Grade 11 Tutorials - Maths Excellence

GRADE 11 TUTORIALS 

LO Topic 

Page 

1 Number patterns and sequences 3 

1 Functions and graphs 6 

2 Algebra and equations 9 

2 Finance 11 

2 Linear Programming 15 

3 Analytical Geometry 21 

3 Transformation 23 

3 Trig / Mensuration 27 

4 Data handling 31 

Grade 11 - 2 - Tutorials

Grade 11 Tutorial 

Number Patterns and Sequences 

Question 1 

Consider the following sequences and in each case: 

• Write down the next three terms. 

• Make a conjecture about the sequence. 

• State whether the general term will be linear, quadratic or neither of the two. 

1.1 3 ; 7 ; 11 ; 15 ; 19 ; ……….. 1.2 50 ; 44 ; 38 ; 32 ; 26 ; …….. 

1.3 2 ; 3 ; 5 ; 8 ; 13 ; ……….. 1.4 1 ; 2 ; 4 ; 8 ;16 ; ……….. 

1 

1 

1.5 2 ; 3 

2 

; 5 ; 6 

2 

; 8 ; ……….. 1.6 1 ; 4 ; 9 ; 16 ; 25 ; ……….. 

1.7 – 27 ; – 22 ; – 17 ; – 12 ; – 7 ; ……….. 1.8 2 ; 6 ; 18 ; 54 ; 162 ; ………… 

1.9 3 ; 6 ; 11 ; 18 ; 27 ; ……….. 

1.10 5 + 7x 

; 7 + 9x 

; 9 + 11x 

; 11+ 13x 

; …………. 

Question 2 

Consider the following sequences and in each case determine: 

• The next three terms. 

• A general algebraic formula that you can use to find any term of the sequence. 

[Check that your formula works] 

2.1 – 4, 1, 6 ; 11 ; 16 ; ................ 2.2 32 ; 16 ; 8 ; 4 ; 2 ; ………….. 

2.3 3 ; 10 ; 21 ; 36 ; 55 ; …………. 2.4 1 ; 8 ; 27 ; 64 ; 125 ; ………. 

2.5 1 ; 3 ; 9 ; 27 ; 81 ; ……………. 2.6 5; 11; 20 ; 32 ; 47 ; …………. 

2.7 

1 

3 

; 

2 

5 

; 

3 

7 

; 

4 

9 

; 

5 

11 

; …………… 

2.8 2 ; 3 ; 5 ; 8 ; 12 ; ………………. 


Question 3 

Once a year the City of Cape Town hosts a “long table dinner”. Small square tables are arranged next to each 

other in a straight row, so that one person sits on each of the available sides of the table. So for example with 2 

tables 6 people can be seated and with 4 tables 10 people can be seated. 

3.1 If 18 tables are arranged as described above, how many people can be seated? 

3.2 How many tables must be packed next to each other if 94 people must be seated? 

(Show all calculations). 

3.3 How many tables must be packed next to each other if 57 people must be seated? 

(Show all calculations). 

3.4 Determine an algebraic formula that would generalize the relationship between the number 

of tables and the number of people that can be seated. 

Question 4 

Thabo links balls with rods in arrangements as shown below: 

Arrangement 1 Arrangement 2 Arrangement 3 Arrangement 4 

1 ball, 4 rods 4 balls, 12 rods 9 balls, 24 rods 16 balls 40 rods 

4.1 Determine the number of balls in the nth arrangement. 

4.2 Determine the number of rods in the nth arrangement. 


Question 5 

Dots are arranged to form patterns as shown below: 

Pattern 1 Pattern 2 Pattern 3 Pattern 4 

5.1 Make a conjecture about the relationship between the number of dots and the Pattern number. 

5.2 Write an algebraic formula that generalizes the relationship in 5.1. 

Question 6 

In each of the following the general term for the sequence is given. Write down the first three terms in each 

case. 

6.1 T n 

= 10 − 4n 

6.2 = n 2 + 3 

T n 

6.3 T n = 

2 n−2 

6.4 T n = 7.3 n−1 

6.5 T n = n 2 – 2n – 3 6.6 T n = – n 2 + n 

6.7 T 1 = 5; T n = T n – 1 + 9 6.8 T 1 = 4; T 2 = 6 ; T n = T n – 2 + T n – 1 

6.9 

T n 

2n −1 

= 

3 n + 1 

Question 7 

Study the following sequences then: 

• Find a formula for the general term (Tn) 

• The number of terms in the sequence. 

7.1 5 ; 9 ; 13 ; 17 ; 21 ; …… ; 217 

7.2 1 : 3 ; 9 ; 27 ; 81 ; …. ; 59049 

7.3 10 ;.20 ; 40 ; 80 ; 160 ; ………; 5242880 

7.4 3 ; 8 ; 15 ; 24 ; 35 ; ……; 10200 



Functions and Graphs 

Question 1 

1.1 Sketch the graph of y = −( 

x −1) 

2 + 4 , showing the co-ordinates of the turning point and the coordinates 

of any intercepts with the axes. 

