wts tutoring 1 wtstutoring · 2017-11-09 · the roots of a quadratic equation is given by √ i....

WTS TUTORING 1

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WTS MATHEMATICS

PAPER ONE

GRADE : 12

COMPILED BY : MR KWV”BABE’SWEMATHS / MASTERMATHS”

DJ MATHS / DR MATHS / PROF/ SOLWAZI SIBIYA

CELL NO. : 082 672 7928

EMAIL : [email protected]

FACEBOOK P. : WTS MATHS & SCEINCE TUTORING

GROUP WHATSAP : 082 672 7928

WTSTUTORING

WTS TUTORING

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WTS TUTORING 2


WHERE TO START MATHS & SCIENCE TUTORING

FINAL EXAMINATION

GRADE 12

SUBJECT MATHEMATICS

PAPER PAPER 1

DURATION OF THE PAPER 3 HOURS

TOTAL MARKS 150

NUMBER OF QUESTIONS 10 – 12

QUESTION PAPER FORMAT

LEVEL 1

questions

Knowledge 20%

LEVEL 2

questions

Routine procedures 35%

LEVEL 3

questions

Complex procedures 30%

LEVEL 4

questions

Problem solving 15%

EXPECTED WORK COVERAGE

ALGEBRA, EQUATIONS AND

INEQUALITIES

25±3 marks

NUMBER PATTERNS 25±3 marks

FINANCE, GROWTH AND DECAY 15±3 marks

FUNCTIONS AND GRAPHS 35±3 marks

DIFFERENTIAL CALCULUS 35±3 marks

PROBABILITY 15±3 marks

WTS TUTORING 3


INSTRUCTIONS AND INFORMATION

Read the following instructions carefully before answering the questions.

1.

2.

3.

4.

5.

6.

7.

8.

This question paper consists of----- questions.

Answer ALL the questions.

Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in

determining your answers.

Answers only will not necessarily be awarded full marks.

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

used, unless stated otherwise.

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

otherwise.

Diagrams are NOT necessarily drawn to scale.

Number the answers correctly according to the numbering system used in this

question paper. Write neatly and legibly.

WTS TUTORING 4


WTS FINISHINING SCHOOL

PLACE

KZN RICHARDS BAY @ MZINGAZI

SUBJECTS

: MATHS, PHYSCS, ACCOUNTING & LIFE SCIENCES

TIME

15:00 TO 17:00

ACCOMMODATION

AVAILABLE

FOR MORE INFO. CALL: 082 672 7928

WTS TUTORING 5


ALGEBRA ± 25 MARKS

A. EQUATIONS

1. Solve for x in each of the following:

(a) ( )

(b) ( )

(c)

(d)

(e)

(f)

(g)

(h) ( )

(i) ( )( )

(j) ( )( )

(k) ( ) ( )

(l) ( ) ( )

(m)

(n)

(o)

(p)

if

WTS TUTORING 6


B. INEQUALITY EQUATIONS


a. ( )

b.

c. ( )

d. ( )( )

e. Given ( )

Determine the values of for which ( ) .

f.

C. EXPONENTIAL EQUATIONS


a.

b.

c.

d.

e.

f.

g.

h.

12792 xxx

WTS TUTORING 7


i.

j.

2. Simplify fully, without using a calculator.

a.

b.

c.

D. SURDS & SQUARING BOTH SIDES


a. √

b. √

c. √

d. √

e. √

2. Simplify, without the use of a calculator:

3. Show that:

3

2

729

1

1

2

2

10.5

m

mm

1

1

2

22.5

y

yy

222 28829872 xxx

4

3223

26

549

WTS TUTORING 8


E. FRACTION EQUATIONS


a.

b. Given:

i. Solve for .

ii. Hence, or otherwise, solve for if √

√

F. SIMULTANEOUS EQUATIONS

1. Solve for x and y simultaneously:

a. and

b. and

c. and

d. and

e. and

f. ( ) ( )

g.

( ) and

2. Calculate the ratio of

for

WTS TUTORING 9


G. NATURE ROOTS

1. The roots of a quadratic equation is given by √

i. Determine the value(s) of for which the equation will have real roots.

ii. Determine the value(s) of for which the equation will have non-real roots.

iii. Determine the value(s) of for which the equation will have equal roots.

2. The roots of a quadratic equation are given by √

,

where * +

i. Write down TWO values of for which the roots will be rational.

ii. Write down ONE value of for which the roots will be irrational.

iii. Write down ONE value of for which the roots will be non-real.

3. For which value of k will the equation have no real roots

4. For what value(s) of p will have equal roots.

5. If 5

2)(

2

x

xxxf , for which values of x is:

i. f (x) = 0?

ii. f (x) undefined?

iii. f (x) real?

6. Given: ( )

For which values of c will f(x) = c have no real roots?

kxx 22

WTS TUTORING 10


H. NON ROUTINE PROBLEMS

a. Calculate, without using a calculator, the value of a and b if a and b are integers and:

√ √ √

b. Calculate, without using a calculator:

c. Calculate the sum of the digits of .

d. Determine the range of the function

, and is real.

e. Write down the domain of ( ) √ .

f. if ( ) √ and ( ) , determine, without ( ( ))

g. Calculate the maximum value of S if

h. if √ calculate without the use of a calculator, the value of √

20112015

2013

55

5

20002009 52

WTS TUTORING 11


NUMBER PATTERNS ± 25 MARKS

A. ARTHMETIC SEQUENCE

1. Prove that:

, ( ) -

2. The sequence is given.

i. Determine the general term

ii. Which term is equal to 71?

iii. Which first term will be greater than 41?

iv. Determine the sum of the first n terms

v. Hence , calculate the sum of the first 40 terms.

3. How many terms of the series 3 + 8 + 13 + … must be added to give a sum of 2265?

4. Consider the following pattern:

Calculate the sum of the terms in the 2010th row.

5. Consider an arithmetic sequence which has the second term equal to 8 and the T5 = 10

i. Determine the first term and common difference of this sequence

ii. Determine the term.

ii. Determine the sum of the first 50 terms

33121110

24987

15654

6321

WTS TUTORING 12


6. Given the finite arithmetic sequence: 5 ; 1 ; –3 ; ... ; –83 ; –87

i. Write down the fourth term (T4) of the sequence.

ii. Calculate the number of terms in the sequence.

iii. Calculate the sum of all the negative numbers in the sequence.

iv. Consider the sequence: 5 ; 1 ; –3 ; ... ; –83 ; –87 ; … ; –4 187.

v. Determine the number of terms in this sequence that will be exactly divisible by 5

7. The following is an arithmetic sequence:

i. Calculate the value of k

ii. Write down the value of a and d

iii. Explain why none of the numbers in this sequence are perfect squares.

8. Determine the value(s) of x in the interval , - for which the sequence

-1 ; 2sin3x ; 5 ;..... will be arithmetic.

WTS TUTORING 13


B. GEOMETRIC SEQUENCE

1. Prove that:

2. Given: ( ) ( ) ( ) as a geometric sequence and x 2

i) Determine the general term of the series in terms of x

ii. Calculate the value of x for which the sequence converges.

iii. Determine the sum to infinity of the series if x = 2,5.

