MATHEMATICS Grade 11 banks/Mathematics/11 Maths... · 2020. 3. 23. · Calculate the effective interest rate. (4) 8.3 The value of a property increased from R145 000 to R221 292,32

Western Cape Education Department

Examination Preparation Learning Resource 2018

MATHEMATICS Grade 11

Razzia Ebrahim

Senior Curriculum Planner for Mathematics

Website: http://www.wcedcurriculum.westerncape.gov.za/index.php/fet-futher-education-training/sciences/mathematics-fet-home

Website: http://wcedeportal.co.za/

mailto:[email protected]/

mailto:[email protected]

http://www.wcedcurriculum.westerncape.gov.za/index.php/component/jdownloads/category/1835-grade-12?Itemid=-1

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Index

Content Page

3

4 – 8

09 – 15

16 – 21

22 – 28

29 – 34

35 – 40

41 – 51

52 – 62

63 – 75

76 – 87

88 – 99

100 – 111

CAPS - GRADE 11

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1. Paper 1 & 2: Instruction & Information

2. 2017 November Paper 1





7. 2013 Exemplar Paper 1






13. 2013 Exemplar Paper 2

Mathematics/P 1

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1. This question paper consists of NINE questions.

2. Answer ALL the questions.

DBE/November...........

3. Number the answers correctly according to the numbering system used m thisquestion paper.

4. Clearly show ALL calculations, diagrams, graphs et cetera that you have used indetermining the answers.

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

6. You may use an approved scientific calculator (non-programmable andnon-graphical), unless stated otherwise.

7. Round off answers to TWO decimal places, unless stated otherwise.

8. Diagrams are NOT necessarily drawn to scale.

9. Write neatly and legibly.

Copyright reserved Please turn over

Mathematics!P2

INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

DBE/November

1. This question paper consists of ............ questions.

2. Answer ALL the questions in the ANSWER BOOK provided.

3. Clearly show ALL calculations, diagrams, graphs et cetera that you used to determine the answers.

4. Answers only will NOT necessarily be awarded full marks.

5. Round off answers to TWO decimal places, unless stated otherwise.

6. Diagrams are NOT necessarily drawn to scale.

7. You may use an approved scientific calculator (non-programmable and non-graphical), unless stated otherwise.

8. Write neatly and legibly.

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Mathematics/P1 DBE/November 2016


QUESTION 1

1.1 Solve for x in each of the following:

1.1.1 places) decimal TWO correct toanswer your (leave 0153 2 xx

(3)

1.1.2 0862 xx

(3)

1.1.3 024 2 xx (4)

1.1.4 1222 313 xx

(4)

1.1.5 431 xx (6)

1.2 Solve for x and y simultaneously:

023 yx and 822 xxy

(6)

1.3 Show that the roots of kxkx 1)2(3 2

are real and rational for all values of k. (4)

[30]

QUESTION 2

2.1 Simplify fully, WITHOUT using a calculator:

2.1.1 2.1010

2.51

22

aa

aa

(5)

2.1.2 6

66

12

4827

m

mm

(3)

2.2 WITHOUT using a calculator, show that 28

8

21

2

(4)

[12]

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QUESTION 3

Consider the quadratic pattern: – 9; – 6 ; 1 ; 12 ; x ; …

3.1 Determine the value of x. (1)

3.2 Determine a formula for the nth

term of the pattern. (4)

3.3 A new pattern, nP , is formed by adding 3 to each term in the given quadratic

pattern. Write down the general term of nP in the form .cbnanPn 2

(1)

3.4 Which term of the sequence found in QUESTION 3.3 has a value of 400? (4)

[10]

QUESTION 4

4.1 Given the linear pattern: 18 ; 14 ; 10 ; …

4.1.1 Write down the fourth term. (1)

4.1.2 Determine a formula for the general term of the pattern. (2)

4.1.3 Which term of the pattern will have a value of – 70? (2)

4.1.4 If this linear pattern forms the first differences of a quadratic pattern,

Qn, determine the first difference between Q509 and Q510.

(2)

4.2 A quadratic pattern has a constant second difference of 2 and 29175 TT .

4.2.1 Does this pattern have a minimum or maximum value? Justify the

answer.

(3)

4.2.2 Determine an expression for the nth

term in the form

.cbnanTn 2

(5)

[15]

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QUESTION 5

Given: f(x) = –2x2 + x + 6

5.1 Calculate the coordinates of the turning point of f. (4)

5.2 Determine the y-intercept of f. (1)

5.3 Determine the x-intercepts of f. (4)

5.4 Sketch the graph of f showing clearly all intercepts with the axes and turning point. (3)

5.5 Determine the values of k such that f(x) = k has equal roots. (2)

5.6 If the graph of f is shifted two units to the right and one unit upwards to form h,

determine the equation h in the form .)( 2 qpxay

(3)

[17]

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QUESTION 6

The diagram below shows the graph of .2

1)( and 1

3

1)( xxg

xxf

The graph of f intersects the x-axis at A and the y-axis at B.

The graph of f and g intersect at points C and D.

6.1 Write down the equations of the asymptotes of f. (2)

6.2 Determine the domain of f. (2)

6.3 Calculate the length of:

6.3.1 OB (2)

6.3.2 OA (3)

6.4 Determine the coordinates of C and D. (6)

6.5 Use the graphs to obtain the solution to: 2

2

3

1

x

x

(4)

[19]

x

y

O

g

f

f

A

CB

D

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QUESTION 7

The sketch below is the graph of ..2)( 1 qbxf x

The graph of f passes through the points A(1 ; 20) and B (–1 ; y).

The line y = 2 is an asymptote of f.

7.1 Show that the equation of f is 2321

xxf (3)

7.2 Calculate the y-coordinate of the point B. (1)

7.3 Determine the average gradient of the curve between the points A and B. (2)

7.4 A new function h is obtained when f is reflected about its asymptote.

Determine the equation of h.

(2)

7.5 Write down the range of h. (1)

[9]

x

y

B(1 ; y)

A(1 ; 20)

f

2

0

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QUESTION 8

8.1 A machine costs R25 000 in 2016. Calculate the book value of the machine after

4 years if it depreciates at 9% p.a. according to the reducing balance method.

(3)

8.2 The nominal interest rate of an investment is 12,35% p.a., compounded monthly.

Calculate the effective interest rate.

(4)

8.3 The value of a property increased from R145 000 to R221 292,32 over 6 years.

Calculate the average annual rate of increase of the property over 6 years.

(4)

8.4 Tebogo made an initial deposit of R15 000 into an account that paid interest at

9,6% p.a., compounded quarterly. Six months later she withdrew R5 000 from the

account. Two years after the initial deposit she deposited another R3 500 into this

account. How much does she have in the account 3 years after her initial deposit?

(5)

[16]

QUESTION 9

9.1 Given: P(A) = 0,2

P(B) = 0,5

P (A or B) = 0, 6 where A and B are two different events

9.1.1 Calculate P(A and B). (2)

9.1.2 Are the events A and B independent ? Show your calculations. (3)

9.2 A survey was conducted amongst 100 learners at a school to establish their

involvement in three codes of sport, soccer, netball and volleyball. The results are

shown below.

55 learners play soccer (S)

21 learners play netball (N)

7 learners play volleyball (V)

3 learners play netball only

2 learners play soccer and volleyball

1 learner plays all 3 sports

The Venn diagram below shows the information above.

9.2.1 Determine the values of a, b, c, d and e. (5)

9.2.2 What is the probability that one of the learners chosen at random from

this group plays netball or volleyball?

(2)

a

S N

V

3

2 1

b

c

d e

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Copyright reserved

9.3 The probability that the first answer in a maths quiz competition will be correct is 0,4.

If the first answer is correct, the probability of getting the next answer correct rises

to 0,5. However, if the first answer is wrong, the probability of getting the next

answer correct is only 0,3.

9.3.1 Represent the information on a tree diagram. Show the probabilities

associated with each branch as well as the possible outcomes.

(3)

9.3.2

Calculate the probability of getting the second answer correct.

(3)

[18]

QUESTION 10

Bongani wants to start a small vegetable garden at his house. He wants to use an existing wall

and 14 m of fencing to enclose a rectangular area for the garden. Calculate the dimensions of

the largest rectangular area that he can enclose.

[4]

TOTAL: 150

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QUESTION 1 1.1 Solve for x in each of the following:

1.1.1 0122 =−+ xx

(3)

1.1.2 112 −=+ xx

(5)

1.1.3 2722 =xx

(4)

1.1.4 0822 <−− xx

(3)

1.2 Given: ( ) 765 2 −+= xxxf 1.2.1 Solve for x if ( ) 0=xf (correct to TWO decimal places).

(4) 1.2.2 Hence, or otherwise, calculate the value of d for which 065 2 =−+ dxx

has equal roots.

(3) 1.3 Solve for x and y simultaneously: 20 and 32 =−=− xyyx (6) [28] QUESTION 2 2.1 Simplify, without using a calculator:

2.1.1 1

12

84.2−

++

n

nn

(3)

2.1.2 12.12 −−−+ xxxx (4) 2.2 Given: P =

325 x

x+

+

2.2.1 For what value(s) of x will P be a real number? (2) 2.2.2 Show that P is rational if x = 3. (2) 2.3 Calculate the sum of the digits of 20192015 52 × .

(4) [15]

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QUESTION 3 3.1 Given the linear pattern: 5 ; – 2 ; – 9 ; … ; – 289 3.1.1 Write down the constant first difference. (1)

3.1.2 Write down the value of 4T .

(1) 3.1.3 Calculate the number of terms in the pattern. (3) 3.2 A linear pattern has a difference of 3 between consecutive terms and its 20th term

is equal to 64 (that is 20T = 64).

