(1.) Let f (t) = sin t and g(t) = 3t − 45.

solve the equation f (g(t)) = 1 for t in the

interval 0 ≤ t ≤ 120◦ .
(2.) These days, due to advancement in technology, the police can determine the speed a
car was travelling from the length of the skid marks it leaves. The function they use is
√
S = f (L) = 2 5L

where S is speed (mph) and L is the length of the skid marks (feet).
(i.) If skid marks of length 125 feet are measured, what was the speed of the car?
(ii.) Find a formula for the inverse function.
(iii.) If the car had been travelling at 80 mph, how long would the skid marks be?
(3.) At a jazz club, the cost of an evening is based on a cover charge of $20 plus a beverage charge of $7
per drink.

(i.) Find a formula for t(x), the total cost for an evening in which x drinks are consumed.
(ii.) If the price of the cover charge is raised by $5, express the new total cost function, n(x), as a
transformation of t(x).
(iii.) The management increases the cover charge to $30, leaves the price of a drink at $7, but includes
the rst two drinks for free. For x ≥ 2, express p(x), the new total cost, as a transformation of
t(x).

(4.) (i.) Given the function f , give a possible formula for f with the following properties.
f has degree ≤ 2, f (0) = f (2) = 0 and f (3) = 3.
(ii.) Use row-echelon form and back-substitution to solve the system of linear equations;
−2x − 2y − 15z = 0
x + 2y + 2z = 18
3x + 3y + 22z = 2

(5.) Hooke's Law for springs states that the distance a spring is stretched or compressed is directly
proportional to the force on the spring. Suppose that a force of 20 pounds stretches a spring 5 inches
as shown in the gure below,

(i.) nd a mathematical model that relates the distance the spring is stretched to the
force applied to the spring.
(ii.) how far will a force of 30 pounds stretch the spring?
 (6.) If you have a string of length 50cm, what are the dimensions of the rectangle of maximum area that
you can form with your string?
[Hint: You may write the function in the vertex form using completing of squares.]

(7.) Suppose that Sam and John travelled to separate university campuses in the same amount of time
to make enquiries. Sam drove 380 miles and John drove 400 miles. John's average speed was 3 miles
per hour faster than Sam's average speed.
(i.) Determine Sam's average speed.
(ii.) What is John's average speed?

(8.) Write a formula for the graph of the function shown below.

(9.) Suppose that f (x) is invertible and that both f and f −1 are dened for all values of x.
Let f (2) = 3 and f −1 (5) = 4. Evaluate the following expressions;
(i.) f −1 (3).
(ii.) f (4).
(10.) (i.) The graph of P = g(t) contains the point (6, −9). What point must lie on the graph of
1 1
R = − g(2t − 4) − ?
2 2

(ii.) An investment is made in a trust fund at an annual interest rate of 8.75%, compounded
continuously. How long will it take for the investment to double?[Use A = P ert ]
(11.) (i.) Simplify the expression
4i3250 + 11i17 − 25i21
2 − i251 + 4i + 5i150
(ii.) Solve the simultaneous equations for the complex numbers A1 and A2 .
A1 + A2 = 2 and (1 − i)A1 + (1 + i)A2 = 0

(iii.) Simplify the rational expression

x3 − 16x
x2 − 2x − 8
(iv.) Solve the logarithmic equation
log3 2x − log3 (x − 3) = 1

(v.) Solve the exponential equation

5 + ex+1 = 20

(12.) Mr. Mensah wants to paint one of his East Legon mansions in readiness for a programme on Christmas
Eve. He determines that the number n of liters of paint needed to complete the painting is n = f (A),
where A is the surface area of the house measured in square meters. Write mathematical expressions
for the following statements.
i. Mr. Mensah gured out how many liters of paint he needed and then bought
two extra liters just in case.
ii. Mr. Mensah bought enough paint to cover his house twice.
iii. He bought enough paint to cover his house and his welcome sign which measures
2 square meters.

(13.) Suppose each cable of the Adomi Bridge is suspended, in the shape of a parabola, between two towers
that are 1, 280 meters apart. The top of each tower is 152 meters above the roadway. The cables
touch the roadway at the midpoint between the towers as shown in the gure below.

(i.) Write an equation that models the cables of the bridge.

(ii.) Find the height of the suspension cables over the roadway at a distance of 320 meters of the
bridge.

(14.) Suppose that you have just purchased a triangular parcel of land near Academic City University
College to be used for the construction of a Students' Hostel and would like to know the exact area of
your land. The pictorial view of the land is shown in the gure below. Use the method of determinants
to nd the area of your land. (Note: The measurements in the gure are in feet.)
 For Questions 15 and 16, use Cramer's rule and row reduction to solve the system of equations.

(15.) Laws that deal with electrical currents are known as Kirchho 's Laws. When Kirchho's Laws are
applied to the electrical network shown in the gure, the currents I1 , I2 and I3 are the solution of the
system

Find the currents.

(16.) When three forces are applied to a beam, Newton's Laws suggest that the forces F1 , F2 and F3 are
the solution of the system

Find the forces.

(17.) Find the vertex of y = 0.03x2 + 1.8x + 2 exactly. Graph the function, labeling all intercepts.
(18.) An investment grows by 5% per year for 20 years. By what percent does it increase over the 20-year
period?
(19.) A sum of $850 is invested for 10 years and the interest is compounded quarterly. There is $1000 in
the account at the end of 10 years. What is the nominal annual rate?