1.2 Write down the equation of 

1.2.1 the reflection of y = −( 

x −1) 

2 + 4 in the y-axis, 

1.2.2 the reflection of y = −( 

x −1) 

2 + 4 in the x-axis and 

1.2.3 the graph formed by translating by 1 unit to the right, the graph of y = −( 

x −1) 

2 + 4 . 

1.3 Does the point (−2; 

3) 

lie on the graph? Why or why not? 

Question 2 

− x+ 

p 

The functions y = f ( x) = 2 and y = g( x) 

= . The graphs intersect at A ( − 1; 4) 

y 

− x+ 

p 

= f ( x) = 2 on the x -axis is B ( 0 ;2) 

q 

x 

, and the intersection of 

^y 

A (−1;4) 

B (0; 2) 

O 

C 

D 

>x 

E 

2.1 Calculate the values of p and q . 

2.2 Write down the domain of = g( x) 

y = 

− x+ 

p 

2.3 Write down the range of y = f ( x) = 2 

− x+ 

p 

2.4 Write down the equation of the asymptote of y f ( x) = 

q 

x 

= 2 . 


q 

y = = . 

x 

2.5 Write down the equations of the axes of symmetry of g( x) 

2.6 CDE is a line perpendicular to the x -axis. Calculate the distance CE if the distance OD = 2 units. 

2.7 Describe in words the difference in the shape and position on a system of axes, of the graphs of 

− x 

− x 

y = 2 , y = 2 . 2 and y = 2 − x + 1 

2.8 Calculate the average gradient of each of the graphs in 2.7 between the points where: 

2.8.1 x = −1 

and x = 0 and 

2.8.2 x = 0 and x = 1. 

2.9 Do your results in 2.8 support any of the differences you mentioned in 2.7? Why or why not? 

Question 3 

Sketched below are graphs of p ( x) = 2x 

2 − x − 3 and q ( x) = −2 x + 3 

D 

^y 

F 

A 

G 

O 

E 

>x 

H 

B 

C 

3.1 Calculate the distance AB. 

3.2 Determine the co-ordinates of C, the turning point of the parabola. 

3.3 Calculate the co-ordinates of D and E, the points of intersection of the two graphs. 

3.4 Calculate the maximum length of FH between D and E where F lies on the straight line, H lies on the 

parabola and FH is parallel to the y -axis. 


Question 4 

0 

On the same system of axes, draw sketch graphs of = tan ( x + 45 ) 

y and y cos x x . 

Show clearly the co-ordinates of all intercepts with the axes, any points of intersection of the two graphs and 

draw any asymptotes, indicating the equation/s where applicable. 

Question 5 

Sketched below is a section of the graphs of f ( x) = a cosbx 

and g( x) csin 

dx 

= for ∈( −135 ) 

0 ; 45 0 

= . 

^y 

A 

(0° ; 1) 

O 

B 

E 

>x 

C 

D 

(90 °; −2 ) 

5.1 Calculate the values of a , b, 

c and d . 

5.2 What is the period of f ( x) = a cosbx 

5.3 What is the range of g( x) = csin 

dx 

5.4 Calculate the co-ordinates of B, C and E. These points are either intercepts with the x axis turning 

points. 

0 

0 

5.5 Determine the value of f ( 45 ) − g( 45 ) 



Algebra and Equations 

1. Solve for x : (Leave irrational answers in simplest surd form) 

x 1.2 ( x − 7 )( x − 5) = 3 

1.1 3 2 −17x + 15 = 0 

1.3 

1 

x = 1+ 

1.4 x 

3 − 2x 

2 − x − 2 0 

x 

= 

1.5 

2 1 

≤ 

x + 1 x −1 

1.6 2x 

2 + x > 6 

1.7 

x 1 3 

3 x 1 

− ≤ 1.8 ( − 3 ) 2 < 40 

− 

x 

x 

1.9 2 = 20 

(correct to 3 decimal places) 1.10 

x 

4 = 

2 

2 

2. Solve for x and y : 

2 

2 

2.1 2 x − y = 5 and x − xy + y + 4 = 0 

2.2 xy +12 = 0 and 2 y = x + 10 

2 

2 

2.3 2 x + 3y 

= 4 and x − xy − y = −1 

3. For which values of k does the straight line y = 2x 

+ k touch the hyperbola xy = −18 

at 

a single point? 

4. The product of two positive integers is 160. The difference between them is 27. What are 

the numbers? 

5. The members of a school photographic club decided to all pay the same amount to buy 

new equipment for R1 000. One of the members negotiated a 27,5% discount on the equipment and 

the amount that each had to pay was calculated. Then 4 more people joined the club. This meant that 

each member had to pay R4 less than was originally announced. How many members were originally in 

the club? 

6. A trip to a meeting 60 km from home took 5 minutes less than the return trip as the peak 

hour traffic reduced the average speed for the return trip by 10 km/h. Calculate the average speed 

travelled on the trip to the meeting. 


7. Michael throws a stone vertically upwards and after t seconds, its height h , in metres 

2 

above the ground, is given by h = 80t 

− 16t 

7.1 At what times was the stone 96 metres above the ground? 

7.2 After how many seconds will the stone hit the ground? 

7.3 Can the stone ever reach 128 metres? 

7.4 What is the maximum height reached by the stone? 

8. Two windmills working simultaneously fill a reservoir in 12 hours. The one windmill takes 

10 hours longer than the other when one or the other is used on its own. How long does each windmill, 

working separately, take to fill the reservoir? 