3. Given the geometric series:

i) If x = 4, then determine the sum to 15 terms of the sequence.

ii) Determine the values of x for which the original series converges.

iii) Determine the values of x for which the original series will be

increasing.

4. Given 2 and –1 as the first two terms of an infinite geometric series. Calculate the

sum of this series.

5. The following information of a geometric pattern is given

and

Determine the following:

i. numerical values of the first three term if r > 0

ii. n-term formula

1;1

)1()...(2

r

r

ratermsntoarara

n

WTS TUTORING 14


6. Mr KWV bought a bonsai (tree) at a nursery, when he bought the tree, its height was 130

mm, thereafter the height of the tree increased each year respectively:

100mm ; 70mm ; 49mm; …

i. During which year will the height of the tree increase by approximately 11,76mm?

ii. Mr KWV plots a graph to represent the height ( ) of the tree (in mm) in years

after he bought it. Determine a formula for ( )

iii. What height will the tree eventually reach?

C.COMBINATION

1. Given the combined arithmetic and constant sequence:

i. Write down the next TWO terms in the sequence.

ii. Calculate the sum of the first 100 terms of the sequence.

iii. Calculate the sum of the first 45 terms of the sequence.

2. Given the sequence:

Determine the value(s) of if the sequence is:

i. Arithmetic

ii. Geometric

...;2;9;2;6;2;3

WTS TUTORING 15


D. QUADRATIC SEQUENCE

1. Given the quadratic sequence:

i. Write down the next TWO terms.

ii. Calculate the nth term of the quadratic sequence.

iii. Determine T10 of the above sequence.

iv. Which term in the sequence is equal to 55?

v. Determine between which two consecutive terms of the quadratic sequence the

first difference will be equal to 2018.

vi. What is the value of the first term of the sequence that is greater than 77?

2. The quadratic pattern is given. Determine the value of .

3. The first four terms of a quadratic number pattern are

i. Calculate the value (s) of .

ii. If , determine the position of the first term in the quadratic number

pattern for which the sum of the first differences will be greater than 250.

WTS TUTORING 16


4. Dots are arranged to form a sequence of patterns as shown below:

i. If the pattern behaves consistently, write down the number of dots in pattern 5

ii. Determine a formula for the number of dots in the nth pattern.

iii. Use your formula to calculate which pattern number has 1 985 dots in it?

E.SIGMA NOTATION

a. The following geometric series is given: up to 5 terms.

i. Write down the series in sigma notation.

ii. Calculate the sum to 5 terms of the series.

b. Given: .

/ .

/

i. For what values of will the series converge?

ii. Hence, determine ∑ .

/

if

c. Calculate the value of n if: 531440321

1

n

k

k

Pattern 1 Pattern 2 Pattern 3 Pattern 4

WTS TUTORING 17


d. The following geometric series is given: x = 5 + 15 + 45 + … to 20 terms.

i. Write the series in sigma notation.

ii calculate the value of x

e. Evaluate:

∑( )

f. For which values of will :

∑

g. Determine the value of if:

∑

∑( )

F. GIVEN SUM FORMULA

1. The sum to n terms of a sequence of numbers is given as:

i. Calculate the first 3 terms

ii. Determine the nth tern

iii. Calculate the sum to 23 terms of the sequence.

iv. Hence, calculate the 23rd term of the sequence.

952

nn

S n

WTS TUTORING 18


G. INTERPRETATIONS

1. The first two terms of a geometric sequence and an arithmetic sequence are the same. The

first term is 12. The sum of the first three terms of the geometric sequence is 3 more than the

sum of the first three terms of the arithmetic sequence. Determine TWO possible values for

the common ratio, r, of the geometric sequence

2. The first two terms of an infinite geometric sequence are 8 and . Prove, without the use

of a calculator, that the sum of the series to infinity is .

3. Three numbers form a geometric sequence. Their sum is 21 and their product is 64.

Find the numbers (show all working)

4. The first two terms of a geometric sequence are the same as the first two terms of an

arithmetic sequence. The first term is 8 and is greater than the second term. The sum of the

first three terms of the arithmetic sequence is 1,125 less than the sum of the first three terms

of the geometric sequence. Determine the first three terms of each sequence.

2

8

2816

WTS TUTORING 19


FINANCIAL MATHS ±15 MARKS

A.COMPOUND & SIMPLE INTEREST

1. How long will it take for a motor car to double in value if the annual inflation rate is 8,5%?

2. A motor car costing R200 000 depreciated at a rate of 8% per annum on the reducing

balance method. Calculate how long it took for the car to depreciate to a value of R90 000

under these conditions.

3. Determine how long, in years, it will take for the value of a motor vehicle to decrease to

20% of its original value if the rate of depreciation, based on the reducing balance method, is

22% per annum.

4. Find the time taken for a certain sum of money to double if the interest rate is 11,2% per

annum compounded quarterly.

5. R1 430,77 was invested in a fund paying i% p.a. compounded monthly. After 18 months

the fund had a value of R1 711,41. Calculate i.

B. FUTURE VALUE

1. Andile decided to start saving money for a period of 8 years starting on 31st December

2009. At the end of January 2010 (in one month‟s time), he deposited R2300 into the savings

plan. Thereafter, he continued making deposits of R2300 at the end of each month for the

planned 8 year period. The interest rate remained fixed at 10% per annum compounded

monthly. How much will he have saved at the end of his 8 year plan which started on the 31st

December 2009?

WTS TUTORING 20


2. Suppose that at the beginning of the month, R2 000 is deposited into a bank. At the end of

the month a further R2 000 is deposited and a further R2 000 at the end of the next month. If

the interest rate is 6% per annum compounded monthly, how much will have been saved after

5 years?

3. Aphile has just turned twenty years old and has a dream of saving R10 000 000 by the time

she reaches the age of 50. She starts to pay equal monthly amounts into a retirement annuity

which pays 8% per annum compounded monthly. Her first payment start on her 20th birthday

and her last payment is made on her 50th birthday. How much will she pay each month?

B. COMPOUND AND FUTURE VALUE

1. Zanoh deposits R4 000 into an account paying 14% per annum compounded half-yearly.

Six months later, he deposits R500 into the account. Six months after this, he deposits a

further R500 into the account. He then continues to make half-yearly deposits of R500 into

the account for a further nine years. Calculate the value of his savings at the end of the

savings period.

2. Nokuthula opened a savings account with a single deposit of R1 000 on 1 April 2011. She

then makes 18 monthly deposits of R700 at the end of every month. Her first payment is

made on 30 April 2011 and her last payment on 30 September 2012. The account earns

interest at 15% per annum compounded monthly.

Determine the amount that should be in her savings account immediately after her last deposit

is made (that is on 30 September 2012)

WTS TUTORING 21


C. LOANS & OUTSTANDING BALANCE

1. May wants to purchase a house that costs R950 000. She is required to pay a 13% deposit

and she will borrow the balance from a bank.

i. Calculate the amount that May must borrow from the bank.

ii. The bank charges interest at 8% per annum, compounded monthly on the loan amount.

