WYNBERG GIRLS’ AND BOYS’ HIGH SCHOOLS

SUBJECT: MATHEMATICS

PAPER: 1 GRADE: 12

DATE: FRIDAY, 30 AUGUST 2013

TIME: 3 HOURS

TOTAL: 150 MARKS

EXAMINER: D BURRELL

MODERATOR: M LERESCHE

THIS EXAM CONSISTS OF 11 QUESTIONS AND 10 PAGES (INCLUDING A FORMULA SHEET)

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 2

INSTRUCTIONS:

1. This paper consists of 11 questions. Answer all the questions.

2. Number questions clearly and correctly according to the numbering system used in this

question paper.

3. A formula sheet has been provided.

4. Diagrams are not drawn to scale.

5. Write neatly and legibly, in blue or black pen. Diagrams should be done in pencil.

6. Be sure to show all relevant calculations and to provide reasons where necessary.

7. An approved non-programmable and non-graphical calculator may be used, unless

otherwise stated.

8. Where necessary round your answers off to 2 decimal places, unless other instructions are given.

________________________________________________________________________________________________

QUESTION 1

1.1 Solve for x:

1.1.1 3x2 = 2(x + 5) (give your answer correct to two decimal places) (4)

1.1.2 3.2x = 36 (3)

1.2 Consider the equation x2 – 6xy + 9y2 = 0.

1.2.1 Calculate the value(s) of the ratio xy (3)

1.2.2 Hence solve for x and y if x + y + 8 = 0. (3)

1.3 Given: f(x) = x2 – 3x.

1.3.1 Solve for x: f(x) ≥ 0 (3)

1.3.2 Hence write down the solutions to f(x – 3) ≥ 0 (2)

1.4 Simplify without using a calculator: 22010 + 22013

22011 +22012 (3)

[21]

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 3

QUESTION 2

2.1 If –19 and 21 are the second and seventh terms respectively of an arithmetic sequence, find the sequence. (6)

2.2 If x is a real number, show that the following sequence cannot be geometric:

1; x + 1; x – 3; … (4)

2.3 A series of diagrams is shown. Each diagram consists of triangles which are grey, white or black.

The table below shows the number of triangles of each colour used in the diagrams.

Diagram 1 2 3 4 n

Grey triangles 2 4 6 a d

White triangles 1 4 9 b e

Black triangles 0 2 6 c f

2.3.1 Write down the values of a, b and c, the number of triangles of each colour in Diagram 4. (1)

2.3.2 Find expressions for d and e, the number of grey and white triangles in the nth diagram. (2)

2.3.3 Show that f = n2 – n, where f is the number of black triangles in the nth diagram. (3)

2.3.4 Hence determine which diagram would have 1 260 black triangles. (4)[20]

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 4

QUESTION 3

3.1 The series 8 + 16 + 24 + 32 + … (to n terms) is given.

3.1.1 Determine an expression for the sum of the first n multiples of 8. (3)

3.1.2 Hence verify that this sum is one less than the square of an odd number. (Hint: An even number can be represented as 2n where n ∈ N ) (3)

3.2 In the diagram ABC is the largest of an infinite series of similar triangles which are inscribed inside each other. Each inscribed triangle has sides which are half the length of the corresponding sides of the triangle which circumscribes it. The sum of the perimeters of all the triangles is 72 cm. Find the perimeter of ΔABC.

(3)[9]

QUESTION 4

The diagram below represents the graph of f(x) = ax - p

+ q

P(-1; 4) is a point on the graph of f.

4.1 Determine the values of a, p and q. (4)

4.2 Write down the range of f. (2)

4.3 Write down the equations of the asymptotes of the graph y = f(x) – 1. (2)[8]

B C

A

33

P(-1;4)

f

f

1 x

y

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 5

QUESTION 5

Consider the function f(x) = 2 x + 1 – 2.

5.1 Calculate the coordinates of the intercepts of f with the axes. (4)

5.2 Write down the equation of the asymptote of f. (1)

5.3 Sketch the graph of f. (3)

5.4 If g is the graph of f shifted 3 units upwards and 1 unit to the right, write down the equation of the new function in the form g(x) = …. (2)

[10]

QUESTION 6

The graphs of f and g intersect at the origin and at point A.