9. A population of insects is diminishing by 20% each week. If the population was estimated 

at 1 000 000 initially, in which week will it drop below 1 000? 

10. Prove that if a two digit number and the number formed by reversing the digits of 

the original number are added, the sum is always divisible by 11. 

For example 43 + 34 = 77 and 77 is divisible by 11. 

11. A small plane is flying from A to B with an air velocity of 200 km / h . There is a 50 km / h 

wind blowing against the plane, which means the actual velocity of the plane is 

150 km / h . Coming back, the wind still has a velocity of 50 km / h and so the actual velocity of the 

plane is 250 km / h . 

11.1 Show that the return trip described above takes longer than it would if there were 

no wind and the actual velocity of the plane were 200 km / h both ways. 

11.2 Show that regardless of the distance travelled and the velocity of the plane and of 

the wind, a return trip, against and then with a wind of constant velocity, will always take longer 

than the same trip with no wind. 

12. Imagine a piece of wire wrapped tightly around the earth along the equator. If the wire 

were lengthened by a single metre, it would not fit so tightly. Would a mouse be able to crawl between 

the earth and the wire, if the centre of the wire circle was the centre of the earth? 



Finance 

Question 1 

1.1 In each of the following, calculate the value of the investment at the end of one year and the total 

amount of interest earned. 

1.1.1 R12 000 invested at 4,5% per annum simple interest. 

1.1.2 R12 000 invested at 4,5% per annum compound. 

1.1.3 R12 000 invested at 4,5% per annum compounded monthly. 

1.1.4 R12 000 invested at 4,5% per annum compounded quarterly. 

1.2 Based on your calculations above, comment on which is type of interest gives the best return on your 

investment. 

Question 2 

Two companies used different methods of calculating the depreciation on capital equipment that they have 

purchased. 

Company A: The equipment is depreciated at a rate of 8% per annum, calculated on the purchase price of the 

item. 

Company B: The equipment is depreciated at a rate of 12% per annum, calculated on diminishing balance. 

2.1 Both companies bought a printer at the beginning of 2000 for an amount of R8 995. Complete the table 

below which shows the depreciated value of the printer at the end of each year. Show all calculations. 

A 8275,40 

B 7915,60 

2000 2001 2002 2003 2004 2005 2006 2007 

2.2 The companies write off their assets when the depreciated value is less than 20% of the purchase 

price. Determine the book value of the printer when it is ready to be written off. 

2.3 During which year will the book value of the printers in both companies be approximately equal? 

2.4 Determine the year in which each company will write off the printer. 

Grade 11 - 11 - Tutorials

Question 3 

Toby buys a sound system at a cost of R3240,00. He puts down a deposit of 15% and pays the balance 

according to a hire purchase agreement paid over two years. 

3.1 Calculate the deposit and the outstanding balance after the deposit is paid. 

3.2 According the hire purchase agreement, 17% simple interest is calculated on the full outstanding 

balance (after the deposit has been paid) for 2 years. Calculate how much Toby will pay in total for the 

sound system. 

3.3 If Toby pays off what he owes on the sound system in 24 equal monthly installments, how much will he 

pay each month. 

Question 4 

Shaheeda wants to buy an MP3 player. The player that she wants currently costs R1500. 

4.1 The MP3 player will increase in cost according to the rate of inflation, which is 6% per annum. How 

much will the MP3 player cost in two year’s time? 

4.2 Shaheeda puts R400 into her savings account at the beginning of every six month period for two years. 

Interest on her savings is paid at 7% per annum, compounded six-monthly. Will she have enough to 

pay for the MP3 player at the end of two years? Show all your work. 

Question 5 

On the day that Clint was born, his grandmother put R10 000 into a savings account for him as a fund to pay for 

university or college education. The day before his 18 th birthday, Clint goes to draw money to pay for his 

enrollment fees. The balance in the account, before Clint makes a withdrawal, is R41 658.37. What is the 

approximate annual compound interest rate earned over the investment period? 

Question 6 

Calculate approximately how long it will take for an investment of R20 000 to double in the following situations: 

(Work correct to a whole compounding period.) 

6.1 9,5% interest per annum, compound. 

6.2 5,5% simple interest, calculated six monthly. 

6.3 8% interest per annum, compounded monthly. 


Question 7 

R10 000 is invested for a period of two years under different investment conditions. In each case, calculate the 

value of the investment at the end of two years and hence determine the effective interest rate, compounded 

annually: 

7.1 10% per annum, compounded monthly. 

7.2 9,5% per annum, compounded daily. 

7.3 10,5% simple interest, calculated annually. 

Question 8 

Talita deposits R15 000 in a savings account. Her money earns 8% per annum interest, compounded quarterly. 

At the end of 6 months, Talita deposits another R7 000 into her account and the interest rate changes to 7,5% 

per annum, compounded quarterly. Calculate the balance in the account at the end of two years. 

Question 9 

A pension fund investment of R576 000 earns 8,75% interest per annum, compounded monthly. At the end of 

fourteen months, the interest rate changes to 9,25% per annum, compounded monthly and then remains 

unchanged for the rest of the investment period, which is 22 months. The investment is withdrawn after a total 

of three years. 