3.2.1 Determine the value of 22T . (1) 3.2.2 Which term in the pattern will be equal to ?23 5 −T

(4) 3.3 Consider the quadratic pattern: 5 ; 12 ; 29 ; 56 ; … 3.3.1 Write down the NEXT TWO terms of the pattern. (2) 3.3.2 Prove that the first differences of this pattern will always be odd. (3)

[15] QUESTION 4 4.1 Consider the quadratic pattern: 3 ; 5 ; 8 ; 12 ; …

Determine the value of 26T .

(6)

4.2 A certain quadratic pattern has the following characteristics:

• pT =1 • 182 =T • 14 4TT = • 3T – 2T = 10

Determine the value of p.

(6) [12]

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QUESTION 5 5.1 The sketch below shows the graph of 2

19)( −−−

=x

xf .

A is the point of intersection of the asymptotes of f.

5.1.1 Write down the coordinates of A. (2) 5.1.2 Determine the coordinates of the x- and y-intercepts of f. (5) 5.1.3 Write down an equation of the axis of symmetry of f that has a negative

gradient.

(2) 5.1.4 Hence, or otherwise, determine the coordinates of a point that lies on f in

the fourth quadrant, which is the closest to point A.

(5) 5.1.5 The graph of f is reflected about the x-axis to obtain the graph of g.

Write down the equation of g in the form ...=y

(2)

5.2 Given: ( ) 124)( += −xxh 5.2.1 Determine the coordinates of the y-intercept of h. (2) 5.2.2 Explain why h does not have an x-intercept. (2) 5.2.3 Draw a sketch graph of h, clearly showing all asymptotes, intercepts

with the axes and at least one other point on h.

(3) 5.2.4 Describe the transformation from h to g if ( ) ( ).224 += −xxg (2)

[25]

A

f

f

x

y

0

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QUESTION 6 The sketch below represents the graphs of two parabolas, f and g.

821)( 2 −= xxf

The turning point of g is C (2 ; 9) and the y-intercept of g is A (0 ; 5). B and D are the x-intercepts of f and g respectively.

f

g

C(2 ; 9)

y

xB DO

A(0 ; 5)

6.1 Show that ( ) 542 ++−= xxxg . (4) 6.2 Calculate the average gradient of g between A and C. (2) 6.3 Calculate the length of BD. (5) 6.4 Use the graphs to solve for x, if:

6.4.1 0)( ≥xf (2) 6.4.2 f and g are both strictly increasing (2)

[15]

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QUESTION 7

The sketch below shows the graphs of 32)( += xxf and .142)( 2 kxxxg ++−= C is any point on f and D any point on g, such that CD is parallel to the y-axis. k is a value such that C lies above D.

7.1 Write down a simplified expression for the length of CD in terms of x and k. (3)

7.2 If the minimum length of CD is 5, calculate the value of k.

(4) [7]

QUESTION 8

8.1 A school buys tablets at a total cost of R140 000. If the average rate of inflation is 6,1% per annum over the next 4 years, determine the cost of replacing these tablets in 4 years' time.

(3)

8.2 An investment earns interest at a rate of 7% per annum, compounded semi-annually. Calculate the effective annual interest rate on this investment.

(3)

8.3 A savings account was opened with an initial deposit of R24 000. Eighteen months later R7 000 was withdrawn from the account. Calculate how much money will be in the savings account at the end of 4 years if the interest rate was 10,5% p.a., compounded monthly.

(5)

8.4 A car costing R198 000 has a book value of R102 755,34 after 3 years. If the value of the car depreciates at r% p.a. on a reducing balance, calculate r.

(5) [16]

f

g

x

y

C

DO

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Copyright reserved

QUESTION 9 9.1 Given: P(A) = 0,6

P(B) = 0,3 P(A or B) = 0,8 where A and B are two different events Are the events A and B mutually exclusive? Justify your answer with appropriate calculations and/or a diagram.

(4)

9.2 The table below shows data on the monthly income of employed people in two

residential areas. Representative samples were used in the collection of the data.

MONTHLY INCOME (IN RANDS) AREA 1 AREA 2 TOTAL

x < 3 200 500 460 960

3 200 ≤ x < 25 600 1 182 340 1 522

x ≥ 25 600 150 14 164

Total 1 832 814 2 646

9.2.1 What is the probability that a person chosen randomly from the entire sample will be:

(a) From Area 1 (2) (b) From Area 2 and earn less than R3 200 per month (1) (c) A person from Area 2 who earns more than or equal to R3 200 (2) 9.2.2 Prove that earning an income of less than R3 200 per month is not

independent of the area in which a person resides.

(5) 9.2.3 Which is more likely: a person from Area 1 earning less than R3 200 or

a person from Area 2 earning less than R3 200? Show calculations to support your answer.

(3) [17]

TOTAL: 150

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QUESTION 1

1.1 Solve for x:

1.1.1 0)73)(2( xx (2)

1.1.2 252 xx (Correct to TWO decimal places) (4)

1.1.3 543 x (4)

1.1.4 0472 2 xx (4)

1.2 Solve the following equations simultaneously:

632

12

2

xyyx

yx

(6)

[20]

QUESTION 2

2.1 Simplify the following fully: x

xx

3.2

33 11

(3)

2.2 Solve for x: 64)2( 3 x

(4)

2.3 Rewrite the following expression as a power of x: 8 7x

xxxx

(4)

[11]

QUESTION 3

ACDF is a rectangle with an area of 822 xx cm2. B is a point on AC and E is a point

on FD such that ABEF is a square with sides of length 2x cm each.

Calculate the length of ED. [5]

C B A

F E D

2x

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QUESTION 4

Consider the following quadratic number pattern: –7 ; 0 ; 9 ; 20 ; …

4.1 Show that the general term of the quadratic number pattern is given by

1242 nnTn .

(4)

4.2 Which term of the quadratic pattern is equal to 128? (4)

4.3 Determine the general term of the first differences. (3)

4.4 Between which TWO terms of the quadratic pattern will the first difference be 599? (3)

[14]

QUESTION 5

Grey and white squares are arranged into patterns as indicated below.

Pattern 1 Pattern 2 Pattern 3

Pattern 1 Pattern 2 Pattern 3

Number of grey squares 5 13 25

The number of grey squares in the thn pattern is given by 122 2 nnTn .

5.1 How many white squares will be in the FOURTH pattern? (2)

5.2 Determine the number of white squares in the th157 pattern. (3)

5.3 Calculate the largest value of n for which the pattern will have less than 613

grey squares.

(4)

5.4 Show that the TOTAL number of squares in the thn pattern is always an odd number. (3)

[12]

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QUESTION 6

Given: 32

8)(

xxf

6.1 Write down the equations of the asymptotes of f. (2)

6.2

Calculate the x- and y-intercepts of f.

(3)

6.3

Sketch the graph of f. Show clearly the intercepts with the axes and the asymptotes.

(3)

6.4 If y = x + k is an equation of the line of symmetry of f, calculate the value of k. (2)

[10]

QUESTION 7

Given: qaxh x 12.)( . The line y = –6 is an asymptote to the graph of h. P is the

y-intercept of h and T is the x-intercept of h.

x

y

P

T

h

- 6

7.1 Write down the value of q.

(1)

7.2 If the graph of h passes through the point

4

15;1 , calculate the value of a.

(4)

7.3

Calculate the average gradient between the x-intercept and the y-intercept of h. (5)

7.4

Determine the equation of p if p(x) = h(x – 2) in the form qaxp x 12.)( . (2)

[12]

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QUESTION 8

The graph of cbxxxf 2)( and the straight line g are sketched below. A and B are the

points of intersection of f and g. A is also the turning point of f. The graph of f intersects

the x-axis at B(3 ; 0) and C. The axis of symmetry of f is x = 1.

f g

OC

A

B x

y

T

P

M

8.1 Write down the coordinates of C. (1)

8.2 Determine the equation of f in the form cbxxy 2 . (3)

8.3 Determine the range of f. (2)

8.4 Calculate the equation of g in the form y = mx + c. (3)

8.5 For which values of x will:

8.5.1 0)( xf (2)

8.5.2 0

)(

)(

xg

xf

(2)

8.5.3 0)(. xfx (2)

8.6 For what values of p will pxx 22 have non-real roots? (2)

8.7 T is a point on the x-axis and M is a point on f such that TM x-axis.

TM intersects g at P. Calculate the maximum length of PM.

(4)

[21]

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QUESTION 9

9.1 A tractor bought for R120 000 depreciates to R11 090,41 after 12 years by using

the reducing balance method. Calculate the rate of depreciation per annum. (The rate

was fixed over the 12 years.)

(4)

9.2 Calculate the effective interest rate if interest is 9,8% p.a., compounded monthly. (3)

9.3 Mrs Pillay invested R80 000 in an account which offers the following:

7,5 % p.a., compounded quarterly, for the first 4 years and thereafter

9,2% p.a., compounded monthly, for the next 3 years

Calculate the total amount of money that will be in the account at the end of 7 years

if no further transactions happen on the account.

(4)

9.4 Exactly 8 years ago Tashil invested R30 000 in an account earning 6,5% per

annum, compounded monthly.

9.4.1 How much will he receive if he withdrew his money today? (3)

9.4.2 Tashil withdrew R10 000 three years after making the initial deposit and

re-invested R10 000 five years after making the initial deposit.

Calculate the difference between the final amount Tashil will now receive

after eight years and the amount he would have received had there not

been any transactions on the account after the initial deposit.

(7)

[21]

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QUESTION 10

A survey was carried out with 240 customers who bought food from a fastfood outlet on a

particular day. The outlet sells cheese burgers (C), bacon burgers (B) and vegetarian burgers (V).

The Venn diagram below shows the number of customers who bought different types of burgers

on the day.

