Question 1 (12 marks) Use a SEPARATE book clearly marked Question 1

(i) Find the remainder when the polynomial

2x3 - 3x2 + 7

is divided by (x - 2) .

(ii) Solve

2w = -7 (1000 - w).

(iii) Simplify

2n ´ 8n+1 .

(iv) Evaluate

32 log3 5 .

(v) If

A = {x : x ³ 2}

and

B = {x : x ³ -1}

find the set

A¢ Ç B .

(vi) Write

x2 + 6x -1

in completed square form.

(vii) Write down the value of x if

log5 x = -2 .

(viii) Solve

x + 1 = 7 .

(ix) Find the gradient of the tangent to the curve

y = x3

at the point where

x = -1.

(x) Sketch the function

y = e-2 x - 1

showing its essential features.

(xi) For the straight line function

f (x) = 3x -1 , write down the equation of the inverse

function

f -1 (x) .

Question 2 (12 marks) Use a SEPARATE book clearly marked Question 2

(i) Find the derivative of each of the following:

(a)

3x 2 - 2 .

x

(b)

x - 1 .

x + 2

(ii) Find the values of x for which the curve

y = x3 - 6x 2 + 5x - 8

is concave up.

(iii) Find

lim

x ®-2

x3 + 8

2x + 4 .

(iv) Find the value of   å24- k  .

k =1

(v) Find the value of the constant k if

x + 1

is a factor of

P(x) = x ⁴ + kx³ + 2 .

(vi) A mountain climber starts from a base camp at an altitude of 1000 metres with the intention of climbing to a summit at an altitude of 6500 metres.

After one day’s climbing he reaches an altitude of 2800 metres. Subsequently

on each day he climbs

2   of the distance climbed on the previous day.  Show 3

that the mountain climber cannot possibly reach the summit.

Question 3 (12 marks) Use a SEPARATE book clearly marked Question 3

(i) Differentiate

y = x2 +1

by first principles.

(ii) Jeffrey invests money in an account paying an annual interest rate of compounded quarterly.

7 × 2 %

(a) Calculate the amount in the account if he invests $2000 for 3 years.

(b) Calculate how much should he invest now in order to have $5000 in 6 years time.

(iii) Consider the curve given by

y = x³ - 6x + 4

(a) Find the coordinates of the stationary points and determine their nature.

(b) Find the coordinates of any points of inflection.

(c) Hence, sketch the function.

Question 4 (12 marks) Use a SEPARATE book clearly marked Question 4

(i) The graph of a function

y = f (x)

is shown below.

(a) Find the domain and range of

y = f (x) .

(b) Find the equation of the piecemeal function f (x) .

(c) On separate diagrams sketch:

(a )

(b )

y = - f (x - 2) .

the inverse of the function

y = f (x) .

(ii) The skirt of a dancer’s costume is to be decorated with 25 rows of sequins. The top  row will contain 16 sequins, the next row  18  sequins, the third row 20  sequins and  so on. Each row after the top row contains two more sequins than the previous row. Find the number of sequins used on the costume.

(iii) (a) Sketch the graph of the hyperbola

y = 1 - 1

2x - 3

showing the

asymptotes and the intercepts on the coordinate axes.

(b) Show that

y = 1 - 1

2x - 3

can be expressed as

y = 4 - 2x .

2x - 3

(c) Hence: (a )

solve the inequation

4 - 2x < 0 .

2x - 3

(b )

find the number of solutions to the equation

4 - 2x = ln x . 2x - 3

Question 5 (12 marks) Use a SEPARATE book clearly marked Question 5

(i) When a number is added to each of Find this number.

2 , 6 , 13,

a geometric progression is formed.

(ii) A man plans to erect a fence around a  972  square metre rectangular storage area next to a building, using the building as one side of the enclosed area. The fencing parallel  to the building will cost $9 per metre installed, while fencing for the other two sides  will cost $6 per metre installed.

(a) Find the length of each type of fence so that the total cost of the fence will be a minimum.

(b) Find the minimum cost.

(iii) 

Diagram not to scale

The graph shown above represents a polynomial P(x) of degree four. When the polynomial P(x) is divided by ( x - 1) , the remainder is equal to 6.

(a) Find an equation for P(x).

(b) Find the remainder when P(x) is divided by ( x + 1) .

(c) How many roots does the equation P( x ) = 2 have?

Question 6 (12 marks) Use a SEPARATE book clearly marked Question 6

(i) The graph of y =

f ( x) is drawn below. Stationary points occur at

x = 1 and

x = 3 .

Draw the graph of y = f ¢( x).

f(x)

x

(ii) One of the following formulas may be of assistance in this question:

é(1 + i) ⁿ - 1ù é1 - (1 + i) ^-n ù

FV  = PMT ê ú

ë i û

PV  = PMT ê ú

ë i û

Ronald Williams takes out a loan of $30 000, repayable by equal

monthly instalments over 25 years. Interest is charged at an annual rate of 15% , compounded monthly.

Find:

(a) the amount of the monthly repayment.

(b) the total interest paid.

(c) the amount still owed 5 years after the loan is taken out.

(d) the amount of interest paid during the fifth year of the loan.