May works out that the loan will carry an effective interest rate of 8,6% per annum. Is her

calculation correct or not? Justify your answer with appropriate calculations.

iii. May takes out a loan from the bank for the balance of the purchase price and agrees to

pay it back over 20 years. Her repayments start one month after her loan is granted.

Determine her monthly instalment if interest is charged at 8% per annum compounded

monthly.

iv. May can afford to repay R8 000 per month. How long will it take her to repay the loan

amount if she chooses to pay R8 000 as a repayment every month?

2. Sbu negotiates a loan of R400 000 with a bank which has to be repaid by means of

monthly payments of R6 000 and a final payment which is less than R6 000. The

repayments start one month after the granting of the loan. Interest is fixed at 12% per

annum, compounded monthly.

i. Calculate the annual effective interest rate of the loan.

ii. Determine the number of payments required to settle the loan.

iii. Calculate the balance outstanding after Sbu has paid the last R6 000.

iv. Calculate the value of the final payment made by Sbu to settle the loan.

v. Calculate the total amount that Sbu repaid to the bank.

WTS TUTORING 22


3. You wish to purchase your first home. The Bank will only allow bond repayments that are

no greater than 30% of your net monthly salary. Your gross salary is R8 250 per month and

you have deductions of 25% per month from your salary.

a. How much do you take home after deductions? (net salary).

b. What is the maximum bond repayment you can afford?.

c. The bank offers a fixed rate of 13, 5% per annum compounded monthly, over a 20

year period. There is a flat that costs R150 000. Can you afford the flat? (show all

your workings).

E. GAP PAYMENTS FOR LOANS

1. Khetha buys future to the value of R20 000. He borrows the money on 1 February 2010

from a financial institution that charges interest at a rate of 9, 5% p.a. compounded monthly.

Khetha agrees to pay monthly instalments of R550. The agreement of the loan allows Khetha

to start paying these equal monthly instalments from 1 August 2010.

i. Calculate the total amount owing to the financial institution on 1 July 2010

ii. How many months will it take Khetha to pay back the loan?

iii. What is the balance of the loan immediately after Khetha has made the 25th payment?

v. Calculate his saving after settling at 25th payment?

WTS TUTORING 23


2. On 1 June 2016 a bank granted Thabiso a loan of R250 000 at an interest rate of 15% p.a.

compounded monthly, to buy a car. Thabiso agreed to repay the loan in monthly instalments

commencing on 1 July 2016 and ending 4 years later on 1 June 2020. However, Thabiso was

unable to make the first two instalments and only commenced with the monthly instalments

on 1 September 2016.

i. Calculate the amount Thabiso owed the bank on 1 August 2016, a month before he

paid his first monthly instalment.

ii. Having paid the first monthly instalment on 1 September 2016, Thabiso will still pay

his last monthly instalment on 1 June 2020. Calculate his monthly instalment.

iii. If Thabiso paid R9 000 as his monthly instalment starting on 1 September 2016, how

many months sooner will he repay the loan?

iv. If Thabiso paid R9 000 as a monthly instalment starting on 1 September 2016,

calculate the final instalment to repay the loan.

F. SINKING FUND

1. Khangelani„s small business called WTS TUTORING purchase a photocopying machine

for R200 000. The photocopy machine depreciates in value at 20% per annum on a reducing

balance. Khangelani‟s business wants to buy a new machine in 5 year‟s time. A new machine

will cost much more in the future and its cost will escalate at 16% per annum effective. The

old machine will be sold at scrap value after 5 years. A sinking is set up immediately in order

to save up for the new machine. The proceeds from the sale of the old machine will be used

together with the sinking fund to buy the new machine.

The small business will pay equal monthly amounts into the sinking fund and interest earned

is 18% per annum compounded monthly. The first payment will be made immediately and

the last payment will be made at the end of the five year period.

i. Find the scrap value of the old machine.

ii. Find the cost of the new machine in five years‟ time.

iii. Find the equal monthly payments made into the sinking fund.

WTS TUTORING 24


iv. Suppose that the business decides to service the machine at the end of each year for

the five year period. R3000 will be withdrawn from the sinking fund at the end of

each year starting one year after the original machine was bought.

v. Calculate the reduced value of the sinking fund at the end of the five year period due

to these withdrawals.

vi. Calculate the increased monthly payment into the sinking fund which will yield the

original sinking fund amount as well as allow for withdrawals from the fund for the

services.

2. On the 2nd day of January 2015 a company bought a new printer for R150 000.

The value of the printer decreases by 20% annually on the reducing-balance method.

When the book value of the printer is R49 152, the company will replace the printer.

a. Calculate the book value of the printer on the 2nd day of January 2017.

b. At the beginning of which year will the company have to replace the printer? Show

ALL calculations.

c. The cost of a similar printer will be R280 000 at the beginning of 2020. The company

will use the R49 152 that it will receive from the sale of the old printer to cover some

of the costs of replacing the printer. The company set up a sinking fund to cover the

balance. The fund pays interest at 8,5% per annum, compounded quarterly. The first

deposit was made on 2 April 2015 and every three months thereafter until 2 January

2020.

Calculate the amount that should be deposited every three months to have enough

money to replace the printer on 2 January 2020.

WTS TUTORING 25


G. ACCUMULATED AMOUNT

1. Mr KWV decides to open savings account for his baby daughter‟s future education. On

opening the account, he immediately deposits R2000 into the account and continues to make

monthly payments at the end of each month thereafter for a period of 16 years. The interest

rate remains fixed at 15% per annum compounded monthly.

i. How much money will he have accumulated at the end of the 16th year?

ii. At the end of the 16-year period, he leaves the money in the account for a further two

year. How much money will he then have accumulated?

H. MISSING PAYMENTS

1. Sakhile decided to buy a house for his family for R800 000. He agreed to pay monthly

instalments of R10 000 on a loan which incurred interest at a rate of 14% p.a. compounded

monthly. The first payment was made at the end of the first month.

i. Show that the loan would be paid off in 234 months.

ii. Calculate the balance outstanding after 119th payments

iii. Suppose the father encountered unexpected expenses and was unable to pay any

instalments at the end of the 120th, 121st, 122nd and 123rd months. At the end of

the 124th month he increased his payment so as to still pay off the loan in 234

months by 111 equal monthly payments.

a. Calculate the accumulated amount.

b. Calculate the value of this new instalment.

WTS TUTORING 26


2. Siphokazi bought a house. She paid a deposit of R102 000, which is equivalent to 12% of

the selling price of the house. She obtained a loan from the bank to pay the balance of the

selling price. The bank charges her interest of 9% per annum, compounded monthly.

a. Determine the selling price of the house.

b. The period of the loan is 20 years and she starts repaying the loan one month after it

was granted. Calculate her monthly instalment.

c. How much interest will she pay over the period of 20 years? Round your answer

correct to the nearest rand.

d. Calculate the balance of her loan immediately after her 85th instalment.

e. She experienced financial difficulties after the 85th instalment and did not pay any

instalments for 4 months (that is months 86 to 89).

Calculate how much Siphokazi owes on her bond at the end of the 89th month.

f. She decides to increase her payments to R8 500 per month from the end of the 90th

month. How many months will it take to repay her bond after the new payment of R8

500 per month?