S

10.1 How many customers did NOT buy burgers on the day? (1)

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

10.3 If a customer from this group is selected at random, determine the probability that

he/she:

10.3.1 Bought only a vegetarian burger (1)

10.3.2 Bought a cheese burger and a bacon burger (1)

10.3.3 Did not buy a cheese burger (3)

10.3.4 Bought a bacon burger or a vegetarian burger (4)

[12]

QUESTION 11

Given: P(A) = 0,12

P(B) = 0,35

P(A or B) = 0,428

Determine whether events A and B are independent or not. Show ALL relevant calculations

used in determining the answer.

[4]

C

B

84

17

52

12

9

3

58

5

V

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Copyright reserved

QUESTION 12

Paballo has a bag containing 80 marbles that are either green, yellow or red in colour. 5

3 of the

marbles are green and 10% of the marbles are yellow. Paballo picks TWO marbles out of the bag,

one at a time and without replacing the first one.

12.1 How many red marbles are in the bag? (2)

12.2 Draw a tree diagram to represent the above situation. (3)

12.3 What is the probability that Paballo will choose a GREEN and a YELLOW marble? (3)

[8]

TOTAL: 150

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Copyright Reserved Please turn over

QUESTION 1 1.1 Solve for x: 1.1.1 253 2 += xx (4) 1.1.2 0422 =−+ xx (Leave your answer correct to TWO decimal places.) (4) 1.1.3 0122 <−+ xx (4) 1.2 Simplify, without the use of a calculator, the following expressions fully:

1.2.1 32

7125

x

x

(3)

1.2.2 272)33( 2 −+ (4) 1.3 Solve for x and y simultaneously:

( ) 01102

2 =+−+

+=

xyxyxy

(6) [25] QUESTION 2 2.1 Given: 46 +=+ xx 2.1.1 Calculate x in the given equation. (5) 2.1.2 Hence, or otherwise, write down the solution to .35 +=+ xx (2)

2.2 Given: ( )93

3−

=x

xf

2.2.1 Determine ( )3f . Leave your answer in simplest surd form. (3) 2.2.2 For which value(s) of x is ( )xf undefined? (3) 2.2.3 For which value(s) of x is ( )xf non-real? (1)

[14]

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QUESTION 3 The hypotenuse of a right-angled triangle is 25 cm and the length of one other side is x cm. The perimeter of the triangle is 60 cm.

3.1 Show that the third side of the triangle is ( )x−35 cm. (1) 3.2 Calculate the lengths of the two shorter sides of the triangle. (5)

[6] QUESTION 4 Sheena receives R1 500 as a gift. She invests her money in a savings account, earning interest at 15% per annum compounded semi-annually.

4.1 How much money does Sheena have in her investment account at the end of 5 years? (4) 4.2 Disa also receives R1 500, but she invests her money in an account which earns

interest annually. If Sheena and Disa have the same amount of money at the end of 5 years, what annual interest rate is Disa earning?

(3) [7]

QUESTION 5 A company bought new machinery for R23 000 at the beginning of 2013. The machinery depreciates on the reducing-balance method at a rate of 13,5% per annum.

5.1 Determine the book value of the machinery at the end of 2017. (2) 5.2 Determine the expected cost of purchasing new machinery at the beginning of 2018 if

the purchase price at the beginning of 2013 increases at 6,6% compounded annually.

(2) 5.3 How much money would the company have had to invest as a lump sum at the

beginning of 2013 if they wanted to pay cash for the new machinery at the beginning of 2018 and the money is invested in a bank account earning interest of 4,7% p.a., compounded monthly?

(6) [10]

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PATTERN 1 PATTERN 2 PATTERN 3

QUESTION 6 Sticks are arranged in patterns as shown below.

Pattern number 1 2 3

Number of sticks 2 7 15

6.1 Write down the number of sticks needed to build Pattern 4 if the patterns are

consistent.

(1) 6.2 Determine a formula to calculate the number of sticks needed to build Pattern n. (4) 6.3 How many sticks would you need to build Pattern 16? (2) 6.4 Calculate the maximum value of n if you have only 126 sticks available to build

Pattern n.

(5) [12]

QUESTION 7

Given the number pattern: y;54;

43;

32;

21 ; …

7.1 Given that the pattern behaves consistently, write down the value of y. (1) 7.2 Determine a formula for nT , the nth term of this pattern. (3)

[4] QUESTION 8 Two number patterns, the one consisting of uneven numbers and the other consisting of even numbers, are combined to form a new number pattern as shown below. 1 ; 2 ; 5 ; 6 ; 9 ; 18 ; 13 ; 54 ; …

8.1 Write down the next TWO terms of the pattern. (2) 8.2 Calculate the 31st term of the pattern. (3)

[5]

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QUESTION 9 The sketch below represents the graphs of 32)( 2 −−= xxxf and cmxxg +=)( . D is a point on f and E is a point on g such that DE is parallel to the y-axis. A and B are the x-intercepts of f. The straight line, g, passes through point A. H is the turning point of the graph of f.

x

y

D

E

A B0

C

H

f

g

9.1 Write down the domain of g. (1) 9.2 Determine the length of AB. (3) 9.3 Determine the average gradient of f between A and C. (3) 9.4 Determine the coordinates of H, the turning point of f. (3) 9.5 Determine the equation of g, if the graph of g is perpendicular to 052 =−− xy . (4) 9.6 For what values of x is 0if0)().( >≥ xxfxg ? (2) 9.7 Determine the positive x-value for which DE = 7,5 units. (5) 9.8 Use the graph to determine for which value(s) of k will kxf =)( have non-real

roots.

(1) [22]

CAPS - GRADE 11

32



QUESTION 10

Given: ( ) 441

+

−=

x

xf

10.1 Write down an equation of the asymptote of f. (1) 10.2 Determine the coordinates of the y-intercept of f. (2) 10.3 Determine the coordinates of the x-intercept of f. (3) 10.4 Sketch a graph of ( )xfy = , clearly indicating the asymptote and the coordinates of

all intercepts with the x- and y-axes.

(4) 10.5 If the graph of f is now reflected in the line y = 4 to create the graph of k,

write down a formula for k in the form y = …

(2) [12]

QUESTION 11

Given: qpx

xf ++

=3)(

11.1 If the asymptotes of f intersect in the point (5 ; 1), determine the values of p and q. (2) 11.2 The graph of f is translated 1 unit right and 4 units up to create the graph of h.

Write down an equation for h in the form y = …

(2) [4]

QUESTION 12 Given: cbxaxxf ++= 2)( (m – 5) and (m + 3) are roots of f. The maximum value of f occurs when x = 2.

12.1 Calculate the value of m. (3)

12.2 Determine the equation of f, in the form cbxaxy ++= 2 , if it is also given that .15)1( =f

(5) [8]

CAPS - GRADE 11

33


Copyright reserved

QUESTION 13 All the students at a certain college undergo annual HIV testing. The results of this year's testing are shown in the table below.

HIV POSITIVE HIV NEGATIVE TOTAL

Male 106 422 b Female a d c TOTAL 192 e 960

13.1 How many students are there at the college? (1) 13.2 Determine the values of a, b, c, d and e. (5) 13.3 Is HIV status independent of gender at this college? Motivate your answer with

relevant calculations.

(4) [10]

QUESTION 14 A retail store did a survey of its customers and found that 30% of the customers were unhappy with the service received. Of those who were unhappy with the service, 74% said that they would not shop at that store again. Of those who were happy with the service, only 5% said that they would not shop at the store again.

14.1 Draw a tree diagram to represent the results of this survey, clearly indicating the

probability of each overall outcome correct to THREE decimal places.

(5) 14.2 Determine the probability that a customer selected at random says that he/she will

shop at the store again.

(2) [7]

QUESTION 15 Two independent relay teams want to qualify for the next Olympic Games. The probability that

the two teams run under the qualifying time, is 94 and

73 respectively. Calculate the probability

that one of the relay teams will run under the qualifying time in their next race.

[4]

TOTAL: 150

CAPS - GRADE 11

34

Mathematics/P1 DBE/2013


QUESTION 1 1.1 Solve for x: 1.1.1 0)5)(12( =+− xx (2) 1.1.2 0142 2 =+− xx (Leave your answer in simplest surd form.) (3) 1.2 Simplify, without the use of a calculator, the following expressions fully:

1.2.1 32

125 (2)

1.2.2 122)(1223( +− ) (3)

1.3 Given: 93

62

−−−

xxx

1.3.1 For which value(s) of x will the expression be undefined? (2) 1.3.2 Simplify the expression fully. (3)

[15] QUESTION 2 2.1 Given: ( )( ) 2332 +−<−+ xxx 2.1.1 Solve for x if: ( )( ) 2332 +−<−+ xxx (4) 2.1.2 Hence or otherwise, determine the sum of all the integers satisfying the

expression 0822 <−+ xx .

(3)

2.2 Given: x

xx

12.1744 11 +− +

2.2.1 Simplify the expression fully. (4)

2.2.2 If tx 43 =− , express x

xx

12.1744 11 +− + in terms of t.

(1)

2.3 Solve for x and y from the given equations:

96and813 2 +−== xxyxy

(7) [19]

CAPS - GRADE 11

35



QUESTION 3

3.1 The solution to a quadratic equation is 4

843 px

−±= where p∈Q.

Determine the value(s) of p such that:

3.1.1 The roots of the equation are equal (2) 3.1.2 The roots of the equation are non-real (2) 3.2 Given: 15 +=− xx 3.2.1 Without solving the equation, show that the solution to the above equation

lies in the interval – 1 5≤≤ x .

