SUBJECT Mathematics PAPER Paper 1

GRADE 11 DATE 24 November 2017

EXAMINER Mrs T Meredith MARKS 150

NAME MODERATOR Maths Department

TEACHER DURATION 3 Hours

QUESTION NO

DESCRIPTION MAXIMUM MARK

ACTUAL MARK

1, 2 & 3 Algebra, Equations and Inequalities 46

4 Patterns and Sequences 17

5 Finance 16

6, 7 & 8 Functions and Graphs 45

9, 10 & 11 Probability 26

TOTAL 150

Grade 11 2 of 12 Mathematics Paper 1

INSTRUCTIONS:

1. This paper consists of 11 questions and 121 pages.

2. Read the questions carefully.

3. Answer all questions.

4. Number your answers clearly and use the same numbering as in the question paper.

5. You may use an approved non-programmable and non-graphical calculator, unless

otherwise stated.

6. Round off your answers to two decimal digits where necessary.

7. All necessary working details must be shown. Answers only, without the relevant

calculations will not be given marks. Equations may not be solved solely with a calculator.

8. It is in your interest to write legibly and present your work neatly.

9. You may detach the Information Sheet at the back of the question paper.

10. Detach the Graph Paper attached at the back of the question paper and hand in with

your answer booklet.

Grade 11 3 of 12 Mathematics Paper 1

QUESTION 1 27 MARKS

Solve for the unknown in the following equations, rounding all decimals to 2 decimal places where necessary.

a) (𝑥 − 3)(𝑥 + 2) = 0 (2)

b) 𝑥2 + 6𝑥 − 2 = 0 by completing the square and leaving your answer in surd form. (5)

c) Solve for 𝑥 and 𝑦 simultaneously if: 3𝑦 + 𝑥 = 2 and 𝑦2 + 𝑥 = 𝑥𝑦 + 𝑦 (6)

d) 6𝑥2 + 13𝑥 − 5 ≥ 0 and represent your answer using a number line and interval

notation. (5)

e) 𝑥2 + 𝑥 +18

𝑥2+𝑥= 11 (9)

QUESTION 2 11 MARKS

a) Given: 𝑓(𝑥) =2𝑥−𝑥2

𝑥+5 . Determine the value(s) of 𝑥 for which:

1) 𝑓(𝑥) = 0 (2)

2) 𝑓(𝑥) is undefined (1)

b) Determine the nature of roots of 𝑓(𝑥) = 𝑥2 − 2𝑥 − 7 (4)

c) For the given equation: 2𝑥2 + 𝑝𝑥 − 2 = 0

1) Find the value of 𝑝 if one root of the equation is 1

2 . (2)

2) Hence find the other root of the equation. (2)

QUESTION 3 8 MARKS

Simplify the following showing all your working.

a) 𝑥3

4⁄ (5𝑥)( √𝑥4

) (2)

b) 4𝑛+1×82𝑛−(23)

2𝑛×4𝑛

162𝑛−1 (6)

Grade 11 4 of 12 Mathematics Paper 1

QUESTION 4 17 MARKS

a) Refer to the number sequence below and answer the questions that follow showing all

your working:

2 ; 6 ; 12 ; 20 ; 30 ; 42 …

1) Determine the 𝑛𝑡ℎ term of the sequence. (4)

2) Determine the value of the 145th term. (1)

b) The first 3 terms of an arithmetic sequence are given below:

(𝟐𝒙 + 𝟏) ; (𝒙 − 𝟑) ; (𝟑𝒙 − 𝟒) …

1) Determine the value of 𝑥. (4)

2) Determine the 𝑛𝑡ℎ term of the sequence. (2)

c) Refer to the number sequence below and answer the questions that follow:

20 ; 5 ; 5

4 ;

5

8 …

1) Determine the 𝑛𝑡ℎ term of the sequence. (2)

2) Determine the position of the value 5

64 in the sequence. (4)

Grade 11 5 of 12 Mathematics Paper 1

QUESTION 5 16 MARKS

Your latest obsession is the Maverick Pro Drone – a drone that can fly up to 60km/hr while transmitting fully stabilized, high quality footage. The Maverick Pro Drone can be purchased online today for R20 000.

a) You decide to begin saving for your drone today in order to buy it in 5 years time. You

deposit R12 000 into a savings account today and 3 years later you add R2 000 to your

savings account. The interest rate for the first 3 years is 12% p.a. compounded quarterly

after which it changes 11% p.a. compounded monthly for the remaining 2 years.

1) Calculate the value of your savings 5 years after opening the account by drawing a

timeline. (5)

2) Calculate the effective interest rate during the first 2 years. (3)

3) If the price of the drone increases at the same rate as inflation (8% p.a.), calculate

the price of the drone in 5 years’ time. (3)

4) Will you be able to afford the drone in 5 years’ time with your savings alone?

Calculate the value you will have in excess or that you will still need. (2)

b) Your parents spoil you and buy you the drone for your birthday. What will the value of the

drone be in 5 years’ time if it is purchased today and the rate of straight-line depreciation

is 6,5% p.a. (3)

QUESTION 6 9 MARKS

Detach the graph paper at the back of your question paper and use it to sketch the following functions on the same Cartesian plane:

a) 𝑚(𝑥) = −2𝑥2 + 4𝑥 (5)

b) 𝑝(𝑥) = −3 (1

2)

𝑥

(4)

Grade 11 6 of 12 Mathematics Paper 1

QUESTION 7 30 MARKS

a) The sketch shows 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 and 𝑔(𝑥) = 𝑚𝑥 + 𝑐.