WTS TUTORING 27


FUNCTIONS

A. HYPERBOLA GRAPH

1. Consider the function

a. Write down the equations of the asymptotes of f.

b. Calculate the intercepts of the graph of f with the axes.

c. Sketch the graph of f .

d. Write down the range and the domain of

e.

f.

g.

h.

i.

Write down the range of y =

Write down the range of ( )

Write down the domain of ( )

Write down the domain of ( )

Write down the range and the domain of ( ) ( )

j. Write down the new equation and asymptotes for the following:

i. ( ) ( )

ii. ( ) ( )

iii. ( ) ( )

iv. If is shifted 2 units to the right and 1 unit downwards

v. If is shifted 2 units to the left and 1 unit upwards.

k. Write down the new equations for the following

i. if reflected across x-axis

.21

3

xxf )(

).(xf

WTS TUTORING 28


ii. If reflected across y-axis

l. Describe, in words the transformation of to if ( )

m. Describe, in words the transformation of to if

n. For which value(s) of will:

i. ( )

ii. ( )

iii. is decreasing

2. The graph of a hyperbola with equation y= f(x) has the following properties

Domain:

Range:

Passes through the point (2; 0)

Determine f(x)

3. Given ( )

Show that the equation can be written as

WTS TUTORING 29


2. The diagram below shows the graph of . The lines x = –1 and y = 1 are

the asymptotes of f . is a point on f and T is the x–intercept of f.

a. Determine the values of and

b. Calculate the coordinates of , the intercept of

c. Determine the equation of symmetry for negative gradient.

d. Determine the equation of symmetry for positive gradient.

e. Determine the coordinates of if is a reflection of about the line .

f. Write down the length of .

g. Calculate the average gradient between and

qpx

axf

)(

)4 ; 2(P

x

y

P(─2 ; 4)

1

─1

O

WTS TUTORING 30


B. PARABOLA GRAPH

1. Given: ( )

a. Rewrite f(x) in the form of ( )

b. Hence, write the coordinates of turning point

c. Write down the coordinate of the y-intercept of f

d. Determine the coordinates of the x –intercepts of f

e. Determine the equation of axis of symmetry.

f. Hence, calculate the maximum value

g. Write down the coordinates of the turning point of f

h. Sketch the graph of y = f (x)

i. Write down the domain and range of f.

j. Determine the equation of , the reflection of in the y –axis.

k. Determine the equation of w , the reflection of in the x –axis.

2. Given and .

Draw graphs of and on the same set of axes.

l. Determine the equation of , the reflection of in the y –axis.

m. Determine the equation of w , the reflection of in the x –axis.

n. Determine the equation of , the reflection of in the x –axis.

k f

f

6)2()( 2 xxf 12

1)(

x

xg

f g

k f

f

l g

WTS TUTORING 31


o. Determine the equation of V , the reflection of in the y–axis.

p. Write down the equation and the new asymptote of :

i. if is translated one unit up.

ii. if is translated two units up.

iii. if is translated one unit left.

iv. if is translated two units right.

v. For ( ) ( )

q. Write down the equation and the new asymptote of :


ii. if is translated two units up.

iii. if is translated one unit left.

iv. if is translated two units right.

vi. For ( ) ( )

r. Write down the coordinates of the new turning point of for the following:


ii. if f is translated two units up.

iii. if f is translated one unit left.

iv. if f is translated two units right.

v. for ( ) ( )

g

,g

g

g

g

g

WTS TUTORING 32


3.The sketch below represent the graphs ( ) and ( ) The point

S(2 ; 4) lies on the axis of symmetry of the graph p and on the line g. A is the turning point of the

graph of p. The graphs of p and g intersect at D and E. The y–intercepts of p and g are C and B

respectively.

a. Show that k = 2, b = 4 and c = 12.

b. Determine the coordinates of A, the turning point of p.

c. Calculate the length of BC.

d. Calculate the length of OB.

e. Calculate the length of AS.

f. Determine the coordinates of D, the point of intersection of p and g.

g. For which value(s) of is ( ) ( ) ?

x

y

E

B

C

A

D

O

S(2 ; 4)

p

g

WTS TUTORING 33


h. For which value(s) of is ( ) ( ) ?

i. For which value(s) of x is ( )

( )

j. For which value(s) of x is ( )

( )

k. Determine the equation of p in the form

l. Write down the maximum value of t(x) if t(x) = 1 – p(x).

m. Solve for x if ( )

n. Solve for x if ( )

o. Determine

p. For which value(s) of is ( ) ( )

q. Show that the coordinates S(3;5) lies on the graph g?

r. Write down the of for which ( ) ( ) .

s. Determine graphically the value(s) of for which:

i. has equal roots

ii. has non-real solutions

iii. has two real, unequal roots

iv. has two positive roots

v. has equal roots

vi. has non-real solutions

vii. has two real, unequal roots

x. has two unequal solutions which differ in signs

.qpxay 2)(

WTS TUTORING 34


4. The sketch below shows the graphs of and . The

graphs intersect at B and E. The graph of g intersects the x – axis at A and B and has a

turning point at C. The graph of h intersects the y – axis at D. The length of CD is 6

units.

a. Determine the coordinates of B and C.

b. Write down the coordinates of D.

c. Write down the values of a and q.

d. Determine the coordinates of E.

e. Determine the value(s) of x for which 0)().( xgxg .

f. Calculate the length of CD in terms of x

g. hence, the length of CD

32)( 2 xxxg qaxxh )(

WTS TUTORING 35


5. The graph of a parabola f has x-intercepts at x = 1 and x = 5. ( ) is a tangent to f

at P, the turning point of f. Sketch the graph of f, clearly showing the intercepts with the

axes and the coordinates of the turning point.

C. EXPONENTIAL GRAPH

1. Consider the function .

a. Is f an increasing or decreasing function? Give a reason for your answer.

b. Determine in the form y = …

c. Sketch the graph of on the same set of axes

d. Write down the range and domain of

e. Write down the equation of the asymptote of ( ) .

f. Write down the range of f(x) – 5.

g. Write down the domain of f(x) – 5.

h. Write down the new equation of for ( ) ( )

i. Write down the new equation of if is reflected across axis.

x

xf

3

1)(

)(1 xf

WTS TUTORING 36


j. Describe the transformation from f to g if .

k. Write down the new equation of if is reflected across:

i. the line y= 0

ii. the line x = 0

l. Write down the new equation for ( ) ( ) and then.

i. Write down the equation of asymptote

ii. Calculate the x intercept

iii. Calculate the y intercept

iv. Sketch the graph of h

v. Write down the range of h

vi. Write down the domain of h

vii. Write down the new equation of ( ) ( )

viii. Write the new equation of if is shifted 6 units up.

ix. If ( ) ( ), write down an equation of the asymptote of

x. Calculate the average gradient of h between the points on the graph where

and

xxg 3log)(

WTS TUTORING 37


2. The graph of , where a > 0 and a ≠ 1, passes through the point .

a. Determine the value of a.

b. Write down the equation of in the form y = … .

c. Determine the value(s) of x for which .

d. Sketch both graph of f and on the same set of axis

e. If , write down the domain of h.

f. Write down the range of

g. Write down the range of

h. Write down the domain of f

i. Write down the domain of

j. Determine the equation of t if .