(3) 3.2.2 Solve the equation. (5) 3.2.3 Without any further calculations, solve the equation – 15 +=− xx . (1)

[13] QUESTION 4 4.1 Melissa has just bought her first car. She paid R145 000 for it. The car's value

depreciates on the straight-line method at a rate of 17% per annum. Calculate the value of Melissa's car 5 years after she bought it.

(2)

4.2 An investment earns interest at a rate of 8% per annum compounded quarterly. 4.2.1 At what rate is interest earned each quarter of the year? (1) 4.2.2 Calculate the effective annual interest rate on this investment. (2) 4.3 R14 000 is invested in an account.

The account earns interest at a rate of 9% per annum compounded semi-annually for the first 18 months and thereafter 7,5% per annum compounded monthly. How much money will be in the account exactly 5 years after the initial deposit?

(5) [10]

CAPS - GRADE 11

36



QUESTION 5 The graphs below represent the growth of two investments, one belonging to Dumisani and one belonging to Astin. Both investments earn interest annually (only).

x

y

15

A(6; 31)

time (in years)

B (12; w)

Astin

Dumisani

5.1 What is the value of both initial investments? (1) 5.2 Does Dumisani's investment earn simple or compound interest? (1) 5.3 Determine Dumisani's interest rate. (2) 5.4 Hence or otherwise, calculate the interest rate on Astin's investment. Give your answer

correct to ONE decimal place.

(4) [8]

QUESTION 6

6.1 Given: 02411;...;

81;

41;

21

6.1.1 Explain how you will determine the 4th term of the sequence. (2) 6.1.2 Write a formula for the nth term of the sequence. (2) 6.1.3 Determine the number of terms in the sequence. (2) 6.2 Given the linear pattern: 156 ; 148 ; 140 ; 132 ; … 6.2.1 Write down the 5th term of this number pattern. (1) 6.2.2 Determine a general formula for the nth term of this pattern. (2) 6.2.3 Which term of this linear number pattern is the first term to be negative? (3) 6.2.4 The given linear number pattern forms the sequence of first differences of

a quadratic number pattern cbnann ++= 2T with T5 = – 24. Determine a general formula for Tn.

(5) [17]

Inve

stmen

t val

ue

(in th

ousa

nds o

f ran

ds)

n

F

CAPS - GRADE 11

37



QUESTION 7 A given quadratic pattern cbnann ++= 2T has 0TT 42 == and a second difference of 12. Determine the value of the 3rd term of the pattern.

[6]

QUESTION 8

The sketch below represents the graphs of 13

2)( −−

=x

xf and edxxg +=)( .

Point B (3 ; 6) lies on the graph of g and the two graphs intersect at points A and C.

x

y

B(3 ; 6)

f

f

g

0

A

C

8.1 Write down the equations of the asymptotes of f. (2) 8.2 Write down the domain of f. (2) 8.3 Determine the values of d and e, correct to the nearest integer, if the graph of g

makes an angle of 76° with the x-axis.

(3) 8.4 Determine the coordinates of A and C. (6) 8.5 For what values of x is )()( xfxg ≥ ? (3) 8.6 Determine an equation for the axis of symmetry of f which has a positive slope. (3)

[19]

CAPS - GRADE 11

38



QUESTION 9 Given: 32)( 2 ++−= xxxf and xxg 21)( −= 9.1 Sketch the graphs of f and g on the same set of axes. (9) 9.2 Determine the average gradient of f between 3−=x and 0=x . (3) 9.3 For which value(s) of x is f(x).g(x) 0≥ ? (3) 9.4 Determine the value of c such that the x-axis will be a tangent to the graph of h,

where ( ) ( ) cxfxh += .

(2) 9.5 Determine the y-intercept of t if t(x) = – g(x) + 1 (2) 9.6 The graph of k is a reflection of g about the y-axis. Write down the equation of k. (1)

[20] QUESTION 10

Sketch the graph of cbxaxxf ++= 2)( if it is also given that: • The range of f is ( ]7;∞− • 0≠a • 0<b • One root of f is positive and the other root of f is negative.

[4] QUESTION 11 Given: 4,0)(P =W 35,0)(P =T P ( ) 14,0 and =WT

11.1 Are the events W and T mutually exclusive? Give reasons for your answer. (2) 11.2 Are the events W and T independent? Give reasons for your answer. (3)

[5]

CAPS - GRADE 11

39


Copyright reserved

QUESTION 12 12.1 A group of 33 learners was surveyed at a school. The following information from the

survey is given: • 2 learners play tennis, hockey and netball • 5 learners play hockey and netball • 7 learners play hockey and tennis • 6 learners play tennis and netball • A total of 18 learners play hockey • A total of 12 learners play tennis • 4 learners play netball ONLY

12.1.1 A Venn diagram representing the survey results is given below. Use the

information provided to determine the values of a, b, c, d and e.

(5) 12.1.2 How many of these learners do not play any of the sports on the survey

(that is netball, tennis or hockey)?

(1) 12.1.3 Write down the probability that a learner selected at random from this

sample plays netball ONLY.

(1) 12.1.4 Determine the probability that a learner selected at random from this

sample plays hockey or netball.

(1) 12.2 In all South African schools, EVERY learner must choose to do either Mathematics

or Mathematical Literacy. At a certain South African school, it is known that 60% of the learners are girls. The probability that a randomly chosen girl at the school does Mathematical Literacy is 55%. The probability that a randomly chosen boy at the school does Mathematical Literacy is 65%. Determine the probability that a learner selected at random from this school does Mathematics.

(6) [14]

TOTAL: 150

a

N H

T

3 4 2

b

c

d

S

e

CAPS - GRADE 11

40

CAPS - GRADE 11

41

CAPS - GRADE 11

42

CAPS - GRADE 11

43

CAPS - GRADE 11

44

CAPS - GRADE 11

45

CAPS - GRADE 11

46

CAPS - GRADE 11

47

CAPS - GRADE 11

48

CAPS - GRADE 11

49

CAPS - GRADE 11

50

CAPS - GRADE 11

51



QUESTION 1

The table below shows the number of cans of food collected by 9 classes during a charity

drive.

5 8 15 20 25 27 31 36 75

1.1 Calculate the range of the data. (1)

1.2 Calculate the standard deviation of the data. (2)

1.3 Determine the median of the data. (1)

1.4 Determine the interquartile range of the data. (3)

1.5 Use the number line provided in the ANSWER BOOK to draw a box and whisker

diagram for the data above.

(3)

1.6 Describe the skewness of the data. (1)

1.7 Identify outliers, if any exist, for the above data. (1)

[12]

QUESTION 2

The table below shows the time (in minutes) that 200 learners spent on their cellphones during

a school day.

TIME SPENT

(IN MINUTES) FREQUENCY

95 < x ≤ 105 15

105 < x ≤ 115 27

115 < x ≤ 125 43

125 < x ≤ 135 52

135 < x ≤ 145 28

145 < x ≤ 155 21

155 < x ≤ 165 10

165 < x ≤ 175 4

2.1 Complete the cumulative frequency column in the table provided in the ANSWER

BOOK.

(2)

2.2 Draw a cumulative frequency graph (ogive) of the data on the grid provided. (3)

2.3 Use the cumulative frequency graph to determine the value of the lower quartile. (2)

2.4 Determine, from the cumulative frequency graph, the number of learners who used

their cellphones for more than 140 minutes.

(2)

[9]

CAPS - GRADE 11

52



QUESTION 3

In the diagram, A(6 ; – 2), B(2 ; 15) and C(– 4 ; 3) are the vertices of ABC.

M is the midpoint of AB. N is a point on CA such that MN BC.

3.1 Determine the coordinates of M, the midpoint of AB. (2)

3.2 Determine the gradient of line MN. (3)

3.3 Hence, or otherwise, determine the equation of line MN, in the form .cmxy (2)

3.4 Calculate, with reasons, the coordinates of point N. (4)

3.5 If ABCD (in that order) is a parallelogram, determine the coordinates of point D. (4)

[15]

x

y

<

<

//

//

M

N

O

A(6 ; –2)

B(2 ; 15)

C(–4 ; 3)

CAPS - GRADE 11

53



QUESTION 4

In the diagram, R and A are the x- and y-intercepts respectively of the straight line AR.

The equation of AR is 42

1 xy . Another straight line cuts the y-axis at P(0 ; 2) and

passes through the points M(k ; 0) and N(3 ; 4).

and are the angles of inclination of the lines MN and AR respectively.

4.1 Given that M, P and N are collinear points, calculate the value of k. (3)

4.2 Determine the size of 𝜃, the obtuse angle between the two lines. (4)

4.3 Calculate the length of MR. (3)

4.4 Calculate the area of MNR. (3)

[13]

x

y

N(3 ; 4)

R

M(k ; 0)

P(0 ; 2)

O

A

<<

CAPS - GRADE 11

54



QUESTION 5

5.1 In the diagram below, P(–8 ; t) is a point in the Cartesian plane such that

OP = 17 units and reflex POX .

