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DURBAN GIRLS' COLLEGE

TRIALS EXAMINATION 2009

MATHEMATICSPAPER I

TIME: 3 HOURS 150 MARKS

EXAMINER: Mrs M Dwyer

MODERATOR: Mrs G Hawkey

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY.

  • This question paper consists of 8 pages. All questions must be answered in the answer book provided.
  • Answer all questions and show all working.
  • Number your answers exactly as the questions are numbered.
  • Do not redraw diagrams.
  • All answers must be given correct to one decimal place, where necessary, unless stated otherwise.
  • Approved calculators may be used, unless stated otherwise.

SECTION A

1. Solve for x and/or y:

a) 2-2x = (3)

b) log x 125 = 3(2)

c) (2x + 3)(x + 1) = 3(4)

d) 2x2 – 3x  0(3)

e) y = 2x –5 and x2 + xy – 2y2 = 0(7)

[19]

2a) For the sequence 4; 7; 12; 19; 28; … determine

i) the common second difference(1)

ii) the formula for Tn(2)

iii) the value of the 100th term.(2)

b) Evaluate (2)

c) How many terms are there in the sequence 22/3 ; 22 ; 66 ….1782? (6) [13]

3a) When John started working, he started saving so that he could become

self-employed. If he invested R1000 per month into an account which paid

13% p.a. compounded monthly, how much will he have saved after 5 years?(4)

b) A businessman bought office equipment for R70 000, which depreciated to

R30 000 at a rate of 15% p.a. on a reducing balance basis. How long did

this take?(4)

[8]

4a) Determine:

i) f’(x) from first principles if f(x) = - x2 + 2.(5)

ii) if y = 3x2 - 2 + (4)

b) g(x) = 1 – x – 3x2. Calculate:

i) g(-2)(1)

ii) g’(-2)(2)

c)

The figure above shows the graph of f(x) = (x + 2)(x – 1)2.

Determine the co-ordinates of A, B and C.(3)

[15]

5. Determine the values of a, p and q in the equations of the following graphs:

a) y = a(x – p)2 + q

(4)

b) y = a cos x + q

(2)

c) Draw rough sketches of the following on separate axes:

  1. y = (2)
  2. y = (2)
  3. y = 4x(2)

d) f(x) = ; g(x) = 2x2 – 2

Calculate:

i) f(3) + g(2)(3)

ii) the zeroes of g(3)

iii) x, if f(x) = 5(3)

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SECTION B

6a)

 

   



Susie comes to school with Flu and infects two friends. The next day, they each then spread the flu to two friends and so on. Assume that this pattern continues on a daily basis.

  1. How many girls will contract Flu on the 5th day?(2)
  2. How long will it take for all 420 of the girls in the High School to be

infected? Give your answer correct to the nearest day.(3)

b i) Show that the series is convergent.(3)

ii) Calculate its sum.(2)

c)Determine the first term of a geometric sequence with a 4th term

of and an 11th term of .(6)

[16]

7. When you apply for a home loan, your repayments may not be more than

30% of your net (after deductions) salary.

a) If your net monthly salary is R12 000, and the home loan is offered at

11,25% p.a. compounded monthly to be paid over 20 years, can you afford

a flat worth R350 000 without paying a deposit?(5)

b) You buy the flat and then get an increase in salary. You decide to pay an

increased instalment of R4 500 per month. How long will it take to pay off

your loan? (5)

[10]

8.

In the figure above, A(-1; 0) and C(!;-4) are the turning points of the graph of f.

B(0; -2) and D(2; 0) are y- and x-intercepts.

i) Without determining the equation of f, give the values of x in each of the

following:

a) f(x)  0(2)

b) f’(x)  0(2)

c) f’(x) = 0(2)

d) f(x). f’(x) > 0 and x < 0(2)

ii) State which of the graphs below is the graph of f’(x). (2)




Justify your answer.

A B C

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9. If y = 4x – 4 is a tangent to y = x2 + ax + b at (1; 0), determine the

values of a and b[4]

10) The slant height of a cone is 6cm.

If you change the shape of the cone,

keeping the slant height at 6cm, the 6

volume changes. h

[Volume of a cone =  area of base x h] r

a) Find r2in terms of the height. (2)

b) Show the volume is given by V = 12h - (2)

c) Find the height for which the volume will be a maximum. Leave your

answer in simplest surd form.(4)

[8]

11.

Above is an architect’s image of the completed Moses Mabhida stadium.

The arch, seen in a side view below, can be approximately modelled by the

equation

y = -x2 + x + 42

where x (in metres) is the horizontal distance from the end of the stadium

and y (in metres) is the height above the stadium floor.

A cable car will carry visitors up the north side of the arch, on the left side of the picture above, to a viewing platform at the top of the arch.

a) Determine an expression to show the rate at which the height of the

cable car changes. (2)

b) At what rate is the height changing when the car has travelled 20m from

A, its starting point?(2)

c) Determine the maximum height above the stadium floor reached by the

cable car.(2)

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a) If B(2; - 1) lies on f, determine the value of a.(2)

b) A, the turning point of g(x) also lies on the graph of f.

Prove that A is ( ½ ; 1)(3)

c) Determine the values of r and t.(5)

d) Determine the equation of f -1(x).(1)

e) Determine the equation of h(x) if h is the reflection of f in the y-axis.(2)

[13]

13a) If , show that a = 4(3)

b) Hence factorise (4)

[7]