a g o
DIVISION OF LIBERAL ARTS AND HUMANDIVISION OF LIBERAL ARTS AND HUMAN SERVICESSERVICES
Mathematics DepartmentFinal Examination
Math 118: Pre-Calculus
TIME: 5:00p.m. -7:45p.m.
DATE: Wednesday 8th May, 2013
CENTRE: City
LECTURER: L. Bridglal
INSTRUCTIONS
Questions may be attempted in any order. They must however be labelled
correctly.
This examination comprises of six (6) questions.
You are required to answer four (4) questions.
The maximum marks for each question is indicated in parentheses to the right of
each question.
The use of a non- programmable electronic calculator is allowed.
Additional sheets of ruled paper are available upon request.
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
1. (a) The polynomial function f is given by f ( x )=x3+x2−10 x+8
State the potential rational zeros of f . (2 marks)
(i) Describe the end behaviour of f and state the maximum number of turning
points that the graph of f could have. (3 marks)
(ii) Show that (x−2) is a factor of f (x) . (3 marks)
(iii) Derive all the zeros
of f , state their respective multiplicities and whether they touch or cross the x
axis. (8 marks)
(b) Three zeros of a polynomial function f of degree four are 3, -1+5i and 2i.
Determine the polynomial f (4 marks)
(c) Use appropriate transformations to sketch the graph of f ( x )=(x−2)4−1
(5 marks)
[Total Marks = 25 marks]
2. (a) Sketch the graph of the function given by f ( x )=1+2x−1. Identify all
asymptotes and intercepts if they exist. Also state the domain and range of the
function. (8 marks)
(b) Find all real solutions to the following equations
(i) ex=e3 x+8
(3 marks)
(ii)log a x−log a( x−2 )=log a( x+4 )(4 marks)
(c) A pizza pan is removed at 5 pm from an oven whose temperature is fixed at 4500F
into a room that is a constant 700 F. The pan cools according to Newton’s Law of Cooling
according to the equation u (t )=T+ (u0−T )ekt ,where T is the constant temperature of the
surrounding medium, u0 is the initial temperature of the heated object and k is a negative
constant.
i. At what time is the temperature of the pan 1350 F (6 marks)
ii. Determine the time that needs to elapse before the pan is 1600 F (4 marks)
[Total Marks = 25 marks]
3. (a) The perimeter of a rectangle is 16 inches and its area is 15 square inches.
(i) Find two equations that describe the information given above. (2 marks)
(ii) Using the method of elimination find the two numbers. (10 marks)
b) Write the partial fraction decomposition of the following rational functions:
i)
x+2x3−2 x2+x (6 marks)
ii)
1( x+1)( x2+4 ) (7 marks)
[Total Marks = 25 marks]
4. (a) A rational function is given by
f ( x )= 3 x+6
x2−49 (i) Find the domain of the function. (2 marks)
(ii) Identify all asymptotes, and intercepts of the graph of f. (5 marks)
(b) A farmer has 300 feet of fence available to enclose a 4500 square feet region in
the shape of adjoining squares with sides of x and y as shown below.
(10 marks)
(c) Solve the following inequality
x3−9 x≤0 (4 marks)
(d) Determine whether or not the following function has a zero in the following interval:
f ( x )=8x5+5 x3+3x2+6 : [−1,0] (4 marks)
[Total marks = 25 marks]
5. (a) Determine the centre and radius of the circle given by the equation below
(3 mark)
(b) Find the equation of the parabola with vertex at (0,0) and focus at (3,0). Graph the equation. (4 marks)
(c) (i) Find an equation of the parabola with vertex at (-2,3) and focus at (0,3)(5
marks)
(ii) Graph the equation of the parabola (5 marks)
(c) A bridge is to be built in the shape of a parabolic arch and is to have a span of 100
feet. The height of the arch at a distance of 40 feet from the center is to be 10 feet. Find
the height of the arch at the center. (8 marks)
[Total marks = 25 marks]
6. (a) (i) Evaluate the sum of the first 75 terms of the sequence 0, -3, -6, -9, ... (5 marks)
(b) Tacoma’s population in 2000 was about 350, 000 and has been growing by about
12% each year.
(i) Find a recursive formula and an explicit formula for the population of Tacoma.
(3 marks)
(ii) If this trend continues what will be the population be in 2019? (3 marks)
(c) (i) Use the binomial theorem to expand (3 x3−2 y )4 . (6 marks)
(ii) What is the coefficient of x4
in the expansion of (2−x2 )9? (4 marks)
(c) How many different 4 letter words (arrangement of letters) can be formed from
TUESDAY? (4 marks)
END OF EXAM