5.1.1 Calculate the value of t. (2)

5.1.2 Determine the value of each of the following WITHOUT using a

calculator:

(a) )cos( (2)

(b) sin1 (2)

5.2 If a17sin , WITHOUT using a calculator, express the following in terms

of a :

5.2.1 17tan (3)

5.2.2 107sin (2)

5.2.3 557sin253cos 22

(4)

5.3 Simplify fully, WITHOUT the use of a calculator:

225tan

330sin135sin).225cos(

(6)

5.4 Prove the identity: xx.xx 22 costan

1

1)1)(cos(cos

1

(4)

5.5 Determine the general solution for .xxx coscos.2sin (6)

[31]

P(– 8 ; t )

y

x O

17

CAPS - GRADE 11

55



QUESTION 6

In the diagram the graphs of xxf cos)( and )sin()( bxxg are drawn for

the interval .x 90180

6.1 Write down the value of b. (1)

6.2 Write down the period of g. (1)

6.3 Write down the value(s) of x in the interval 90180 x for which

.0)()( xgxf

(2)

6.4 For which values of x in the interval 90180 x is )()90sin( xgx ? (3)

6.5 The graph of h is obtained by shifting f 3 units upwards. Determine the range of h. (2)

[9]

-180 -150 -120 -90 -60 -30 30 60 90

-1.5

-1

-0.5

0.5

1

1.5

x

y

f

g

0

CAPS - GRADE 11

56



QUESTION 7

7.1 In the figure below, acute-angled ABC is drawn having C at the origin.

7.1.1 Prove that .cosC2abbac 222 (6)

7.1.2 Hence, deduce that

ab

cbacbaC

2

))((cos1

(4)

7.2 Quadrilateral ABCD is drawn with m 235BC and m. 90,52 AB It is also

given that 31,23BDA ; 109,16BAD and .ˆ 48,88DBC

Determine the length of:

7.2.1

7.2.2

BD

CD

(3)

(3)

[16]

B

C A

A

B

D

C

90,52 m

235 m

109,16°

31,23°

48,88°

CAPS - GRADE 11

57



QUESTION 8

The diagram below shows a water tank which is made up of a cylinder and cone having equal

radii. The height of the tank is 1,8 m and the radius is 0,5 m. The angle between the

perpendicular height, AB, and the slant height, AC, of the conical section is 35,5°.

8.1 Calculate the perpendicular height, AB, of the cone. (2)

8.2 When the tank is full, an electric pump switches on and pumps the water from the

tank into an irrigation system at a rate of 0,52 m2/h. The pump automatically

switches off when the tank is 4

1

full.

Calculate how long, in hours, the pump feeds water into the irrigation system.

(4)

[6]

Volume of cone =

Total surface area of cone =

Volume of cylinder =

Total surface area of cylinder =

1,8 m

0,5 m

35,5°

A

B C

CAPS - GRADE 11

58



T

N

S P

R O

60°

Give reasons for your statements and calculations in QUESTIONS 9, 10, 11 and 12.

QUESTION 9

9.1 Complete the statement so that it is TRUE:

The angle subtended by an arc at the centre of a circle is ...

(2)

9.2 O is the centre of circle TNSPR. 60SOP and PS = NT.

Calculate the size of:

9.2.1 SRP

(2)

9.2.2 TSN

(2)

[6]

CAPS - GRADE 11

59



QUESTION 10

D, E, F, G and H are points on the circumference of the circle.

20G1 x and 102H x . DE FG.

10.1 Determine the size of GED in terms of x . (2)

10.2 Calculate the size of G.HDˆ (4)

[6]

E

D

H

G

F

2

1

1

2

x + 20°

2x + 10°

CAPS - GRADE 11

60



QUESTION 11

O is the centre of the circle PTR. N is a point on chord RP such that ON PR.

RS and PS are tangents to the circle at R and P respectively.

RS = 15 units; TS = 9 units; .ˆ 42,83 SPR

11.1 Calculate the size of R.ON (5)

11.2 Calculate the length of the radius of the circle. (4)

[9]

S

9

O

P

42,83

N

R

15

T

CAPS - GRADE 11

61


Copyright reserved

QUESTION 12

12.1 Use the diagram below to prove the theorem which states that F.DEGFE

(5)

12.2 In the diagram below, BOC is a diameter of the circle. AP is a tangent to the circle

at A and AE = EC.

Prove that:

12.2.1 BA OD (4)

12.2.2 AOCD is a cyclic quadrilateral (5)

12.2.3 DC is a tangent to the circle at C (4)

[18]

TOTAL: 150

E

D

F G

C

O

A D

B

1

1

1 1 2

2

2

2

4

1 3

2

3

3

3

3

E

P

CAPS - GRADE 11

62



QUESTION 1 The table below shows the weight (to the nearest kilogram) of each of the 27 participants in a weight-loss programme.

1.1 Calculate the range of the data. (2) 1.2 Write down the mode of the data. (1) 1.3 Determine the median of the data. (1) 1.4 Determine the interquartile range of the data. (3) 1.5 Use the number line provided in the ANSWER BOOK to draw a box and whisker

diagram for the data above.

(2) 1.6 Determine the standard deviation of the data. (2) 1.7 The person weighing 127 kg states that she weighs more than one standard deviation

above the mean. Do you agree with this person? Motivate your answer with calculations.

(3)

[14]

56 68 69 71 71 72 82 84 85

88 89 90 92 93 94 96 97 99

102 103 127 128 134 135 137 144 156

CAPS - GRADE 11

63



QUESTION 2 The table below shows the weight (in grams) that each of the 27 participants in the weight-loss programme lost in total over the first 4 weeks.

WEIGHT LOSS OVER 4 WEEKS

(IN GRAMS) FREQUENCY

1 000 < x ≤ 1 500 2

1 500 < x ≤ 2 000 3

2 000 < x ≤ 2 500 3

2 500 < x ≤ 3 000 4

3 000 < x ≤ 3 500 5

3 500 < x ≤ 4 000 7

4 000 < x ≤ 4 500 2

4 500 < x ≤ 5 000 1

2.1 Estimate the average weight loss, in grams, of the participants over the first 4 weeks. (2) 2.2 Draw an ogive (cumulative frequency graph) of the data on the grid provided. (4) 2.3 The weight-loss programme guarantees a loss of 800 g per week if a person follows

the programme without cheating. Hence, determine how many of the participants had an average weight loss of 800 g or more per week over the first 4 weeks.

(2)

[8]

CAPS - GRADE 11

64



QUESTION 3 In the diagram, A(–2 ; 3), C(10 ; 11) and D(5 ; –1) are the vertices of ∆ACD. CA intersects the y-axis in F and CA produced cuts the x-axis in G. The straight line DE is drawn parallel to CA. .OFC α=

3.1 Calculate the gradient of the line AC. (2) 3.2 Determine the equation of line DE in the form y = mx + c. (3) 3.3 Calculate the size of α. (3) 3.4 B is a point in the first quadrant such that ABDE, in that order, forms a rectangle.

Calculate, giving reasons, the:

3.4.1 Coordinates of M, the midpoint of BE (3) 3.4.2 Length of diagonal BE (3) [14]

A(–2 ; 3)

C(10 ; 11)

D(5 ; –1)

E

G O

F α

y

x

CAPS - GRADE 11

65



QUESTION 4 In the diagram, the straight line SP is drawn having S and P as its x- and y-intercepts

respectively. The equation of SP is x + ay – a = 0, a > 0. It is also given that OS = 3OP.

The straight line RT is drawn with R on SP and RT ⊥ PS. RT cuts the y-axis in

T

−

325;0 .

x

y

4.1 Calculate the coordinates of P. (2) 4.2 Calculate the value of a. (2)

4.3 Determine the equation of RT in the form y = mx + c if it is given that a = 3. (3) 4.4 Calculate the coordinates of R, the point where PS and TR meet. (4)

4.5 Calculate the area of ∆PRT if it is given that R

31;2 .

(3)

4.6 Calculate, giving reasons, the radius of a circle passing through the points P, R

and T.

(2) [16]

O

P R

T

−

325;0

S

CAPS - GRADE 11

66



QUESTION 5

5.1 In the diagram below, P(x ; 24) is a point such that OP = 25 and β=POR , where β is an obtuse angle.

5.1.1 Calculate the value of x. (2) 5.1.2 Determine the value of each of the following WITHOUT using a

calculator:

(a) βsin (1) (b) )180cos( β−° (2) (c) )tan( β− (2) 5.1.3 T is a point on OP such that OT = 15. Determine the coordinates of T

WITHOUT using a calculator.

(4) 5.2 Determine the value of the following expression:

xxxx

tan)tan1(cos.sin2 2+

(4)

5.3 Consider:

)A90cos(4Acos1 2

+°−

5.3.1 Simplify the expression to a single trigonometric term. (3)

5.3.2 Hence, determine the general solution of 21,0)290cos(4

2cos1 2=

+°−

xx .

(6)

[24]

• T

R

y

x β

25

P(x ; 24)

O •

CAPS - GRADE 11

67



QUESTION 6 6.1 In the diagram, the graph of bxxf tan)( = is drawn for the interval .13590 °≤≤°− x

6.1.1 Determine the value of b. (1) 6.1.2 Determine the values of x in the interval °≤≤° 1350 x for which

1)( −≤xf .

(2) 6.1.3 Graph h is defined as )55(tan)( °+= xbxh . Write down the equations

of the asymptotes of h in the interval °≤≤°− 13590 x .

(2)

f

0° 45° 90° 135° –45° –90° x

y

f f 1

2

–1

–2

CAPS - GRADE 11

68



6.2 In the diagram, the graph of )60cos()( °+= xxg is drawn for the interval

°≤≤°− 120150 x .

x

y

6.2.1 On the same system of axes, draw the graph of xxk sin)( −= for the

interval °≤≤°− 120150 x . Show ALL the intercepts with the axes as well as the coordinates of the turning points and end points of the graph.

(4) 6.2.2 Determine the minimum value of 3)60cos()( −°+= xxh . (2) 6.2.3 Solve the equation 0sin)60cos( =+°+ xx for the interval

°≤≤°− 120150 x .

(6) 6.2.4 Determine the values of x for the interval °≤≤°− 120150 x , for which

0sin)60cos( >+°+ xx .

(2) 6.2.5 The function g can also be defined as )sin( θ−−= xy , where θ is

an acute angle. Determine the value of θ.

(2) [21]

g

1

21

0

21−

–1

–150° –60° 30° 120°

CAPS - GRADE 11

69



QUESTION 7 In the diagram, PR is the diameter of the circle. Triangle PQR is drawn with vertex Q outside the circle. R = θ, PR = QR = 2y and PQ = y.

