Grade 11 November Examination 2018 Mathematics: Paper 1 ... 11 Papers/Herschel/Gr 11...Grade 11...

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Grade 11

November Examination 2018

Mathematics: Paper 1

Time: 3 hours Marks: 150

Instructions and Information:

Read the following instructions carefully before answering the questions.

1. This question paper consists of 9 questions. Answer ALL the questions.

2. Clearly show ALL calculations, diagrams and graphs that you have used in

determining your answers. All working should be shown in its proper place.

3. Answers only will not necessarily be awarded full marks.

4. An approved scientific calculator (non-programmable and non-graphical) may

be used, unless stated otherwise.

5. If necessary, answers should be rounded off to TWO decimal places, unless

stated otherwise.

6. Diagrams are not necessarily drawn to scale.

7. A marking rubric is provided. Please attach this to the front of your answer

sheet and write your name and the name of your Maths teacher on it clearly.

8. Number your answers according to the numbering system used in this

question paper.

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

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Question 1 [33]

1.1 Solve for x in each of the following:

1.1.1 2 3 5 0x x (2)

1.1.2 432 xx correct to 2 decimal place. (4)

1.1.3 42 xx (6)

1.1.4 𝑥2 + 3𝑥 − 10 ≤ 0 (3)

1.1.5

124232 1212 xxx (5)

1.2 The roots of a quadratic equation are 𝑥 = 3 ± √18 − 2𝑎2. For which real

values of a will the roots be equal.

(2)

1.3 Solve simultaneously for x and y:

2𝑥 − 𝑦 = 7

𝑥2 + 𝑥𝑦 = 21 − 𝑦2

(7)

1.4 Given 𝑚 = 12. Show that 65. 48. 24. 38 = 6𝑚12 without using a calculator. (4)

Question 2 [17]

NO CALCULATOR ALLOWED in this question. Show all necessary working.

2.1 Simplify:

2.1.1 2

1

81

16

(3)

2.1.2

9𝑥+32𝑥+1

18𝑥.2−𝑥

(5)

2.2 If 14

√63−√28 can be written in the form 𝑎√𝑏, determine, without using a

calculator, the value(s) of a and b, where a and b are integers.

(4)

2.3 Erin had to find the product of 22007 and 52000 and then calculate the sum of

the digits of the answer. Erin arrived at an answer of 11.

Is she correct? Show ALL the calculations to motivate your answer.

(5)

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Question 3 [15]

3.1 Given the linear pattern: 156; 148; 140; 132; …

3.1.1 Write down the 5th term of this pattern. (1)

3.1.2 Determine a general formula for the nth term of this pattern (2)

3.1.3 Which term of this linear number pattern is the first term to be negative? (3)

3.1.4 The given linear number pattern forms the sequence of first differences

of a quadratic number pattern 𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐 where 𝑇5 = −24.

Determine the general formula (𝑇𝑛) of this quadratic pattern. (5)

3.2 A plant starts growing from a single shoot. At the end of the month, a flower

blooms at the tip of the shoot. The next month, two new shoots branch off

from the first one, and at the end of that month, a flower blooms at the tip of

each new shoot. This growth pattern is shown below.

Month 1 2 3

Flowers 1 2 4

Shoots 1 3 7

3.2.1 How many flowers will appear on the plant during the 4th month? (1)

3.2.2 Write down an expression for the number of flowers that will bloom

during month n. (1)

3.2.3 How many shoots will there be on a 5-month-old-plant? (2)

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Question 4 [9]

The number of new businesses in a small town is growing according to a pattern. Examine

the table below and answer the questions that follow.

Year 2001 2002 2003 2004

New businesses formed

3 8 17 30

4.1 How many new businesses will be formed in 2005 if the pattern continues? (2)

4.2 If there were no businesses in the town at the beginning of 2001, how many

businesses were there in the town by the end of 2003?

(1)

4.3 Find a general term for the number of new businesses formed in the nth year,

where 1n corresponds to the year 2001.

(4)

4.4 Use your model from 4.3 to determine how many new businesses

were formed in 2010.

(2)

Question 5 [14]

5.1 A machine costs R45 000 today and has a scrap value of R9 000 after 10

years. Determine the annual rate of depreciation if it is calculated on the

reducing balance method.

(5)

5.2.1 Lisa invested R14 500 in a bank for a period of 8 years at 6,5% p.a.

compounded monthly for the first 5 years and at 8,4% p.a. effective for the

remaining period. If Lisa runs into financial difficulties and withdraws R3 000

after 6 years, calculate how much her investment is worth at the end of the 8

years.