3. Determine the equation of ( )in the form , if A(1; 18) and the

equation of the horizontal line is .

xaxf )(

8

27;3

1f

1)(1 xf

1f

)5()( xfxh

1f

1f

)()( 1 xfxt

WTS TUTORING 38


4. The graph of an increasing exponential function with equation ( ) has the

following properties:

• Range: y > -3

• The points (0; – 2) and (1; – 1) lie on the graph of f.

Determine the equation that defines f

WTS TUTORING 39


4. The graph of ( ) is sketched

alongside.

a. Give the coordinates of A.

b. Write down the equation of in the form y = …

c. For which value(s) of x will ?

d. For which value(s) of x will ?

e For which value(s) of x will ( ) ?

f. For which value(s) of x will ( ) ?

g. For which value(s) of x will ( ) ?

h. Write down the equation of the asymptote of ( )

i. Write down the equation of the asymptote of ( ) + 3

j. Write down the equation of the asymptote of ( )

x

y

0

f

A

WTS TUTORING 40


i. Determine the equation of f in the form y = …

ii. Hence, write down the equation of in the form y = …

iii. Give the coordinates of the turning point of g(x) = .

D. LOGARITHM GRAPH

1. Given

i. Write down the equation of , the inverse of p, in the form y = …

ii. Sketch the graphs of p and on the same system of axes. Show clearly all the

intercepts with the axes and at least one other point on each graph.

iii. Determine the values of x for which

iv. Write down the x intercept of h if h(x) = p(–x).

v. Solve for x if ( )

1f

1)3(1 xf

xxp 3log)(

1p

1p

2)( xp

5. Sketched is the graph of ( ) √

, the inverse of a restricted parabola. The point

A(8 ;2) lies on the graph of f

x

y

O

A(8 ; 2)

WTS TUTORING 41


2. Sketched below is the graph of ( )

a. Write down the domain of f.

b. Write down the equation of in the form y = …

c. Write down the equation of the asymptote of ,

d. Explain how, using the graph of f, you would sketch the graphs of:

i. ( )

ii. ( )

e. Use the graph of f to solve for x where .

WTS TUTORING 42


E. INVERSE FUNCTIONS

1. Given

a. Sketch the graph of h

b. Determine the inverse of in the form …..

c. Give a reason why the inverse of is not a function.

d. Write down TWO ways in which you could restrict the domain and range of so that

its inverse is a function?

e. Hence, sketch the two graphs of the function on separate sets of axes.

2. The graphs of ( ) √ ; and ( )

for are given.

a. Determine the coordinates of A, the point of intersection between f and g.

b. Determine the equation of .

c. Give the equation of the graph which is obtained when ( ) is reflected in the line

.

2)( xxh

h y

h

h

1h

WTS TUTORING 43


3. The graph of g is defined by the equation ( ) √ . The point (8; 4) lies on g.

a. Calculate the value of a.

b. If g(x) > 0, for what values of x will g be defined?

c. Determine the range of g.

d. Write down the equation of , the inverse of g, in the form y = ...

e. If ( ) is drawn, determine ALGEBRAICALLY the point(s) of

intersection of h and g.

f. Hence, or otherwise, determine the values of x for which ( ) ( )

WTS TUTORING 44


4. The graph of ( ) is sketched below. The point P (– 6 ; – 8) lies on the

graph of f.

a. Calculate the value of a.

b. Determine the equation of , in the form y = …

c. Write down the range of .

d. Draw the graph of . Indicate the coordinates of a point on the graph

different from (0 ; 0).

e. The graph of f is reflected across the line y = x and thereafter it is reflected

across the x-axis. Determine the equation of the new function in the form y = …

WTS TUTORING 45


CALCULUS ±35MARKS

A. FIRST PRINCIPLES

1. A. Calculate the derivative of the following from first principles:

a. ( )

b. ( )

c. ( )

d. ( )

e. ( )

f. ( )

g. ( )

h. ( )

i. ( )

j. ( )

k. ( )

l.

m. ( )

n. ( )

o. ( ) √

WTS TUTORING 46


2. Given: ( )

i. Determine ( ) from the first principle

ii. A( ) where at B ( ) are points on the graph of Calculate the

numerical value of average gradient of between A and B.

3. Sakhile determines ( ) the derivative of a certain function f at and arrives at the

answer

√

, write down the equation of f and the value of b.

B. RULES

1. Determine each of the following:

a. Calculate 0

1

b. ( )

; ( )

c.

( ) if ( )

d. ( ) if ( )

e. ( )

; ( )

f. ( )

; ( )

g.

if

√

h.

if √

i. ( ) ( √ ) ( )

WTS TUTORING 47


j.

if √

k.

if .

/

l.

[√

√ ]

m.

n.

o.

,( ) √

-

p. ( )

q. If 3

8

xy and

y

yz

12 , determine:

i. dx

dy

ii. dy

dx

iii. dy

dz

iv. dx

dz

WTS TUTORING 48


C.TANGENT EQUATIONS

1. Given: ( )

a. Determine .

b. Determine .

c.

d.

e.

Determine the equation of the tangent to g at x = –2 in the form y = mx + c.

Calculate the coordinates of the point of inflection of g

Show that g is increasing for all real value(s) of x.

2. Determine the equation of the tangent to the curve of ( ) at the point on

the curve when .

3. The tangent to the curve of ( ) has the equation .

a. Show that ( ) is the point of contact of the tangent to the graph.

b. Hence, or otherwise, calculate the value of p and q.

4. The curve with equation has a gradient of at the point (1:8) on the

curve. Determine the values of and .

5. The equation of a tangent to the curve of ( ) and .

If the point of contact is ( ). Calculate the values of a and b

)2(g

)2(g

WTS TUTORING 49


6. The tangent to the curve of is perpendicular to the line

.

Find the equation of the tangent.

7. Consider ( )

Determine the -intercept of the tangents to f that has a slope of (at where is an integer)

8. Given: ( ) .

Determine the value of a if it is given that ( ) ( ).

D. CUBIC FUNCTION

1. P is the function defined by:

(i)

(ii)

(iii) ( )( )( )

(iv)

Determine the following of each function:

a. Write down coordinates of ( ) and ( ) or intercepts with the axes.

b. Calculate the coordinates of the turning point

c. Hence, sketch the graph of p.

WTS TUTORING 50


d. Find the coordinate of inflection point/ point at which ( ) is a maximum

e. For which values of will;

i. ( )

ii. ( )

iii. ( )

iv. ( )

v. ( )

vi. ( )

vii. ( ) ( )

viii. ( ) ( )

f. For which value(s) of x the concavity of the graph will:

i. Concave up?

ii. Concave down?

g. Use the graph to determine the values of x for which the equation:

i. ( ) , have one real root, equal roots and 3 distinct roots.

ii. ( ) , have one real root, equal roots and 3 distinct roots.

iii. , have one real root, equal roots and 3 distinct roots.

iv. , have one real root, equal roots and 3 distinct roots.