7.1 Determine the value of θcos . (4) 7.2 If QR cuts the circumference of the circle at T, determine PT in terms of y

and θ.

(3) [7]

P

Q R

2y y

θ

2y

CAPS - GRADE 11

70



QUESTION 8 A cylindrical aerosol can has a lid in the shape of a hemisphere that fits exactly on the top of the can. The height of the can is 16 cm and the radius of the base of the can is 2,9 cm.

8.1 Calculate the surface area of the can with the lid in place, as shown in FIGURE 1. (5) 8.2 If the lid is 80% filled with a liquid, as shown in FIGURE 2, calculate the volume of

the liquid in the lid.

(3) [8]

FIGURE 2

Volume of sphere = 3

34 rπ

Surface area of sphere = 4π r2

2,9

16

FIGURE 1

CAPS - GRADE 11

71



Give reasons for your statements and calculations in QUESTIONS 9, 10 and 11. QUESTION 9 In the diagram, O is the centre of the circle. Diameter LR subtends RKL at the circumference of the circle. N is another point on the circumference and chords LN and KN are drawn. °= 58L1 .

Calculate, giving reasons, the size of: 9.1 RKL

(2) 9.2 R

(2) 9.3 N (2) [6]

L

K

O

N

R

58°

2

1

1 2

CAPS - GRADE 11

72



QUESTION 10 10.1 In the diagram, O is the centre of the circle. A, B and C are points on the

circumference of the circle. Chords AC and BC and radii AO, BO and CO are drawn. x=A and y=B .

10.1.1 Determine the size of 1O in terms of x. (3) 10.1.2 Hence, prove the theorem that states that the angle subtended by an arc

at the centre is equal to twice the angle subtended by the same arc at the circumference, that is B.C2ABOA =

(3)

A B

O

C

x

2

y

1

1 2

CAPS - GRADE 11

73



10.2 In the diagram, PQ is a common chord of the two circles. The centre, M, of the

larger circle lies on the circumference of the smaller circle. PMNQ is a cyclic quadrilateral in the smaller circle. QN is produced to R, a point on the larger circle. NM produced meets the chord PR at S. .P2 x=

10.2.1 Give a reason why 2N = x. (1) 10.2.2 Write down another angle equal in size to x. Give a reason. (2) 10.2.3 Determine the size of R in terms of x. (3) 10.2.4 Prove that PS = SR. (3) [15]

x

2

P

Q

R

M

S

N 1

1

1

1

2

2

2 1

CAPS - GRADE 11

74


Copyright reserved

QUESTION 11 In the diagram, the vertices A, B and C of ∆ABC are concyclic. EB and EC are tangents to the circle at B and C respectively. T is a point on AB such that TE | | AC. BC cuts TE in F.

11.1 Prove that 31 TB = . (4) 11.2 Prove that TBEC is a cyclic quadrilateral. (4) 11.3 Prove that ET bisects CTB . (2) 11.4 If it is given that TB is a tangent to the circle through B, F and E, prove that

TB = TC.

(4) 11.5 Hence, prove that T is the centre of the circle through A, B and C. (3) [17]

TOTAL: 150

A

B

C

E F T 1

1

1

1

2

2 2

2

3

3

CAPS - GRADE 11

75



QUESTION 1

1.1 The number of delivery trucks making daily deliveries to neighbouring supermarkets,

Supermarket A and Supermarket B, in a two-week period are represented in the box-

and-whisker diagrams below.

1.1.1 Calculate the interquartile range of the data for Supermarket A. (2)

1.1.2 Describe the skewness in the data of Supermarket A. (1)

1.1.3 Calculate the range of the data for Supermarket B. (2)

1.1.4 During the two-week period, which supermarket receives 25 or more

deliveries per day on more days? Explain your answer.

(2)

1.2 The number of delivery trucks that made deliveries to Supermarket A each day during

the two-week period was recorded. The data is shown below.

If the mean of the number of delivery trucks that made deliveries to supermarket A is

24,5 during these two weeks, calculate the value of x.

(3)

[10]

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

10 15 20 x 30 35 15 31 32 21 x 27 28 29

Supermarket

B

Supermarket

A

CAPS - GRADE 11

76



QUESTION 2

The 2012 Summer Olympic Games was held in London. The average daily temperature, in

degrees Celsius, was recorded for the duration of the Games. A cumulative frequency graph

(ogive) of this data is shown below.

2.1 Over how many days was the 2012 Summer Olympic Games held? (1)

2.2 Estimate the percentage of days that the average daily temperature was less

than 24 °C.

(2)

2.3 Complete the frequency table for the data in the SPESCIAL ANSWER BOOK. (3)

2.4 Hence, use the grid provided in the SPECIAL ANSWER BOOK to draw a frequency

polygon of the data.

(4)

[10]

0

5

10

15

20

25

30

17 19 21 23 25 27 29 31 33 35

Cu

mu

lati

ve

freq

uen

cy

Average daily temperature (in degrees Celsius)

Cumulative frequency graph of average daily temperature

recorded

CAPS - GRADE 11

77



QUESTION 3

In the diagram A(–9 ; 12), B(9 ; 9) and C(–3 ; –9) are the vertices of ABC. N(a ; 7) is a

point such that BN = 55 . R is a point on AB and S is a point on BC such that RNS is

parallel to AC and RNS passes through the origin. T lies on the x-axis to the right of

point P.

θBCA , OMA and TPB .

x

y

3.1 Calculate the gradient of the line AC. (2)

3.2 Determine the equation of line RNS in the form y = mx + c. (2)

3.3 Calculate the value of a. (4)

3.4 Calculate the size of . (5)

[13]

A(–9 ; 12)

B(9 ; 9)

C(–3 ; –9)

N(a ; 7)

R

O M P T

S

55

CAPS - GRADE 11

78



QUESTION 4

In the diagram A(–8 ; 6), B, C and D(3 ; 9) are the vertices of a rhombus. The equation of

BD is 3x – y = 0. The diagonals of the rhombus intersect at point K.

4.1 Calculate the perimeter of ABCD. Leave your answer in simplest surd form. (3)

4.2 Determine the equation of diagonal AC in the form y = mx + c. (4)

4.3 Calculate the coordinates of K if the equation of AC is x + 3y = 10. (3)

4.4 Calculate the coordinates of B. (2)

4.5 Determine, showing ALL your calculations, whether rhombus ABCD is a square

or not.

(5)

[17]

A(–8 ; 6)

D(3 ; 9)

K

B

C

O

CAPS - GRADE 11

79



QUESTION 5

5.1 If cos 23° = p, express, without the use of a calculator, the following in terms of p:

5.1.1 cos 203° (2)

5.1.2 sin 293° (3)

5.2 Simplify the following expression to a single trigonometric term:

)AcosA).(sin180cos(

)tan().360sin(22

x

xx

(6)

5.3 5.3.1 Prove the identity: xx

x

x

x

cos

2

cos

sin1

sin1

cos

(5)

5.3.2 For which values of x in the interval 0° ≤ x ≤ 360° will the identity in

QUESTION 5.3.1 be undefined?

(2)

5.4 Determine the general solution of: xx 2cos42sin (5)

5.5 In the diagram below )3;(P x is a point on the Cartesian plane such that OP = 2.

Q(a ; b) is a point such that QOT and OQ = 20. 90QOP .

5.5.1 Calculate the value of x. (2)

5.5.2 Hence, calculate the size of α. (3)

5.5.3 Determine the coordinates of Q. (5)

[33]

O x

P( )3;x

Q(a ; b)

2

y

20

T

CAPS - GRADE 11

80



QUESTION 6

In the diagram below the graphs of f(x) = a cos bx and g(x) = sin (x + p) are drawn for

x [–180° ; 180°].

6.1 Write down the values of a, b and p. (3)

6.2 For which values of x in the given interval does the graph of f increase as the graph

of g increases?

(2)

6.3 Write down the period of f (2x). (2)

6.4 Determine the minimum value of h if h(x) = 3f(x) – 1. (2)

6.5 Describe how the graph g must be transformed to form the graph k, where

k(x) = – cos x.

(2)

[11]

f g

0

CAPS - GRADE 11

81



QUESTION 7

7.1 In the diagram, the base of the pyramid is an obtuse-angled ABC with A = 110°,

B = 40° and BC = 6 metres. The perpendicular height of the pyramid is 8 metres.

7.1.1 Calculate the length of AB. (3)

7.1.2 Calculate the area of the base, that is ABC. (2)

7.1.3 Calculate the volume of the pyramid. (3)

Surface area = Srr 2 where S is the slant height.

Volume = 3

1 area of base perpendicular height

Volume = hr 2

3

1

A B

C

40° 110°

6

8

T

CAPS - GRADE 11

82



7.2 The perpendicular height, AC, of the cone below is 2 metres and the radius is r.

AB is the slant height.

36CAB

Calculate the total surface area of the cone. (6)

[14]

A

2

r

36°

B C

CAPS - GRADE 11

83



GIVE REASONS FOR YOUR STATEMENTS AND CALCULATIONS IN QUESTIONS 8,

9 AND 10.

QUESTION 8

8.1 In the diagram below, PT is a diameter of the circle with centre O. M and S are

points on the circle on either side of PT.

MP, MT, MS and OS are drawn.

2M

= 37°

Calculate, with reasons, the size of:

8.1.1 1M

(2)

8.1.2 1O

(2)

M

P

S

T

O 2

1

1

2 37°

CAPS - GRADE 11

84



8.2 In the diagram O is the centre of the circle. KM and LM are tangents to the circle

at K and L respectively. T is a point on the circumference of the circle. KT and

TL are joined. 1O = 106°.