(7)

5.2.2 Calculate the effective interest rate which is equivalent to a nominal interest

rate of 6,5% p.a. compounded monthly.

(2)

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Question 6 [23]

6.1 The graph of 𝑓(𝑥) = 𝑎. 2𝑥 + 1 (a is a constant) passes through the origin as shown

below.

6.1.1 Show that 𝑎 = −1 (2)

6.1.2 Determine the value of 𝑓(−15) correct to FIVE decimal places. (1)

6.1.3 Determine the value of x, if (𝑥; 0,5) lies on the graph of f. (3)

6.1.4 Give the range of f. (2)

6.2 Given the functions:

ℎ(𝑥) =3

𝑥−2+ 3 and 𝑔(𝑥) = −𝑥 − 3

6.2.1 Use the set of axes provided on the diagram sheet to draw neat sketch

graphs of h and g. Label each graph and show any intercepts with the

axes and any asymptotes. Also show at least one other point on each

graph. (6)

6.2.2 Write down the domain of h. (2)

6.2.3 Determine the equation of function k which is the reflection of h in the

y-axis. (2)

6.2.4 If the graph of h is reflected across the line with the equation

𝑦 = −𝑥 + 𝑐, the new graph coincides with the graph of 𝑦 = ℎ(𝑥).

Determine the value of c. (2)

6.2.5 Using your graph, give the value(s) of x for which 𝑥. ℎ(𝑥) ≤ 0. (3)

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Question 7 [14] The sketch alongside represents the graphs of functions f

and g where:

𝑓(𝑥) = 𝑥2 − 2𝑥 + 4 and 𝑔(𝑥) = 𝑥 + 1

7.1 Calculate the co-ordinates of the turning point of f. (3)

7.2 Calculate the co-ordinates of D if DM is parallel to the x-axis and 𝑂𝑀 = 7 units. (4)

7.3 Calculate the minimum length of AB if AB is parallel to the y-axis, with A on the

parabola and B on the straight line. (5)

7.4 For which value(s) of x is 1

𝑓(𝑥).𝑔(𝑥) undefined? (2)

x

y

D M A

B

f

g

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Question 8 [7]

Amy, an enthusiastic basketball player, is practising her shooting.

She throws from a point 1,7m from the floor. Each throw follows the path of a parabola. On

one of her throws, the ball reaches its maximum height of 3,1625m when it has covered a

horizontal distance of 3m. Unfortunately the ball does not go into the basket but hits the rim

which is 3m above the floor.

Determine how far Amy’s hand is from the front of the rim at the moment she

releases the ball (i.e. the length of AB).

A

B

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Question 9 [18]

9.1 For events A and B:

P(A) = P(B) = 0,4

P(A and B) = 0,1

9.1.1 Are events A and B mutually exclusive? Give a reason for your answer. (2)

9.1.2 Show whether events A and B are independent or not. Give a reason

for your answer.

(2)

9.1.3 Determine P(A or B). (2)

9.2 In a survey, a group of 283 workers were asked what mode of transport they use to get to work. The results of the survey are summarized below.

x workers take a car (C), a bus (B) and a taxi (T) to get to work.

110 workers take a car and a taxi.

38 workers take a taxi and a bus.

32 workers take a car and a bus but not a taxi.

60 get to work by taxi only.

110 workers take a bus.

172 workers take a car.

No workers use any other mode of transport.

9.2.1 Draw a Venn diagram to represent the information above. (8)

9.2.2 Determine the number of workers who take a car, a bus and a taxi to

work (i.e. x).

(2)

9.2.3 What is the probability that a worker picked from the sample takes a

car and a bus to work? Give your answer as a fraction.

(1)

9.2.4 What is the probability that a worker picked from the sample takes a

bus but not a car? Give your answer as a fraction.

(1)

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T O T A L : 1 5 0 M A R K S Grade 11 Paper 1 DIAGRAM SHEET November 2018 Name: ____________________________________Maths Teacher:____________________________

Q 1 Q 2 Q 3 Q 4 Q 5 Q 6 Q 7 Q8 Q 9

TOTAL Equations Algebra Patterns &

Sequences Patterns & Sequences

Finance Functions Functions Functions Probability

33 17 15 9 14 23 14 7 18 150

Question 6.2.1

−8 −6 −4 −2 2 4 6 8

−8

−6

−4

−2

2

4

6

8

x

y

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INFORMATION SHEET: MATHEMATICS

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Grade 11 November Examination 2018 Mathematics: Paper 1 ... 11 Papers/Herschel/Gr 11...Grade 11...

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