WTS TUTORING 51


v. Determine the value(s) of k for which p = k has negative roots only

h. Calculate the average gradient between the turning points.

i. Determine the equation of the tangent to p at

j. Write down the coordinate of turning point of:

i. ( )

ii. ( )

iii. ( )

iv. ( )

k. Write down the coordinates of a turning points and equations of if is defined by

( ) ( ) ( ) .

(ii) Reflected across x-axis / ( ) ( )

(iii) Reflected across y-axis / ( ) ( )

(iv) Reflected about the line

(v) Reflected about the line

WTS TUTORING 52


2. Given ( ) and ( )

the graph of f intersects the x-axis at x = -2 ; x = 1 and x = 3 . the turning points of f

are at A points B respectively, where Line PQ is perpendicular to the x-axis,

with point P on f and point Q on g.

a. Show that the equation of f can be given as ( )

b. Calculate the coordinates of points A and B

c. Calculate the maximum length of line PQ , for the interval

d. The graph of f is concave down for x < k , calculate the value(s) of k.

e. Determine the equation to f at the point of inflection in the form

WTS TUTORING 53


E. EQUATION OF CUBIC FUNCTION

1. The graph of a cubic function with equation ( ) is drawn.

f has a local maximum at B and a local minimum at x = 4.

a. Show that

b. Calculate the x-coordinate of the point at which is a maximum.

c. Determine the value of x for which f is strictly increasing.

0)4()1( ff

.16and24,9 cba

y

WTS TUTORING 54


3. The graph below represents the functions f and g with. ( ) and

( ) . A and D(– 1; 0) are the x-intercepts of f . The graphs of f and g

intersect at A and C.

a. Determine the coordinates of A.

b. Show by calculation that and .

c. Calculate the coordinates of C

d. Calculate the average gradient between B and D

D

WTS TUTORING 55


3. The function ( ) is sketched below.

The turning points of the graph of f are T (2:-9) and S (5:18).

y

x

a. Show that and

b. Write down the coordinates of the turning points of ( ) ( )

c. Write down the coordinates of the turning points of ( ) ( )

4. The function defined by ( ) is sketched below P (-1:-1) and

R are the turning points of f.

A(x: y)

a) Show that and .

0

T (2:-9)

S(5:18)

P (-1:-1)

WTS TUTORING 56


F. DERIVED GRAPH

1. The sketch represents the curve of ( ) with ( ) .

-1 5

a. What is the slope of the tangent to f at the point where ?

b. Give the x-intercept of the curve

c. Show that

is the x-coordinate of the inflection point of f.

d. For which values of x is f decreasing?

e. For which values of x is ( )

e. Write down the value(s) of x that give local maximum and local minimum.

f. Hence, sketch the graph of ( ).

g. Determine the equation of ( )

2

WTS TUTORING 57


G. PROPERTIES GIVEN

1. The following information about a cubic polynomial, ( ) is given:

( )

( )

( )

( )

( ) ( )

( )

( )

a. Draw a neat sketch graph of

b. For which value(s) of x is strictly decreasing and increasing?

c. Use your graph to determine the x-coordinate of the point of inflection of .

d. For which value(s) of x is concave up?

e. For which value(s) of x is ( )

2. Given: ( ) .

Draw a possible sketch of ( ) if and are all negative real numbers.

3. A cubic function has the following properties:

• .

/ ( ) ( )

• ( ) .

/

• decreases for x ,

- only.

Draw a possible sketch graph of , clearly indicating the x-coordinates of the turning

points and ALL the x-intercepts.

f

f

f

f

WTS TUTORING 58


x

H. APPLICATION OF CALCULUS

TO SHAPES

1. A rectangular box has a length of 5x units, breadth of units and its height of x

units.

a. Show that the volume (V) of the box is given by .

b. Determine the value of x for which the box will have maximum volume.

c. Hence, calculate the maximum volume.

d. Calculate the total surface area.

e. Determine the value of x for which the box will have maximum surface area.

f. Hence, calculate the maximum surface area.

)29( x

5x (9 – 2x)

WTS TUTORING 59


2. A container shaped in the form of a cylinder with no top has a volume of 340 ml.

It has a radius of x cm and a height of h cm. Note: 1 ml = 1 cm3

a. Write down the height (h) in terms of x.

b. Show that the surface area (S) of the cylinder with no top is given by

.

c. Calculate the value of x for which the surface area of the cylinder will be a minimum.

Determine the rate of change of the volume of water flowing into the tank when the depth is 5 cm.

3. A water tank in the shape of a right circular cone has a height of h cm. The top rim of the tank

is a circle with radius of r cm. The ratio of the height to the radius is 5:2. Water is being

pumped into the tank at a constant rate.

x

h

Surface Area of Cone =

Volume of Cone

WTS TUTORING 60


COST

1. A crate used on fruit farms in the Ping River valley is in the form of a rectangular prism

which is open on top. It has a volume of 1 cubic metre. The length and the breadth of its

base is 2x, and x metres respectively. The height is h metres. The material used to

manufacture the base of this container costs R200 per square metre and for the sides, R120

per square metre

2x

a. Express h in terms of x

b. Show that the cost, C, of the material is given by: ( )

c. Calculate the value of x for which the cost of the material will be a minimum.

d. Hence, calculate the minimum cost of the material.

2x

WTS TUTORING 61


TO GRAPH

1. A farmer has a piece of land in the shape of a right-angled triangle OMN, as shown in the

figure below. He allocates a rectangular piece of land PTOR to his daughter, giving her the

freedom to choose P anywhere along the boundary MN. Let OM = a, ON = b and P(x ; y) be

any point on MN.

( )

T ( )

( )

O

a. Calculate the gradient of MN

b. Determine an equation of MN in terms of a and b.

c. Calculate the midpoint of MN

d. Prove that the daughter's land will have a maximum area if she chooses P at the

midpoint of MN.

WTS TUTORING 62


2. The rectangle PQRS is drawn as shown in the sketch, with P a point on the curve y = x2

and SR the line x = 6.

a. Write down the coordinates of Q, P and R

b. Express the length, QR, and breadth, SR, of the rectangle in terms of x.

c. Show that the area of the rectangle can be given as A = – x3 + 6x2.

d. Hence, calculate the area of the largest rectangle PQRS which can be drawn.

x

y

P S

Q R

WTS TUTORING 63


TO RATE

1. A stone is thrown vertically upward and its height (in metres) above the ground at (in

seconds) is given by ( )

a. Find its initial height above the ground.

b. Find the initial speed with which it was thrown.

c. Determine the rate of change at .

d. Calculate the time at which the rate of change will be minimum.

2. A tourist travels in a car over a mountainous pass during his trip. The height above sea

level of the car, after t minutes, is given as ( ) meters . The

journey lasts 8 minutes.

a. How high is the car above sea level when it starts its journey on the mountainous pass?

b. Calculate the car's rate of change of height above sea level with respect to time, 4 minutes

after starting the journey on the mountainous pass.

c. Interpret your answer to QUESTION. b.

d. How many minutes after the journey has started will the rate of change of height with

respect to time be a minimum?