8.2.1 Calculate, with reasons, the size of 1T . (3)

8.2.2 Prove that quadrilateral OKML is a kite. (3)

8.2.3 Prove that quadrilateral OKML is a cyclic quadrilateral. (3)

8.2.4 Calculate, with reasons, the size of M . (2)

[15]

1 1

K

L

M O

T 2

106°

CAPS - GRADE 11

85



QUESTION 9

In the diagram M is the centre of the circle passing through points L, N and P.

PM is produced to K. KLMN is a cyclic quadrilateral in the larger circle having KL = MN.

LP is joined. 20 LMK .

9.1 Write down, with a reason, the size of MKN . (2)

9.2 Give a reason why KN | | LM. (1)

9.3 Prove that KL = LM. (2)

9.4 Calculate, with reasons, the size of:

9.4.1 MNK (4)

9.4.2 NPL (3)

[12]

1

1

1

1

2

2

2 2

2

1

3

4

K

L

M

N P

20°

CAPS - GRADE 11

86


Copyright reserved

QUESTION 10

10.1 Use the sketch in the SPECIAL ANSWER BOOK to prove the theorem which states

that CTAB .

(6)

10.2 In the diagram PQ is a tangent to the circle QST at Q such that QT is a chord of

the circle and TS produced meets the tangent at P. R is a point on QT such that

PQRS is a cyclic quadrilateral in another circle. PR, QS and RS are joined.

10.2.1 Give a reason for each statement. Write down only the reason next to the

question number in the SPECIAL ANSWER BOOK.

Statement Reason

TQ1 10.2.1 (a)

22 PQ 10.2.1 (b)

(2)

10.2.2 Prove that PQR is an isosceles triangle. (4)

10.2.3 Prove that PR is a tangent to the circle RST at point R. (3)

[15]

TOTAL: 150

P

Q R

S

T

1

1 1

1

2

2 2

2 3

3

P A T

B C

O

CAPS - GRADE 11

87



QUESTION 1 The 100th Tour de France took place from 29 June 2013 to 21 July 2013. The race was made up of 21 stages of varying distances. The distance, in kilometres, covered in each stage is given in the table below:

Stage Distance Stage Distance Stage Distance

1 213 8 195 15 247 2 156 9 168 16 168

3 145 10 197 17 32

4 25 11 33 18 172 5 228 12 218 19 204

6 176 13 173 20 125

7 205 14 191 21 133 [Source: www.letour.fr.le-tour/2013/us] 1.1 Calculate the mean distance. (3) 1.2 Calculate the standard deviation of the distances. (2) 1.3 Determine the number of stages that lie beyond ONE standard deviation of the mean. (2) 1.4 The distance covered in each stage has been rearranged in ascending order and is

shown below. Determine the five-number summary of this data.

25 32 33 125 133 145 156

168 168 172 173 176 191 195

197 204 205 213 218 228 247 (4)

1.5 Use the scaled line provided in DIAGRAM SHEET 1 to draw a box and whisker

diagram to represent the distance covered in each stage.

(2) 1.6 Are there any outliers in the data set? Explain. (2)

[15]

CAPS - GRADE 11

88



QUESTION 2 A manufacturer recorded how far a minibus taxi travels before it needs new tyres. He recorded the distances, in 1 000s of kilometres, covered by a number of taxis that travelled the same route. This information is shown in the cumulative frequency graph (ogive) below.

2.1 How many times did they record the distance travelled by a minibus taxi before it

needed new tyres?

(1) 2.2 Write down the modal class of the data. (1) 2.3 Estimate the median distance travelled before new tyres are needed. (1) 2.4 Estimate the inter-quartile range for this data. (3)

[6]

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

105

0 8 16 24 32 40 48 56 64 72

Cum

ulat

ive

Freq

uenc

y

Distance travelled (in 1 000s of kilometres)

Cumulative frequency curve showing the distance travelled by a minibus taxi before it needs new tyres

CAPS - GRADE 11

89



QUESTION 3 In the diagram below, A(4 ; –1), B(–14 ; –10) and C are the vertices of a triangle. E is a point on AC such that BE ⊥ AC. The point D(–8 ; –4) lies on BE. The equation of the line BC is 4y – 5x – 30 = 0.

3.1 Calculate the gradient of BD. (2) 3.2 Hence, write down the gradient of AC. (1) 3.3 Determine the equation of AC in the form y = mx + c. (2) 3.4 The point G(p ; –5) lies on AB. Calculate the value of p. (3) 3.5 Calculate the coordinates of C. (4)

[12]

O

y

x

C

A(4 ; –1)

B(–14 ; –10)

D(–8 ; –4)

E

CAPS - GRADE 11

90



QUESTION 4 A(3 ; 2), B(0 ; k), C(–8 ; 0) and D are the vertices of a rectangle. AB = 5 units. The angle of inclination of AD is θ, as shown in the diagram.

4.1 Calculate the length of AC. (2) 4.2 Calculate the value of k. (4) 4.3 Determine the equation of BC in the form y = mx + c. (3) 4.4 Calculate the size of θ. (3) 4.5 Calculate the area of ABCD. (3)

4.6 Calculate the size of CAB . (2) [17]

θ

y

x

A(3 ; 2)

B(0 ; k)

C(–8 ; 0)

D

5

CAPS - GRADE 11

91



QUESTION 5 5.1 In the diagram, P(–15 ; y) is a point in the Cartesian plane.

OP = 17 units and reflex α=POM .

Determine the value of the following without using a calculator: 5.1.1 y (2) 5.1.2 sin (90° + α) (2) 5.1.3 tan β, if α + β = 540° (3) 5.2 Simplify the following expression to a single trigonometric ratio:

)cos()360(cos2cos)90cos(2)180sin(

2 xxxxx

−−+°−°−−°

(6)

5.3 5.3.1 Prove that xxxx

xx

sincossincos

tan1tan1

+−

=+−

(3)

5.3.2 For which value(s) of x in the interval °≤≤° 1800 x is the identity in

QUESTION 5.3.1 undefined?

(2) 5.4 Determine the general solution of the following equation: 2 tan x = 5 sin x (8)

[26]

y

17

P(–15 ; y)

x

•

O

α • M

CAPS - GRADE 11

92



QUESTION 6 6.1 Use the system of axes provided on DIAGRAM SHEET 1 to draw the graphs of

xxf 2cos)( = and 1sin)( +−= xxg for the interval °≤≤°− 180180 x . Show clearly ALL intercepts with the axes, turning points and end points.

(6)

6.2 Write down the period of f. (1)

6.3 For which value(s) of x in the interval °≤≤°− 180180 x will )()( xfxg − be a

maximum?

(1)

6.4 The graph f is shifted 45° to the right to obtain a new graph h. Write down the equation of h in its simplest form.

(2) [10]

QUESTION 7 7.1 Prove that in any acute-angled ∆ABC, c2 = a2 + b2 – 2ab cos C. (6)

7.2 In ∆ABC, AB = 60 cm, BC = 160 cm and CBA = 60°. BD is the bisector of AC with D a point on AC.

7.2.1 Calculate the length of AC. (3) 7.2.2 Determine the value of sin A. Leave the answer in its simplest surd form. (3) 7.2.3 Calculate the area of ∆ABD. Give your answer correct to ONE decimal

place.

(3)

A

B C

60°

60 cm

160 cm

D

CAPS - GRADE 11

93



7.3 In the diagram, O is the centre of a semi-circle.

PQRS is a rectangle drawn inside the semi-circle such that O lies on RS. α SOP = .

Calculate the size of α for which PQRS will be a square. (3)

[18] QUESTION 8 A spherical glass ball is tightly packed in a box. The box is in the shape of a cube, as shown in the picture on the LEFT. The radius of the ball is 6 cm. The diagram on the RIGHT shows the cross-section of the glass ball placed in the box.

What volume of the box remains after the glass ball is placed in it? [5]

P Q

S R O α

6 cm

CAPS - GRADE 11

94



Give reasons for your statements in QUESTIONS 9, 10, 11 and 12. QUESTION 9 In the diagram, O is the centre of the circle. A, B, C and D are points on the circumference of the circle. Chord DC is produced to E. AC is drawn. °= 80COD and °= 80ECB .

9.1 Calculate the size of the following angles:

9.1.1 CAD (2)

9.1.2 BAD (2)

9.1.3 CAB (1) 9.2 Hence, or otherwise, prove that DC = BC. (2)

[7]

1 2

1 2

A B

C

D

E O

1 2

80° 80°

CAPS - GRADE 11

95



QUESTION 10 In the diagram, PQRS is a cyclic quadrilateral. PS and QR are produced and meet at T. PR bisects SPQ . Also, °= 92RSP and SPQ = 68°.

Calculate the size of the following angles:

10.1 TPR

(1)

10.2 SQT

(2)

10.3 SQP

(3)

10.4 T

(4) [10]

1 2

1 2

1 2 3

1

T

P

Q

R

S

M

68° 92°

CAPS - GRADE 11

96



QUESTION 11 11.1 In the diagram, O is the centre of the circle and AB is a chord. D is a point on AB

such that OD ⊥ AB. Use Euclidean geometry methods to prove the theorem which states that AD = DB.

(5) 11.2 In the diagram, PN is a diameter of the circle with centre O. RT is a tangent to the

circle at R. RT produced and PN produced meet at M. OT is perpendicular to NR. NT and OR are drawn.

11.2.1 Prove that TO || RP. (3)

11.2.2 It is further given that x=NRT . Name TWO other angles each equal to x.

(3)

11.2.3 Prove that NTRO is a cyclic quadrilateral. (2) 11.2.4 Calculate the size of M in terms of x. (3) 11.2.5 Show that NT is a tangent to the circle at N. (3) [19]

• O

A B D

O

P

N

M

R

T S

1 2 3

1 2 3

1 2 3

1 2

3

CAPS - GRADE 11

97


Copyright reserved

QUESTION 12 In the diagram, ABCF is a cyclic quadrilateral. AB is a tangent to circle BCD at B.