The turning point of the parabola is (1; 8) and both 𝑓(𝑥) and 𝑔(𝑥) pass through the point

(0; 6). T is a point on the x-axis.

1) Determine the equation of 𝑓(𝑥) and hence show that:

𝑎 = −2 ; 𝑏 = 4 ; 𝑐 = 6 . (5)

2) Determine the coordinate of point T and hence determine the equation of 𝑔(𝑥). (6)

3) If PQ is a line parallel to the y-axis with point P on 𝑓(𝑥) and Q on 𝑔(𝑥), determine

an expression for the length of PQ in terms of 𝑥. (3)

4) Determine the maximum length of PQ. (3)

𝑦

𝑔 (1 ; 8)

(0 ; 6)

𝑓

𝑥 T

P

Q

Grade 11 7 of 12 Mathematics Paper 1

b) Sketched below is the parabola 𝑓, with equation 𝑓(𝑥) = −𝑥2 + 4𝑥 − 3 and hyperbola 𝑔,

with equation 𝑔(𝑥) =3

𝑥+𝑝+ 𝑞.

B, the turning point of 𝑓, lies at the point of intersection of the asymptotes of 𝑔.

A (−1 ; 0) is the x-intercept of 𝑔(𝑥).

1) Show that the coordinates of point B are (2 ; 1). (4)

2) Write down the range of graph 𝑓. (2)

3) Determine for which values of 𝑥 will 𝑔(𝑥) ≥ 0. (2)

4) Determine the values of 𝑥 for which 𝑓(𝑥) increases. (2)

5) Determine the values of 𝑝 and 𝑞 in the equation of 𝑔(𝑥). (2)

6) Determine the equation of the vertical asymptote of the graph ℎ if ℎ(𝑥) = 𝑔(𝑥 + 4).

(1)

𝑥

𝑦

Grade 11 8 of 12 Mathematics Paper 1

Grade 11 9 of 12 Mathematics Paper 1

QUESTION 8 7 MARKS

Given below is a sketch of 𝑦 = 𝑏𝑥 .

a) Write down the coordinate of point P. (1)

b) Use the sketch of 𝑦 = 𝑏𝑥 to help you determine which of the given graphs best fit each of

the following equations:

1) 𝑦 = 2𝑏𝑥 (2)

2) 𝑦 = 𝑏𝑥 − 1 (2)

3) 𝑦 = 𝑏𝑥+3 (2)

𝑥

𝑦

𝑛

𝑚

P

𝑥

𝑦

2𝑛

2𝑚

A

𝑥

𝑦

𝑛 − 1

𝑚

B

𝑥

𝑦

𝑛

𝑚 + 3

C

𝑥

𝑦

2𝑛

𝑚

D

𝑥

𝑦

𝑛

𝑚 − 3

E

Grade 11 10 of 12 Mathematics Paper 1

QUESTION 9 11 MARKS

Your granny has the best sweet bowl in town – and this morning you were allowed to take 2 treats from the bowl! A summary of all the contents of the sweet bowl as well as a partially completed tree diagram depicting your random choice is given below.

a) Using the information given above, calculate the probability of the following choices for

your lunchbox:

1) 2 boxes of Smarties. (2)

2) A box of Smarties first and then a packet of Jelly Tots or Wine Gums. (3)

3) A packet of Wine Gums and a box of Smarties. (3)

4) A box of Smarties and a packet of Jelly Tots or a 2nd box of Smarties. (3)

Treat Type

Box of Smarties 3

Packet of Jelly Tots 5

Packet of Wine Gums 4

S

JT

WG

S

JT

WG

S

JT

WG

S

JT

WG

312⁄

512⁄

412⁄

211⁄

511⁄

411⁄

311⁄

411⁄

411⁄

311⁄

511⁄

311⁄

Grade 11 11 of 12 Mathematics Paper 1

QUESTION 10 7 MARKS

a) Postal codes in South Arica consist of 4 digits.

For example, the suburb of Brooklyn in Pretoria has the code 0011.

Calculate:

1) The total number of postal codes that can be created. (2)

2) The probability that Tom selects the correct code for Bruma in Johannesburg if he

knows that Johannesburg’s codes start with 2, and he randomly guesses the other

digits. (2)

b) Calculate the number of unique words that can be created from the letters of the word

BANANAS. (3)

Grade 11 12 of 12 Mathematics Paper 1

QUESTION 11 8 MARKS

A survey of 100 grade 11 learners showed the following information about the online multiplayer games they played.

8 students did not play any games.

45 students played League of Legends

35 played Call of Duty

44 played Overwatch

13 students played League of Legends and Call of Duty

18 played League of Legends and Overwatch

11 played Call of Duty and Overwatch

Some students played all 3 games

A partially completed Venn Diagram is given below:

a) Calculate the missing values for the following in terms of 𝑥, showing all your working.

1) A (1)

2) B (2)

3) C (2)

4) D (1)

b) How many students played all 3 games? (2)

League of

Legends

Call of

Duty Overwatch

13 - 𝑥

14 + 𝑥

11 + 𝑥

𝑥

A

B

C

D