WTS TUTORING 64


PROBABILITY ± 15 MARKS

A. INDEPENDENT& DEPENDENT EVENTS VS MUTUAL EXCLUSIVE

1. P (A) = 0,3 and P(B) = 0,5. Calculate P(A or B) if:

a. A and B are mutually exclusive events.

b. A and B are independent events.

2. Events A and B are mutually exclusive. It is given that:

• P(B) = 2P(A)

• P(A or B) = 0,57

Calculate P(B).

3. If ( )

and ( )

, Find:

a. ( ) if A and B are mutually exclusive events.

b ( ) if and are independent events.

4. The events A, B and C are such: A and B are independent, B and C are independent and A and C

are mutually exclusive. Their probabilities are;

P(A) = 0,3

P(B) = 0,4

P(C) = 0,2.

Calculate the probability of the following events occurring:

a. Both A and C occur.

b. Both B and C occur.

WTS TUTORING 65


c. At least one of A or B occur.

5. Given that A and B are independent events. Determine the values of x and y if:

P(B only) = 0,3

P( A and B) = 0,2

P(A only) = x

P(not A or B) = y

B.VENN DIAGRAMS

1. In a group of 50 learners, 35 take Mathematics and 30 take History. 12 learners do not take

Mathematics or History.

a. Draw a Venn diagram to represent this information.

b. If a learner is chosen at random from this group, what is the probability that he takes:

(i) Both Mathematics and History?

(ii) Mathematics only?

(iii) History only?

(iv) Mathematics or History?

WTS TUTORING 66


2. A group of 65 learners were surveyed on their choice of movies, namely Comedy (C),

Action (A) and Drama (D). These were their responses.

30 learners enjoyed Comedy movies.

38 learners enjoyed Action movies.

7 learners enjoyed Drama and Comedy only.

11 learners enjoyed Comedy and Action only.

13 enjoyed Drama only.

9 learners enjoyed Action and Drama.

There were x number of learners who enjoyed all three types of movies.

a. Use the above information and the Venn diagram on Annexure A to

calculate the value of x.

b. Write down the probability that a learner selected at random enjoys Action

movies ONLY.

c. Determine the probability that a learner selected at random enjoys both

Action and Drama.

WTS TUTORING 67


3. A school organised a camp for 103 Grade 12 learners. The learners were asked which food they prefer

for the camp. They had to choose from chicken (C), vegetables (V) and fish (F).

The following information was collected:

2 learners do not eat chicken, fish or vegetables

5 learners eat only vegetables

2 learners only eat chicken

21 learners do not eat fish

3 learners eat only fish

66 learners eat chicken and fish

75 learners eat vegetables and fish

Let the number of learners who eat chicken, vegetables and fish be x.

b) Draw a Venn diagram to represent the information.

c) Calculate x.

d) Calculate the probability that a learner, chosen at random:

e) Eats only chicken and fish, and no vegetables.

f) Eat only chicken

g) Eat only fish

h) Eats any TWO of the given food choices: chicken, vegetables and fish.

WTS TUTORING 68


C. TREE DIAGRAMS

1. Zanoh and Aphile enter a competition that involves Running (R) and Swimming (S). The

probability of Zanoh choosing to run is 0,4 while the probability of Aphile choosing to

swim is 0,3 .

a. Complete the tree diagram below.

b. Determine the probability of Zanoh choosing both events at the end of the second year in any order of

choice.

c. Determine the probability of Aphile choosing both events at the end of the second year in any order of

choice.

d. Determine the probability of Zanoh choosing both events at the end of the second year in an order of

both swimming.

e. Determine the probability of Aphile choosing both events at the end of the second year in an order of

both running.

R

S

R

S

R

S

R

S

R

R

S

S

Z

A

0,3

0,4

WTS TUTORING 69


2. A drawer contains 20 envelopes. 8 of the envelopes each contain 5 blue and 3 red sheets of paper.

The other 12 envelopes each contain 6 blue and 2 red sheets of paper. One envelope is chosen at random.

A sheet of paper is chosen at random from it.

a. Draw a tree diagram

b. What is the probability that this sheet of paper is red?

3. In a factory, three machines, A, B and C, are used to manufacture plastic bottles. They produce 20%,

30% and 50% respectively of the total production. 1%, 2% and 6% respectively of the plastic bottles

produced by machines A, B and C are defective.

a. Represent the information by means of a tree diagram. Clearly indicate the probability associated

with each branch of the tree diagram and write down all the outcomes.

b. A plastic bottle is selected at random from the total production.

i. What is the probability that it was produced by machine B and it is not defective?

ii. What is the probability that the bottle is defective?

WTS TUTORING 70


D. CONTINGENCY TABLES

The hair colour of 50 learners was recorded. Girls with black, Brown and Blond hair 10, 8

and 6 respectively: Boys with black, Brown and Blond hair 12, 9 and 5 respectively.

a) Represents the information on the table below

GIRLS BOYS TOTAL

BLACK

BROWN

BLOND

TOTAL

b. Calculate the probability that learner chosen at random:

i. will have brown hair

ii. will have blond hair

iii. will have black hair or brown hair

xi. will have blond hair or brown hair or black hair

xii. will have Girls and Black

xiii. will have Boys and Brown

c. Are the events brown and Boys for the hair, independent or dependent? Support your

answer with the appropriate calculations.

d. Are the events of Girls and Black mutually exclusive? Explain your answer

WTS TUTORING 71


2. In a survey 1 530 skydivers were asked if they had broken a limb. The results of the survey

were as follows:

Broken a limb Not broken a limb TOTAL

Male 463 B 782

Female A C D

TOTAL 913 617 e

a. Calculate the values of a, b, c, d and e.

b. Calculate the probability of choosing at random in the survey, a female skydiver who

has not broken a limb.

c. Is being a female skydiver and having broken a limb independent? Use calculations,

correct to TWO decimal places, to motivate your answer.

WTS TUTORING 72


E. COUNTING PRINCIPLES

1. How many different outfits can be combined using a shirt and a pair of pants from?

3 shirts (red, white or black) and 2 pairs of pants (black or green)

2. The Matric Dance Committee has decided on the menu below for the 2017 Matric Dance.

A person attending the dance must choose only ONE item from each category,that is starters,

main course and dessert.

MENU

STARTERS

Garlic Bread: Crumbed Mushrooms: Fish

MAIN COURSE

Vegetable Curry: Chicken Curry: Fried Chicken: Beef Bolognaise

DESSERT

Malva Pudding: Ice-cream

a. How many different meal combinations can be chosen?

b. A particular person wishes to have chicken as his main course. How many different

meal combinations does he have?

WTS TUTORING 73


F. ARRANGEMENTS

1. Consider the word MAY.

How many word arrangements can be made with the word MAY?

2. Consider the word MOM.

How many word arrangements can be made with the word MOM?

a) If the repeated letters are treated as different letters?

b) How many word arrangements can be made with the word MOM if the repeated

letters are treated as the same letters?