Prove that CDEF is a cyclic quadrilateral. [5]

TOTAL: 150

A

B

C D

E

F

1 2

1 1

1

2 2

2

CAPS - GRADE 11

98

Mathematics/P2 DBE/November 2013 CAPS – Grade 11

Copyright reserved

LEARNER'S NAME:

DIAGRAM SHEET 1 QUESTION 1.5

QUESTION 6.1

QUESTION 9

20 40 60 80 100 120 140 160 180 200 220 240 260

1 2

1 2

A B

C

D

E O

1 2

80° 80°

x

y

1

2

3

–1

–2

–3

0 90° 180° –90° –180°

CAPS - GRADE 11

99

Mathematics/P2 DBE/2013 Exemplar


QUESTION 1

The data below shows the number of people visiting a local clinic per day to be vaccinated against measles.

5 12 19 29 35 23 15 33 37 21 26 18 23 18 13 21 18 22 20

1.1 Determine the mean of the given data. (2)

1.2 Calculate the standard deviation of the data. (2)

1.3 Determine the number of people vaccinated against measles that lies within ONE standard deviation of the mean. (2)

1.4 Determine the interquartile range for the data. (3)

1.5 Draw a box and whisker diagram to represent the data. (3)

1.6 Identify any outliers in the data set. Substantiate your answer. (2) [14]

CAPS - GRADE 11

100

Mathematics/P2


QUESTION 2

A group of Grade 11 learners were interviewed about using a certain application to send SMS messages. The number of SMS messages, m, sent by each learner was summarised in the histogram below.

2.1 Complete the cumulative frequency table provided in DIAGRAM SHEET 1. (2)

2.2 Use the grid provided in DIAGRAM SHEET 2 to draw an ogive (cumulative frequency curve) to represent the data. (3)

2.3 Use the ogive to identify the median for the data. (1)

2.4 Estimate the percentage of the learners who sent more than 11 messages using this application. (2)

2.5 In which direction is the data skewed? (1) [9]

0

5

10

15

20

25

30

35

40

45

50

0 2 4 6 8 10 12 14 16 18

Freq

uenc

y

Number of SMS messages (m)

Histogram showing the number of SMS messages sent by learners

7

15

26 29

36

14

31

2

CAPS - GRADE 11

101

DBE/2013 Exemplar

Mathematics/P2


QUESTION 3

A(1 ; 6), B(3 ; 0), C(12 ; 3) and D are the vertices of a trapezium with AD || BC. E is the midpoint of BC. The angle of inclination of the straight line BC is θ, as shown in the diagram.

3.1 Calculate the coordinates of E. (2)

3.2 Determine the gradient of the line BC. (2)

3.3 Calculate the magnitude of θ. (2)

3.4 Prove that AD is perpendicular to AB. (3)

3.5 A straight line passing through vertex A does not pass through any of the sides of the trapezium. This line makes an angle of 45° with side AD of the trapezium. Determine the equation of this straight line. (5)

[14]

y

x

C(12 ; 3)

A(1 ; 6)

B(3 ; 0)

D

E

θ

CAPS - GRADE 11

102

DBE/2013 Exemplar

Mathematics/P2


QUESTION 4

In the diagram below, P(–3 ; 17), Q, O and S are the vertices of a parallelogram. The sides OS and OQ are defined by the equations xy 6= and xy −= respectively. α SOQ = .

4.1 Determine the equation of QP in the form cmxy += . (3)

4.2 Hence, determine the coordinates of Q. (4)

4.3 Calculate the length of OQ. Leave your answer in simplified surd form. (2)

4.4 Calculate the size of α. (3)

4.5 If 148 OS= units, calculate the length of QS. (3) [15]

P(–3 ; 17)

S

O

Q

y

x

α

CAPS - GRADE 11

103

DBE/2013 Exemplar

Mathematics/P2


QUESTION 5

5.1 In the figure below, the point P(–5 ; b) is plotted on the Cartesian plane. OP = 13 units and α POR = .

Without using a calculator, determine the value of the following:

5.1.1 αcos (1)

5.1.2 )180tan( α−° (3)

5.2 Consider: )90cos(

)tan()90sin()360sin(θ

θθθ+°

−−°°−

5.2.1 Simplify )90cos(

)tan()90sin()360sin(θ

θθθ+°

−−°°− to a single trigonometric ratio. (5)

5.2.2 Hence, or otherwise, without using a calculator, solve for θ if °≤≤° 3600 θ :

5,0)90cos(

)tan()90sin()360sin(=

+°−−°°−

θθθθ (3)

5.3 5.3.1 Prove that AAA cos1

4cos14

sin82 −

=+

− . (5)

5.3.2 For which value(s) of A in the interval °≤≤° 3600 A is the identity in QUESTION 5.3.1 undefined? (3)

5.4 Determine the general solution of 01cos2cos8 2 =−− xx . (6) [26]

13

P(–5 ; b)

y

x R•

•

O

α

CAPS - GRADE 11

104

DBE/2013 Exemplar

Mathematics/P2


QUESTION 6

In the diagram below, the graphs of )cos()( pxxf += and xqxg sin)( = are shown for the interval °≤≤°− 180180 x .

6.1 Determine the values of p and q. (2)

6.2 The graphs intersect at A(–22,5° ; 0,38) and B. Determine the coordinates of B. (2)

6.3 Determine the value(s) of x in the interval °≤≤°− 180180 x for which0)()( <− xgxf . (2)

6.4 The graph f is shifted 30° to the left to obtain a new graph h.

6.4.1 Write down the equation of h in its simplest form. (2)

6.4.2 Write down the value of x for which h has a minimum in the interval °≤≤°− 180180 x . (1)

[9]

x

y

A

B

g

-180° -90° 0° 90° 180°

f

45° 135° -135° -45°

1

0,5

- 0,5

-1

x

y

CAPS - GRADE 11

105

DBE/2013 Exemplar

Mathematics/P2


QUESTION 7

7.1 Prove that in any acute-angled ∆ABC, c

Ca

A sinsin= . (5)

7.2 In ∆PQR, ,132P °= PQ = 27,2 cm and QR = 73,2 cm.

7.2.1 Calculate the size of R . (3)

7.2.2 Calculate the area of ∆PQR. (3)

7.3 In the figure below, a QPS = , b SQP = and PQ = h. PQ and SR are perpendicular to RQ.

7.3.1 Determine the distance SQ in terms of a, b and h. (3)

7.3.2 Hence show that )sin(

cossin RSba

bah+

= . (3) [17]

P

Q R

132° 27,2 cm

73,2 cm

S

R Q

P a

b

h

CAPS - GRADE 11

106

DBE/2013 Exemplar

Mathematics/P2


QUESTION 8

A solid metallic hemisphere has a radius of 3 cm. It is made of metal A. To reduce its weight a conical hole is drilled into the hemisphere (as shown in the diagram) and it is completely filled

with a lighter metal B. The conical hole has a radius of 1,5 cm and a depth of 98 cm.

Calculate the ratio of the volume of metal A to the volume of metal B. [6]

QUESTION 9

9.1 Complete the statement so that it is valid:

The line drawn from the centre of the circle perpendicular to the chord … (1)

9.2 In the diagram, O is the centre of the circle. The diameter DE is perpendicular to the chord PQ at C. DE = 20 cm and CE = 2 cm.

Calculate the length of the following with reasons:

9.2.1 OC (2)

9.2.2 PQ (4) [7]

D

O

E

C

P

Q

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QUESTION 10

10.1 In the diagram, O is the centre of the circle and A, B and D are points on the circle. Use Euclidean geometry methods to prove the theorem which states that

BD2ABOA = .

(5)

10.2 In the diagram, M is the centre of the circle. A, B, C, K and T lie on the circle. AT produced and CK produced meet in N. Also NA = NC and °= 38 B .

10.2.1 Calculate, with reasons, the size of the following angles:

(a) AMK (2)

(b) 2T (2)

(c) C (2)

(d) 4K (2)

10.2.2 Show that NK = NT. (2)

10.2.3 Prove that AMKN is a cyclic quadrilateral. (3) [18]

M

A

B

C

K

T

N

2 1

1 2

3

4

38°

B

D

O

A

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QUESTION 11

11.1 Complete the following statement so that it is valid:

The angle between a chord and a tangent at the point of contact is … (1)

11.2 In the diagram, EA is a tangent to circle ABCD at A. AC is a tangent to circle CDFG at C. CE and AG intersect in D.

If ,E and A 11 yx == prove the following with reasons:

11.2.1 BCG || AE (5)

11.2.2 AE is a tangent to circle FED (5)

11.2.3 AB = AC (4) [15]

TOTAL: 150

x

y

A

B

C

D

E

F

G

1 5

1

1

1

1

1

2

2

2

2

2

2 3

3

3

4

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NAME OF LEARNER:

DIAGRAM SHEET 1

QUESTION 2.1

CLASS FREQUENCY CUMULATIVE FREQUENCY

20 <≤ m

42 <≤ m

64 <≤ m

86 <≤ m

108 <≤ m

1210 <≤ m

1412 <≤ m

1614 <≤ m

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NAME OF LEARNER:

DIAGRAM SHEET 2

QUESTION 2.2

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Cum

ulat

ive

Freq

uenc

y

Number of SMS messages

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MATHEMATICS Grade 11 banks/Mathematics/11 Maths... · 2020. 3. 23. · Calculate the effective interest rate. (4) 8.3 The value of a property increased from R145 000 to R221 292,32