6.1 Which of the graphs, f or g, is not a function? Give a reason for your answer. (2)

6.2 If f(x) = ax2 and the point (1; 2) is on f, show that a = 2 . (1)

6.3 If g = f -1, write down the equation of g in the form y = … (3)

6.4 Find the coordinates of A. (5)

6.5 If h is the reflection of f in the x-axis, write down the equation of h in the form y = … (1)

6.6 If the graph of f is shifted down by 3 units and left by 1 unit, give the equation of the new graph in the form y = ax2 + bx + c. (3)

[15]

g

f

(1;2)

A

y

x

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 6

QUESTION 7

7.1 R5 000 was invested at an interest rate of 6% per annum, compounded monthly. How long (in years and months) did it take for the amount to double? (5)

7.2 On 31 January 2012, Alex paid R250 into an account bearing interest at 8% per annum, compounded monthly. He continued to deposit R250 on the last day of every month until 31 December. He was hoping to have R3 250 by 1 January 2013 so that he could buy an air ticket to Namibia. Did he have enough money to buy the ticket? (4)

7.3 Nicky has been working for several years and she has decided to buy a flat for R625 000. She pays a deposit of 15% and takes out a housing loan from her bank for the balance. The interest rate on the loan is 12,5% per annum compounded monthly.

7.3.1 What is the value of Nicky’s loan from the bank? (1)

7.3.2 Calculate her monthly repayment if the loan is to be paid back over 20 years with the first repayment due on 31 August 2013. (5)

[15]

QUESTION 8

8.1 If f(x) = 2 – x2, determine f ‘(x) from first principles. (5)

8.2 Determine the following:

8.2.1 dydx if y = √x3−1

2 x2 (3)

8.2.2 Dr [π r2+r π+ π ] (2)

[10]

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 7

QUESTION 9

The graph of f(x) = x3 – 4x2 – 11x + 30 is shown.

9.1 Determine the coordinates of A and B, the turning points of the graph. (5)

9.2 Write down the turning points of g if g(x) = f(x – 2). (2)

9.3 Determine the average gradient of f between the points A and B. (3)

9.4 Show that the equation of the tangent to the graph at the point where x = 1 is y = –16x + 32 (4)

9.5 Determine the x-coordinate of the point at which the tangent in Question 9.4 cuts the graph of f again. (4)

9.6 For which value(s) of k will the equation x3 – 4x2 – 11x + 30 = k have only one real solution? (2)

[20]

B

A

f

y

x

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 8

QUESTION 10

A rectangular box of volume 32 cm3 has a length which is twice its width.

10.1 If the width is x cm, find the height (h) in terms of x. (2)

10.2 Show that the surface area (A) of the box is given by

the formula A = 4x2 + 96x (3)

10.3 Calculate the value of x if the surface area of the box is to be a minimum. (4)

[9]

QUESTION 11

11.1 A feasible region is described by the following set of constraints:

x + y ≥ 15 (1)x + 2y ≥ 18 (2)6x + 2y ≤ 36 (3)

x ≥ 0, y ≥ 0

Refer to the diagram alongside and answer the questions.

11.1.1 Determine which of the regions marked A to G is the feasible region. (2)

11.1.2 State which of constraints (1), (2) or (3) does not affect this feasible region. (2)

11.1.3 If it is further given that x ε N , determine the largest possible value of x that satisfies all of the constraints. (1)

h

2xx

xx

G

F E

D

C

B

A 15

10

5

2010 15 5

y

x

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 9

11.2 The quadrilateral in the diagram below represents the feasible region of a linear programming problem. It has vertices at A(2; 3), B(1; 5), C(3; 7) and D(6;2).

11.2.1 Which point in the feasible region will maximise the objective function P = 2x + y? (2)

11.2.2 The minimum value of a different objective function Q = 3x + y is obtained at B. Calculate the minimum value of Q. (2)

11.2.3 Given that point A is where the minimum value of objective function R = mx + k is found, determine the value(s) of m. (4)

[13]

Total: 150 marks______________________________________________________________________________

WGHS/WBHS Gr 12 Maths P1 – Aug 2013 Page 10

FORMULA SHEET