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o f Tr i n i d a d a n d To b a g o

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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