+KING DAVID HIGH SCHOOL LINKSFIELD

MATHEMATICS PAPER 1

GRADE 12

PRELIMINARY EXAMINATION

AUGUST 2015

Total: 150 marks

Reading Time: 10 minutes Writing Time: 3 hours

Name:

This paper comprises a question paper of 14 pages and a separate data sheet

Check that your paper is complete

Write your name on the A4 paper.

Please read the following instructions carefully:

1. Number all questions exactly as they appear on the question paper.

2. Pay careful attention to time management and mark allocation.

3. Write legibly and not in pencil.

4. Non programmable calculators may be used unless otherwise instructed.

5. All answers to be given to 2 decimal places where appropriate.

6. All necessary calculations must be clearly shown. You will NOT receive full credit if you write down only the answers and show no working out.

Q1[5]

Q2[13]

Q3[20]

Q4[8]

Q5[18]

Q6[14]

Q7[13]

Q8[11]

Q9[10]

Q10[9]

Q11[13]

Q12[6]

Q13[10] TOTAL: = %

150

Question 1

Page 2 of 14

Solve for x

√5−2 x−2 x=1 [5]

Question 2

(a) Mr I M Owen plans to buy a house for two million rand. He has to pay a deposit of 15% and take a loan from the bank for the balance owing. Investor Bank offers him an interest rate of 9,5% per annum compounded monthly.

(i) If he pays off the loan over 20 years what are his monthly payments? (5)

(ii) What is the effective interest rate? (3)

(b) R6000,00 is invested at 11,6% per annum compounded quarterly. After how many years will the investment be worth R46000,00.

Give your answer correct to the nearest whole number. (5)

[13]

Question 3

Page 3 of 14

The sketch below (not drawn to scale) represents the graphs of

f ( x )=k x+q and g( x )=ax2+bx+c .

B is the turning point of g. The graphs intersect at B ( 12;3)

The co-ordinates of A and D are (0;2) and (-1;0) respectively.

(a) Write down the equation of the asymptote of f (1)

(b) Determine the equation of f (3)

(c) Show that the numerical value of a is −43 (4)

(d) Determine the co-ordinates of C (2)

(e) Determine the equation of f−1

in the form f−1( x )=.. .. (3)

(f) Write down the domain of f−1

(2)

(g) Determine the values of x for which

(i) g( x )≥f ( x ) ifx≥0 (2)

(ii) f ( x ). g( x )≤0 (3)

[20]

Page 4 of 14

Question 4

(a)

(i) Determine the equation of the curve f drawn above if the sketch represents

f ( x )= a

x−p+q

(3)

(ii) Determine the co-ordinates of the x-intercept of f. (2)

(b) Sketch a curve of y=ax2+bx+c if

a < 0, c < 0 and b2 = 4ac (3)

[8]

Page 5 of 14

Question 5

(a) A simple musical instrument has many strings, attached as shown in the

diagram. The difference between the lengths of adjacent strings is

constant. The shortest string is 30 cm long and the longest string is 50 cm.

The sum of the lengths of all the strings is 1240 cm.

Find the number of strings. (3)

(b) Consider the sequence

3; 1; 3; 2; 3; 4; 3; 8…

Calculate the sum of the first 20 terms of the sequence. (4)

(c) Consider the sequence

27(x+3)2; 9(x+3)3; 3(x+3)4; …. x ≠ −3

(i) Determine the values(s) for x for which the series converges(3)

(ii) Determine the sum to infinity of the series if x = −1(3)

Page 6 of 14

(d) Given that ∑k=1

n

T k=3n2+2

Determine

(i) S6 (2)

(ii) T6 (3)

[18]

Question 6

(a) Given

x−1x

1+ 1x

=1

, solve for x. (5)

(b) If x 2 +4x + 5 – p 2 = 0, find the value for p so that the equation has non- real roots. (5)

(c) Without using a calculator, calculate the exact value of

√72015√72013−√72011 (4)

[14]

Page 7 of 14

Question 7

(a) Find the derivative of each of the following functions:

(i) f ( x ) = 5x .√x (2)

(ii)g( x ) = x2−5

x2 , x ≠ 0 . (3)

(b) Find

dkdx if x

2 = 20 x − 4 k . (3)

(c) Calculate the value of m for which the graph defined by

y = −2x3 + 6 x2 + mx + 4 will have only one stationary point. (5)

[13]

Page 8 of 14

Question 8

The graph of f is defined as f(x) = ax3 + bx2 + cx + d.

f cuts the x-axis at the points (-4; 0) ; (-1; 0) and (2; 0) and the y-axis at (0; 8).

(a) Determine the values of a, b, c and d. (5)

(b) Determine the point of inflection. (Show the necessary calculations.) (4)

(c) A copy of this graph appears on the attached answer sheet. Draw a rough

sketch of flon the same system of axes which clearly indicates the

relationship between f andfl. Do not work out the co-ordinates of the

important points (eg stationary points, intercepts) (2)

[11]

f

- 4 2

8

-1 x

y

Page 9 of 14

Question 9

A display case, in the form of a rectangular prism, is made of thin glass. The case has no bottom since it will be place over a display on a raised platform. The dimensions of the base are 4 xm and x m. The height isym.

(a) Express the volume, V , of the box, in terms of x and y . (1)

(b) The total surface area of the box is 6m2.

Prove that y = 3

5x− 2 x5.

(4)

(c) Find the value of x which gives a maximum volume. (5)

[10]

y

x

4 x

Page 10 of 14

Question 10

(a) Use the sketch of f in the diagram below to answer the following questions:

(i) Is f a function? Explain. (2)

(ii) State the range of f(x) + 2 (2)

(iii) State the range of f(x+2) (2)

(iv)Give a restriction on the domain of f that would ensure f−1

is a function. (1)

Page 11 of 14

Question 10 (continued)

(b) Part of the graph of the function y= f ( x ) is

shown opposite.

Which of the following graphs could represent the

related function y=−f ( x )−3 ?

A B

C D

(2)

−6

o x

y −5

6

5 xo

y

(5 ,−6)

o x

y

(−5 , 6)

o x

y

x

y= f ( x )

(5, 3)

−3

o

y

Page 12 of 14

[9]

Question 11

(a) A soccer star is suffering from an injury. The probability that the player will be

fit to play in the upcoming match is

13. When he plays the probability that his

team will win is

34 but otherwise it is only

12 .

(i) Set up a tree diagram to illustrate all the possible outcomes. (3)

(ii) Hence (or otherwise) calculate the probability that the team will win the next game. (3)

(b) (i) In how many different ways can the letters of the word PARABOLA

be arranged? (3)

(ii) Determine the probability that an arrangement will have all the A’s

together.(4)

[13]

Question 12

A and Bare two independent events.

P(A) = x and P(B) = y

P( A∩B ) = 13 and

P (A∪B ) = 910

Page 13 of 14

Show that 30 y2+10=37 y [6]

Question 13

Congruent triangles made of matchsticks are stacked as shown in the figures below.

Fig. 1 Fig. 2 Fig.3 Fig.4

(a) Complete the table:(write the answers on your folio paper – not on this sheet)

Figure 1 2 3 4 5

No.of matches

3 (i) 18 30 (ii)

(2)

(b) Determine a formula for the number of matches in the nth figure. (5)

(c) Would it be possible to make a stack of triangles using exactly 135 matches? (3)

[10]

Page 14 of 14

Answer Sheet

Question 8(c)

Name:………………………………………………………………………………………

f

- 4 2

8

-1x

y