9.1 Calculate the value of the investment at the end of three years. 

9.2 What is the effective annual compound interest rate for the investment over the three year period? 

9.3 What is the nominal annual interest rate for the investment smoothed over the period of the 

investment? 


Question 10 

The graph below represents a comparison of investments. In each case, R15 000 was invested for a period of 

5 years at an interest rate of 9%. 

R 24,000.00 

Investment Comparison 

R 23,000.00 

R 22,000.00 

R 21,000.00 

Simple Interest 

Annual Compound 

Quarterly Compound 

Daily Compound 

Balance (R) 

R 20,000.00 

R 19,000.00 

R 18,000.00 

R 17,000.00 

R 16,000.00 

R 15,000.00 

0 1 2 3 4 5 

Number of Years 

10.1 Is there a significant difference in the investments after 1 year? Justify your answer. 

10.2 Which two investments are the most similar? 

10.3 If you were advising someone about interest on investments, which interest payment would you advise 

him/her to avoid and why? 

10.4 Give the equations of each of the lines above, using n to represent the number of years and F to 

represent the value of the investment after n years. 



Linear Programming 

Question 1 

Use the variables x and y and write down the equation or inequality that represents each of the following 

situations. State clearly, in each case, what x and y represent. 

1.1 When mixing green paint, the amount of yellow pigment must be at least double the amount 

of blue pigment. 

1.2 The number of hours spent on Maths and English homework each day should not exceed two. 

1.3 The sum of two numbers must be greater than 12. 

1.4 The profit made on selling an ice cream must be one-and-a-half times the profit made on 

selling a chocolate. 

1.5 In a hotel, the rooms can accommodate either two people or three people. The total number 

of guests in the hotel must be less than 250. 

1.6 M is the sum of two numbers. 

1.7 M is the sum of one number and double another number. 

Question 2 

Represent each of the following systems of inequalities graphically: 

2.1 

x ≤ 15 

2.2 

− 2 ≤ x ≤ 2 

2.3 

150x 

+ 60y 

≤ 30000 

y ≤ 12 

−1 

≤ 

y ≤ 5 

50x 

+ 60y 

≤ 13000 

x + y ≤ 20 

2x 

+ y ≤ 4 

10x 

+ 20y 

≤ 5000 

2.4 

x ≤ 200 

3x 

+ 2y 

≤ 2160 

5x 

+ 2y 

≤ 4800 

y 

x 

≥ 

3 

2 


Question 3 

Represent each of the following systems of graphs symbolically in terms of x and y: 

3.1 3.2 

300 

y 

y 

250 

200 

250 

200 

150 

150 

100 

100 

50 

50 

50 100 150 200 250 300 

x 

50 100 150 200 250 300 

x 

Question 4 

On the grid provided, draw the graph of 2 y + x = 12 , where x ≥ 0 and y ≥ 0 

4.1 Give three ordered pairs that are a 

solution to the equation 2 y + x = 12 . 

4.2 What is the gradient of the graph? 

4.3 State the x and y intercepts of the graph. 

4.4 Using a blue pen, shade the region 

which represents 2 y + x < 12 

4.5 Give three ordered pairs that are a 

solution to the equation 2 y + x < 12 


Question 5 

Nomhle and Shaun are playing a game with dice. There is one red dice and one blue dice. A throw (which is a 

throw of both dice) is awarded a point if it meets the following conditions: 

• Neither the red nor the blue dice must be 6. 

• The number on the red dice must be greater than the number on the blue dice. 

• The sum of the numbers of the two dice must not exceed 7. 

5.1 If x represents the number on the red dice and y represents the number on the blue dice, set up 

equations that represent the rules of the game as given above. 

5.2 On graph paper and on the same system of axes, draw the graphs of the equations that you 

have set up. 

5.3 Using your graphs, list all the throws that for which a point is awarded. 

5.4 If two points are awarded for an admissible throw where the sum of the two numbers on the dice 

is a maximum, which throws will be awarded two points? 

Question 6 

At SmoothSmoothies, the SuperSmoothies contain a base of yoghurt and banana. Customers may choose to 

add strawberry and melon, subject the following constraints, which have also been expressed mathematically: 

Using x to represent the scoops of strawberry and y to represent the scoops of melon, the diagram below 

represents the linear constraints in making a SuperSmoothie. 

6 

a) The added fruit may not exceed six scoops: 

( x + y ≤ 6 ) 

4 

b) At least one scoop of each fruit must be added. 

( x ≥ 1 and y ≥ 1) 

2 

5 

c) The number of scoops of strawberry may not be 

more than double the number of scoops of melon. 

( x ≤ 2y 

) 

d) No fractions of a scoop are allowed. ( x, y ∈ Z ) 

6.1 Use the letters (a), (b), (c) and (d) to match the constraints to their graphical representation 

in the diagram above. 

6.2 Write down any implicit constraints or constraints that are not represented in the diagram. 

6.3 On the diagram, shade the feasible region. 

6.4 What is the maximum number of scoops of strawberry in a SuperSmoothie? 

6.5 A customer orders a smoothie with 2 scoops of strawberry and 5 scoops of melon. Explain 

to the customer why it is not possible to have this option in a smoothie. 


