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Mathematics
Paper 1
FORM 4
November 2014
TIME: 3 hours TOTAL: 150 marks
Examiner: Mrs A Gunning / Moderated: Mrs B Philpot
Questions and mark allocation / Q1 (16) / Q2 (16) / Q3 (9) / Q4 (13)
Q5 (10) / Q6 (13) / Q7 (5) / Q8 (5) / Q9 (16)
Q10 (9) / Q11 (14) / Q12 (16) / Q13 (8) / Total:
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY BEFORE ANSWERING THE QUESTIONS.
  • This question paper consists of 8 pages including an information sheet. Please check that your question paper is complete.
  • Read and answer all questions carefully.
  • It is in your own interest to write legibly and to present your work neatly.
  • Number your answers exactly as the questions are numbered.
  • All necessary working which you have used in determining your answers must be clearly shown.
  • Approved non-programmable calculators may be used except where otherwise stated. Where necessary give answers correct to 2 decimal places unless otherwise stated.
  • Diagrams have not necessarily been drawn to scale.

SECTION A

QUESTION 1

Solve for x in each of the following, leaving your answers correct to 2 decimal places where relevant:

(a)(4)

(b)(3)

(c)(4)

(d)(5)

[16]

QUESTION 2

(a)Given that , calculate:

  1. (1)
  2. if (4)

(b)The equation has no real roots. Determine the value of m(3)

(c)Determine, without using a calculator the value of y, given that

and (5)

(d)Simplify:showing all relevant working detail

(3)

[16]

QUESTION 3

(a)By completing the square, solve for x in the following equation: (4)

(b)Hence, or otherwise give the coordinates of the turning point of

(2)

(c) State the domain and range of(3)

[9]

QUESTION 4

(a) Given

  1. If the graph has the line as its axis of symmetry, determine the value of b.

(1)

  1. And if , what is the range of f ?(1)

(b)If is any function, what is the relationship between:

  1. and (1)
  2. and (1)
  3. and (1)

(c)Given

Determine the co-ordinates of the turning points of:

  1. (2)
  2. (2)

(d)Match the equation in Column 1 with an option from Column 2 which best describes the graph. Write ONLY the letter of choice next to the number.

COLUMN 1 / COLUMN 2
/
  1. Vertical line

/
  1. Horizontal line

/
  1. A line with a positive gradient

/
  1. A line with a negative gradient

  1. Asymptote

  1. Hyperbola

  1. Parabola

  1. Exponenial

(4)

[13]

QUESTION 5

(a)There are 5 blue smarties and 4 pink smarties in a box. 1 smartie is taken out and eaten, then a second. Determine the probability of the following. (Using a tree diagram may help you).

  1. Both smarties being blue(2)
  2. One of each colour being chosen.(2)
  3. At least one pink smartie being chosen.(2)

(b)It is given that and . Showing all relevant working detail and giving reasons,

  1. State whether or not C and D are independent.(2)
  2. Hence, or otherwise, give the value of .(2)

[10]

QUESTION 6

Given:

(a)For what value of is ?(2)

(b)Determine the value of if the point B(0 ;) lies of the graph.(2)

(c)Write down the equations of the asymptotes of .(2)

(d)Draw the graph of on the set of axes provided on the annexure, clearly indicating the asymptotes and intercepts. (3)

(e)The equation of 1 of the axes of symmetry for is .Determine, showing all working, the value of . (4)

[13]

QUESTION 7

In the sketch the following points are given: B (5 ; 0), C ( 0 ; -4) , D ( k ; -4 )

The parabola’s axis of symmetry has the equation

(a)Determine the coordinates of A.(1)

(b)Determine the value of k.(1)

(c)Determine the equation of the parabola.(3)

[5]

QUESTION 8

Given:

(a)Write down the equation of the asymptote of h.(1)

(b)Determine any intercepts with the axes, showing clearly all relevant working detail.(4)

[5]

QUESTION 9

A bank offers interest on investments at a rate of 15% per annum compounded monthly.

(a)Calculate the effective interest rate equivalent to this.(2)

(b)Determine how much a person must invest now (as a lump sum) so that they have R20000 in four years time. (2)

(c)Should a person follow the following savings plan (in the bank with 15% per annum compounded monthly):

1 September 2014:Deposit R3500

1 February 2016Deposit R8100

1 January 2017Withdrawal R4200

1 July 2017Deposit R8500

Determine if the person would have more than R20000 in her account by the 1st January 2018. Show all relevant working. (7)

(d)This same person has a new trailer worth R8500. The depreciation rate on this item is 10% per annum using a reducing balance method. Calculate the value of his trailer after 4 years. (2)

(e)Draw a sketch graph showing the reducing value of the trailer over the four year period. (3)

[16]

QUESTION 10

In a herd of 200 Nguni cattle, the field ranger counted

  • 130 cattle with black patches
  • 110 cattle with brown patches
  • cattle with both black and brown patches
  • 5 cattle with neither brown nor black patches

(a)Draw a Venn diagram to represent the above information, showing values in terms of x. (4)

(b)Calculate how many cattle have both black and brown patches(3)

(c)Determine the probability that if the field ranger randomly selected a single Nguni from the 200 cattle, it would only have brown patches. (2)

[9]

SECTION B

QUESTION 11

(a)Given: , determine the values of for which C is a real number.(3)

(b)Solve for x:

  1. using a suitable substitution. Where relevant, leave your answers in the simplest surd form. (7)
  2. (4)

[14]

QUESTION 12

A field ranger did an analysis of information about the Eastern Cape Aloe.

End of 1st year / End of 2nd year / End of 3rd year / End of 4th year
Number of leaves on the Aloe / 2 / x / 2x+1 / 4x

(a)She suspects that the pattern of the number of leaves, as shown in the table she drew up, has a constant second difference. Use this fact to calculate how many leaves are on the Aloe at the end of the 4th year. (5)

(b)Determine an expression for the number of leaves on the Aloe at the end of the nth year. (6)

(c)Determine the end of which year the Aloe will have 572 leaves.(5)

[16]

QUESTION 13

James takes off from the top of a skateboard ramp.

The ramp can be modelled by , while James’ path can be modelled by with units in metres.

(a)Calculate the coordinates where James:

  1. Takes off (A)
  2. Lands (B)(5)

(b)Determine the highest point that James reaches above the ramp while airborne.(3)

[8]

INFORMATION SHEET

Gunning 124