6.6 If the customer wants 3 scoops of melon, how many scoops of strawberry can the customer have? 

6.7 If the customer wants six scoops of fruit, but scoops of strawberry are cheaper than scoops of melon, 

which is the cheapest option? (i.e. how many scoops of melon and how many scoops of strawberry?) 

Question 7 

A toy factory makes two different types of wooden toys –coloured blocks (x) and mobiles (y). According to its 

worker contracts, the factory guarantees each department a minimum amount of work per day. 

The cutting department must have at least 480 minutes of work per day. A set of coloured blocks takes 20 

minutes to cut and a mobile takes 10 minutes to cut. 

The decorating department must have at least 600 minutes of work per day. A mobile takes 20 minutes to 

decorate and a set of wooden blocks take 10 minutes to decorate. 

The assembly department must have at least 1080 minutes of work per day. A set of coloured blocks takes 10 

minutes to assembly and a mobile takes 60 minutes to assemple. 

7. If x represents the number of set of blocks and y represents the number of mobiles, present the 

information given above as a set of inequalities. 

7.2 On graph paper, graph the set of inequalities and indicate the feasible region. 

7.3 If materials for a set of blocks costs R15 and the materials for a mobile cost R45, set up an equation 

which represent the cost (C) of producing paint. 

7.4 Plot the cost equation on your graph and use it to determine which vertex will result in the most 

effective production schedule that minimises costs. 

7.5 Find the co-ordinates of this vertex by solving the appropriate two boundary lines. 


Question 8 

An organic farmer grows lavender and vegetables. He has a plot of land with an area of 9 hectares. In order to 

fulfil his market contracts the farmer has to plant at least 2 hectares of lavender and 1 hectare of vegetables. In 

order to minimise the damage done by insects, the area under lavender should be at least equal to the area 

under vegetables. In order to maintain the nitrogen balance of the soil, the area under lavender should not be 

more than double the area under vegetables. 

8.1 Below is a system of equations that models the scenario described. If x is the hectares under lavender 

and y is the hectares under vegetables, extract from the text the words that have resulted in the 

constraint being modelled (i.e. write down the words from the text that belong to each constraint). 

8.1.1 x ≥ y 

8.1.2 x ≥ 2 

8.1.3 x ≤ 2y 

8.1.4 x + y ≤ 9 

8.1.5 y ≥ 1 

8.2 Using graph paper, draw the system of equations and shade the feasible region. 

8.3 Set up a Profit equation if the profit on a hectare of lavender is R10 000 and the profit on a 

hectare of vegetables is R8 000. 

8.4 Graph the profit equation and use it to assist you in determining accurately the planting schedule 

that will result in the farmer maximizing profit. 

8.5 If the profit margins changed so that the profit on vegetables and lavender were equal, would it be 

necessary for the farmer to change his planting schedule? Explain your answer. 

Question 9 

Creative-cards is a small company that makes two types of cards, Card X and Card Y. With the available labour 

and material, the company cannot make more than 150 of Card X and not more than 120 of Card Y per week. 

Altogether they cannot make more than 200 cards per week. 

There is an order for at least 40 of Card X and 10 of Card Y cards per week. 

Creative-cards makes a profit of R5 for each of Card X and R10 for each of Card Y . 

Let the number of Card X manufactured per week be x and the number of Card Y manufactured per week be y. 

9.1 State all the constraint inequalities that represent the above situation. 

9.2 Represent the constraints graphically and shade the feasible region. 

9.3 Write the equation that represents the profit P (the objective function), in terms of x and y. 

9.4 On your graph, draw a straight line which will help you to determine how many of each type of card 

must be made weekly to produce the maximum profit. 

9.5 Calculate the maximum weekly profit. 


Question 10 

Vitamins B and E are vital ingredients of two types of 

health drinks, Energex and Vitagex 

• You require 3 g of Vitamin B and 4 g of Vitamin E to produce 1 litre of Energex 

• You require 9 g of Vitamin B and 6 g of Vitamin E to produce 1 litre of Vitagex 

• The company has 27 g of Vitamin B and 30 g of Vitamin E per day 

The above information is summarized in the table below: 

Ingredients Energex Vitagex Maximum grams 

Vitamin B 3 9 27 

Vitamin E 4 6 30 

• Furthermore at least 3 litres of Energex needs to be produced per day. 

Let x and y be the number of litres of Energex and Vitagex respectively that are produced per day. 

10.1 State algebraically, in terms of x and y, the constraints that apply to this problem for a day. 

10.2 Represent the constraints graphically on the graph paper provided and shade the feasible region. 

10.3 If the profit on 1 l of Energex is R30 and the profit on 1 l of Vitagex is R50, express the 

profit, P, in terms of x and y. 

10.4 Determine, by making use of a searchline, how many litres of each health drink must be 

produced in a day to ensure a maximum profit. 

10.5 Calculate the maximum possible profit. 



Analytical Geometry 

Question 1 

P is the point (5; 2) and Q is the point (3;6). Calculate the: 

1.1 length of the line segment PQ (correct to 1 decimal place); 

1.2 coordinates of the midpoint M of the line segment PQ; 

1.3 equation of the line through P and Q; 

1.4 equation of the perpendicular bisector of PQ. 

Question 2 

In the given sketch, ( − 4; 

−1) 

A , (1;2 ) 

B and ( 4; 

−3) 

C are the vertices of ∆ ABC . 

y ^ 

Β(1; 2 ) 

α 

>x 

Α(−4; −1 ) 

C(4; −3 ) 

2.1 Show that the triangle is right-angled at B. 

2.2 Calculate the size of , the inclination of AB (correct to 1 decimal place). 

2.3 Calculate the equation of the line parallel to AC, through the point B 

2.4 Calculate the area of ABC ∆ (remember, don’t round off numbers until you get to 

your final answer) 


Question 3 

3.1 Determine the values of x and y if; 

3.1.1 (1, 3) is the midpoint of the line segment joining (4, 5) and (x; y) 

3.1.2 (-1, y) is the midpoint of the line segment joining (0, -2) and (x; 8) 

3.1.3 (x; 3) is the centre of a circle with diameter MN where M is the point (5; -2) and 

N is the point (-7: y) 

3.2 P(-3: 1), Q(-5: -3) and R(1: -5) are 3 vertices of a triangle. 

3.2.1 Find the gradient of QR 

3.2.2 Find the equation of the altitude from P to QR. 

3.2.3 Find the length of PP’ if P’ is the reflection of P about the line y = x 

3.2.4 Find the midpoint of PP’ 

Question 4 

o 

The inclination of a line VW is 

1; 

− 3 

1; 

− 1 . 

Are the lines VW and XY parallel, perpendicular or neither parallel nor perpendicular? 

Show all working. 

45 . X is the point ( − ) and Y is the point ( ) 

Question 5 

Find the size of 

BA 

ˆ C 

in the figure alongside 

y 

A ( -2 ; 3 ) 

B ( 4 ; -1 ) 

x 

C ( -3 ; -2 ) 



Transformation Geometry 

Question 1 

T 

6 

y 

E 

R 

S 

4 

D 

C 

Q 

P 

2 

A 

B 

x 

-5 5 

V 

U 

-2 

W 

X 

-4 

Y 

-6 

1.1 State the general rule for the coordinates of any point representing the transformation of 

polygon ABCDE to polygon PQRST. 

1.2 Describe two possible transformations of polygon ABCDE to polygon UVWXY 

1.3 Give the coordinates of the reflection of point D in the line y = −x 

1.4 Describe the transformation in words if R is mapped to the point R’( 8 ; - 4 ) 

1.5 Give the coordinates of W’ if W is first rotated 

in the y-axis. 

o 

90 anti-clockwise and then reflected 


Question 2 

P ( -1 ; 2), Q (2 ; - 1 ) and R ( - 3 ; - 2 ) are the vertices of 

∆ PQR . 

P 

2 

y 

x 

-5 

Q 

R 

-2 

2.1 ∆ PQR is to be enlarged by a factor of 2 

2.2 ∆PQR 

2.1.1 Use the grid on the diagram sheet to draw this enlargement and clearly indicate 

the vertices P’Q’R’. 

2.1.2 Give the coordinates of vertices P’ and R’ of the enlargement. 

2.1.3 If the area of PQR is x square units, determine the area of the enlargement 

P’Q’R’. 

is rotated 

o 

90 in an anti-clockwise direction through the origin. 

2.2.1 State the general rule for the coordinates of a point satisfying this type of rotation. 

2.2.2 Give the coordinates of the vertices of P’’Q’’R’’ for this rotation. 

2.2.3 Write down the coordinates of R’’’ if R is reflected in the line x y = and then 

translated by the translation (-2 ; 3) 


Question 3 

P 

y 

8 

6 

S 

R 

Q 

4 

T 

2 

-5 5 10 

x 

-2 

-4 

-6 

-8 

U 

3.1 Describe the transformation in words that would transform ∆ PQR to ∆ STU 

3.2 Give the general rule for the coordinates of any point undergoing this transformation. 

3.4 Give the coordinates of S’ if S is transformed in this same manner again. 

3.3 Give the coordinates of P’, if P’ is the reflection of P in the line y = −x 

3.5 Give the coordinates of S’’, T’’ and U’’ if ∆ STU is translated to ∆ S’’T’’U’’ by the 

translation of (-2 ; 3). 

3.6 Write down the coordinates of T’’’ if T is reflected about the x-axis and then reflected 

about the line y = x . 

3.7 Write the coordinates of P’’ if P is translated by the rule ( x + 1; y − 2 ) and is then 

reflected about the y axis 


Diagram Sheet 

Question 2 

6 

y 

4 

P 

2 

x 

-5 5 

Q 

R 

-2 

-4 

-6 



Trig / Mensuration 

Question 1 

1.1 In which quadrant does θ lie if: 

1.1.1 sin θ < 0 and tan θ < 0 

1.1.2 cos θ > 0 and sin θ < 0 

1.1.3 tan θ > 0 and cos θ > 0 

1.2 If 

2 

cosθ = − and 180° ≤ θ ≤ 360° 

, use a sketch to determine the value of: 

13 

1.2.1 tan θ 

1.2.2 sin θ cosθ 

1.3 If tan θ = t and θ is acute, determine sin θ in terms of t. 

Question 2 

2.1 If x = 87, 6° 

and y = 240, 2° 

, use a calculator to evaluate each of the following correct to two decimal 

places: 

2.1.1 cos( x + y) 

2.1.2 

2 

sin( 2x 

− y) 

+ tan x 

sin y 

2.1.3 + 3tan 2x 

2.1.4 

cos x 

cos y 

2 

2.2 Without using a calculator, find the value of: 

2.2.1 tan 30° .sin 60° 

2.2.2 cos 2 45° + sin 30° 

2.2.3 cos 30° + sin 60° 

Question 3 

3.1 Reduce each of the following to a trigonometric ratio of x: 

3.1.1 sin( 180° + x) 

3.1.2 tan( 90° 

+ x) 

3.1.3 cos( 360° − x) 

3.1.4 

cos(90° − x) 

sin(360° − x) 

3.1.5 sin x − cos(90° − x) 

− tan(180° 

− x) 


3.2 Evaluate without using a calculator: 

3.2.1 tan 120° 

3.2.2 cos 630° 

3.2.3 sin 150° + tan 330° 

.cos30° 

3.2.4 

1 

3.3 Prove that: sin 240° 

tan 300° + cos330° 

= ( 3 + 3) 

Question 4 

4.1 Use basic trigonometric identities to simplify the following: 

2 

tan 315° + cos300° 

sin150° + tan135° 

2 2 

2 2 

4.1.1 tan y. 

cos y 

4.1.2 tan y sin y + tan y cos y 

4.1.3 

2 

1− 

cos y 

sin y 

4.2 Prove the following identities: 

2 

sin x 

4.2.1 1− = cos x 4.2.2 

1+ 

cos x 

cos x 

+ tan x = 

1+ 

sin x 

1 

cos 

x 

Question 5 

Find the general solution to the following equations. Give answers correct to two decimal places: 

5.1 sin θ = 0, 515 

5.2 3 − tanθ 

= 2, 4 

5.3 cos( θ + 20° 

) = −0, 

242 

5.4 2 sin( θ −15° 

) + 1 = 0 

5.5 cos 2θ 

= tan 24° 

Question 6 

On the same system of axes, draw sketch graphs of: 

f ( x) 

= sin x and g ( x) 

= cos(90° 

+ x) 

for the interval −180° 

≤ x ≤ 180° 

. Use the graphs to answer the 

following questions. 

6.1 Describe g (x) 

in terms of a reflection of f (x). 

6.2 Explain why f ( x) 

+ g( 

x) 

= 0 for all values of x. 


B 

Question 7 

68 o 

Refer to the diagram. 

7.1 Calculate the measurement of AB (correct to two 

decimal places. 

41 o 

C 

7.2 Calculate the area of the triangle. 

A 

11,2cm 

Question 8 

Refer to the diagram. 

8.1 Calculate the length of BC. 

8,5cm 

B 

8.2 Calculate the size of angle B. 

A 

17 o 

14,2cm 

C 

Question 9 

A surveyor is calculating the width of a river that is to have a 

bridge built across it. He takes measurements as follows: 

The distance from point P to point Q, on the same side of the 

river, is 64 metres. R P ˆQ = 62, 4° 

, B P ˆQ = 32, 1° 

and 

ˆQ = 59, 3° 

Calculate the width of the river. 

P 

R 

river 

32,1 o 62,4 o 64m 

59,3 o 

B 

Q 


Question 10 

In ∆ FGH , I is a point on FH. G HI ˆ = a, 

FGI ˆ = b , 

GH = f and FG = h . 

F 

Show that: 

10.1 

sin H 

= 

hsin( 

a − b) 

f 

h 

I 

a 

10.2 

GI 

= 

hsin( a − b) 

sin a 

b 

10.3 

Area ∆ FGI 

= 

h 

2 

sin( a − b)sin 

b 

2sin a 

G 

f 

H 



Data Handling 

Question 1 

Fifteen households were surveyed in Saldanha, Oudtshoorn and Stellenbosch with regard to the monthly 

amount spent on electricity. The results are recorded in the table below: 

Saldanha Oudtshoorn Stellenbosch 

R 52.00 R 103.00 R 246.00 

R 112.00 R 92.00 R 126.00 

R 83.00 R 67.00 R 226.00 

R 256.00 R 140.00 R 101.00 

R 412.00 R 136.00 R 92.00 

R 61.00 R 183.00 R 67.00 

R 54.00 R 214.00 R 63.00 

R 81.00 R 87.00 R 71.00 

R 147.00 R 145.00 R 167.00 

R 134.00 R 135.00 R 129.00 

R 61.00 R 164.00 R 117.00 

R 78.00 R 103.00 R 135.00 

R 225.00 R 129.00 R 129.00 

R 23.00 R 152.00 R 168.00 

R 189.00 R 118.00 R 131.00 

1.1 Calculate the following for each of the municipalities: 

1.1.1 The mean 

1.1.2 The median 

1.2 Comment on how these measures are similar or vary in 

the three sets of data. 

1.3 Explain why the mode is not a suitable measure to 

describe any of the sets of data. 

1.4 Calculate the following for each of the municipalities: 

1.4.1 The maximum value 

1.4.2 The minimum value 

1.4.3 The range 

1.4.4 The upper quartile 

1.4.5 The lower quartile 

1.4.6 The interquartile range 

1.5 Using the 5 number summary for each of the sets of data, draw box and whisker diagrams for 

each of the municipalities. Use the grid provided below: 

0 50 100 150 200 250 300 350 450 500 


1.6 Using the any of the statistical measurements you have calculated, as well as the box and whisker 

diagrams, write a short paragraph comparing the household expenditure on electricity in Saldanha, 

Oudtshoorn and Stellenbosch. 

1.7 The table below gives grouped data for the household electricity expenditure in Saldanha, Oudtshoorn 

and Stellenbosch. 

Expenditure in R 

Number of households 

Saldanha Oudtshoorn Stellenbosch 

Question 2 

A group of people were surveyed with respect to how long it took them to travel to work each day. The results 

of the survey is summarized in the chart below. 

Time taken to travel to work 

2.1 What is the range? 

Number of people 

450 

400 

350 

300 

250 

200 

150 

100 

2.2 What is the modal class? 

2.3 Which class contains the 

median? 

2.4 Calculate the estimated 

mean. Show your 

calculations and work 

correct to the nearest 

minute. 

50 

0 

0 - 14 minutes 






90 - 104minutes 

105 - 119 

minutes 

120 - 134 

minutes 

Time (minutes) 

Question 3 

Oudtshoorn 

Oudtshoorn Monthly 

Monthly Electricity 

Household Income (R) 

Expenditure (R) 

R 5,534.00 R 92.00 

R 5,886.00 R 103.00 

R 6,231.00 R 107.00 

R 6,671.00 R 139.00 

R 7,004.00 R 118.00 

R 7,421.00 R 135.00 

R 7,821.00 R 140.00 

R 7,974.00 R 145.00 

R 8,023.00 R 146.00 

R 8,368.00 R 164.00 

R 8,541.00 R 152.00 

R 8,718.00 R 168.00 

R 9,687.00 R 177.00 

R 10,355.00 R 183.00 

R 12,076.00 R 214.00 

The table alongside gives the monthly household income 

and monthly electricity expenditure for 15 families in 

Oudsthoorn. 

3.1 Calculate the mean monthly household income. 


3.2 Complete the table below and then 

calculate the standard deviation of the 

monthly household income. 

3.3 Calculate the percentage household 

incomes that fall within one standard 

deviation of the mean. 

3.4 Use the data in the table at the beginning of 

question 2 to draw a scatterplot of monthly 

household income against monthly 

electricity expenditure. 

Question 4 

3.4.1 Draw a line of best fit on your 

graph. 

3.4.2 Is the relationship between 

household income linear or 

exponential? Justify your answer. 

3.4.3 Use your graph to predict the 

electricity expenditure in a 

household with an income of R3500 

per month. 

3.4.4 The electricity supplier has 

requested that residents in Saldanha cut their electricity consumption by 10%. Assuming the 

residents cooperate with this request, how will the line of best fit be affected? 

The housing department recently conducted a survey on the number of bedrooms in 200 houses in Saldanha. 

The following data was obtained: 

No. of bedrooms 1 2 3 4 5 6 7 8 9 10 

No. of houses 25 40 42 28 22 18 12 7 4 2 

4.1 Set up a cumulative frequency table for the given data. 

4.2 Hence draw an ogive (cumulative frequency graph) for this data. 

4.3 Use the ogive to determine the median, lower quartile (Q1) and upper quartile (Q3). 

4.4 Draw a box and whisker diagram for this data. 

4.5 What is the value of the interquartile and semi-interquartile range? 

Income (R) ( x i 

− x) 

R 5,534.00 

R 5,886.00 

R 6,231.00 

R 6,671.00 

R 7,004.00 

R 7,421.00 

R 7,821.00 

R 7,974.00 

R 8,023.00 

R 8,368.00 

R 8,541.00 

R 8,718.00 

R 9,687.00 

R 10,355.00 

R 12,076.00 

n 

∑ 

i= 

1 

( x i 

− x) 

2 

2 

( x i 

− x) 


Question 5 

The monthly household income for families in Oudtshoorn is represented in the graph below. 

3 sd 

2 sd 

1 sd 

R2600 

1 sd 2 sd 3 sd 

The results suggest that the household income is symmetrically distributed with a mean of R2600 per month 

and a standard deviation of R950 per month. Research has suggested that if the monthly household income is 

below R1650 the family’s income would be deemed to be below the poverty line. 

It is also known that : 

68% of the monthly household income recorded is within one standard deviation of the mean: 34% above and 

34% below; 

96% are within two standard deviations of the mean: 48% above and 48% below ; 

100% are within three standard deviations of the mean: 50% above and 50% below. 

Estimate the percentage of families: 

5.1 Whose income is below the poverty line 

5.2 Earning more than R3550 per month 

5.3 Earning less than R700 per month.

Grade 11 Tutorials - Maths Excellence

Create successful ePaper yourself

Delete template?

Save as template?