3. Determine the number of arrangement that can be formed from all the letters of the word

a. WILMOTH

b. KHANGELANI

4. Determine the number of arrangement that can be formed from all the letters of the word

KHANGELANI, if ;

a. the letters may be repeated?

b. the letters may not be repeated?

c. last letters must be A.

d. the first and last letters must be A.

e. all the As have to be next to each other

WTS TUTORING 74


f. What is the probability that a random arrangement of the letters of KHANGELANI if;

i. Last letters must be A.

ii. Starts and ends with an „A‟

iii. All the As have to be next to each other

5. In how many ways can 10 people be seated in 6 places?

i. If repetition is not allowed

ii. If repetition is allowed

6. How many different ways are there of predicting the results of 5 PSL matches where each

match can end in either a win, lose or draw?

7. Suppose that a number plate is formed using three letters of the alphabet, excluding the

vowels and Q followed by any three digits. Calculate the probability that a number plate,

chosen at random,

a. starts with a „B‟ and ends with a „5‟

b. has exactly one „B‟

c. has at least one „5‟?

WTS TUTORING 75


G. SITTING ARRANGEMNENT

1. Six players of a volleyball team stand at random positions in a row before the game

begins. X and Y are two players in this team. Determine the probability that:

X and Y will not stand next to each other.

2. Consider: Five boys and four girls are to be seated randomly in a row.

In how ways can:

a. five boys and four girls sit in the row.

b. they sit in a row if a boy and his girlfriend must sit together?

c. they sit in a row if the boys and girls are each to sit together?

d. they sit in a row if just the girls are to sit together?

e. they sit in a row if just the boys are to sit together?

ii. what is the probability that:

a. The row has a girl at each end?

b. The row has girls and boys sitting in alternate positions?

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H. DIGITS

NUMBER

1. Determine how many 4 – digit numbers can be formed from 10 digits 0 to 9 if:

a. repetition of digits is allowed.

b. repetition of digits is not allowed.

c. the last digit must be 0 and repetition of digits is allowed.

d. the last digit must be 0 and repetition of digit is not allowed

e. the last digit must be 2 and repetition of digits is allowed.

f. the last digit must be 2 and repetition of digits is not allowed.

CODE

A code is made using the format XYY, where the X is any letter in the alphabet and Y

represents any digit from 0 to 9.

a. How many possible codes can be formed if the letters and digits are repeated?

b. How many possible codes can be formed if the letters and digits are not repeated?

2. The digits 0, 1, 2, 3, 4, 5 and 6 are used to make 3 digit codes.

a. How many unique codes are possible if digits can be repeated?

b. How many unique codes are possible if the digits cannot be repeated?

c. In the case where digits may be repeated, how many codes are numbers that are

greater than 300 and exactly divisible by 5?

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3. The digits 0, 1 , 2 , 3 , 4 , 5 , 6 , 7 and 8 are used to make 4 digit codes.

a. how many unique codes are possible different if the digits may be repeated?

b. how many unique codes are possible different if the digits may not be repeated?

c. In the case where digits may be repeated, how many codes are numbers that are greater

than 2 000 and even?

d. In the case where digits cannot be repeated, how many codes are numbers that are greater

than 2 000 and divisible by 4?

e. What is the probability that a code will contain at least one 7? The digits may be repeated.

f. What is the probability that a code will contain at least one 7? The digits may not be

repeated.

g. how many codes can be formed between 4 000 and 5 000? The digits may be repeated.

4. The digits 1 to 7 are used to create a four-digit code to enter a locked room. How many

different codes are possible if the digits may not be repeated and the code must be an even

number bigger than 5000?

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MATHEMATICS

EXAMINATION GUIDELINES

GRADE 12

1. INTRODUCTION

The Curriculum and Assessment Policy Statement (CAPS) for Mathematics outlines the

nature and purpose of the subject Mathematics. This guides the philosophy underlying the

teaching and assessment of the subject in Grade 12.

The purpose of these Examination Guidelines is to:

• Provide clarity on the depth and scope of the content to be assessed in the Grade 12

National Senior Certificate (NSC) Examination in Mathematics.

• Assist teachers to adequately prepare learners for the examinations.

This document deals with the final Grade 12 external examinations. It does not deal in any

depth with the School-Based Assessment (SBA).

These Examination Guidelines should be read in conjunction with:

• The National Curriculum Statement (NCS) Curriculum and Assessment Policy Statement

(CAPS): Mathematics

• The National Protocol of Assessment: An addendum to the policy document, the National

Senior Certificate: A qualification at Level 4 on the National Qualifications Framework

(NQF), regarding the National Protocol for Assessment (Grades R–12)

• The national policy pertaining to the programme and promotion requirements of the

National Curriculum Statement, Grades R–12

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TOPICS

FUNCTIONS

1. Candidates must be able to use and interpret functional notation. In the teaching process

learners must be able to understand how f has been transformed

2. Trigonometric functions will ONLY be examined in Paper 2.

NUMBER PATTERNS, SEQUENCES AND SERIES

1. The sequence of first differences of a quadratic number pattern is linear. Therefore,

knowledge of linear patterns can be tested in the context of quadratic number patterns.

2. Recursive patterns will not be examined explicitly.

3. Links must be clearly established between patterns done in earlier grades.

FINANCE, GROWTH AND DECAY

1. Understand the difference between nominal and effective interest rates and convert fluently

between them for the following compounding periods: monthly, quarterly and half-yearly or

semi-annually.

2. With the exception of calculating i in the Fv and Pv formulae, candidates are expected to

calculate the value of any of the other variables.

3. Pyramid schemes will not be examined in the examination.

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ALGEBRA

1. Solving quadratic equations by completing the square will not be examined.

2. Solving quadratic equations using the substitution method (k-method) is examinable.

3. Equations involving surds that lead to a quadratic equation are examinable.

4. Solution of non-quadratic inequalities should be seen in the context of functions.

5. Nature of the roots will be tested intuitively with the solution of quadratic equations and in

all the prescribed functions.

DIFFERENTIAL CALCULUS

1. The following notations for differentiation can be used: ( )

2. In respect of cubic functions, candidates are expected to be able to:

• Determine the equation of a cubic function from a given graph.

• Discuss the nature of stationary points including local maximum, local minimum and points

of inflection.

• Apply knowledge of transformations on a given function to obtain its image.

3. Candidates are expected to be able to draw and interpret the graph of the derivative of a

function.

4. Surface area and volume will be examined in the context of optimisation.

5. Candidates must know the formulae for the surface area and volume of the right prisms.

These formulae will not be provided on the formula sheet

6. If the optimisation question is based on the surface area and/or volume of the cone, sphere

and/or pyramid, a list of the relevant formulae will be provided in that question. Candidates

will be expected to select the correct formula from this list.

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PROBABILITY

1. Dependent events are examinable but conditional probabilities are not part of the syllabus.

2. Dependent events in which an object is not replaced are examinable.

3. Questions that require the learner to count the different number of ways that objects may

be arranged in a circle and/or the use of combinations are not in the spirit of the curriculum.

4. In respect of word arrangements, letters that are repeated in the word can be treated as the

same (indistinguishable) or different (distinguishable). The question will be specific in this

regard.

NB:

WE ALSO HAVE COMPILATION FOR ALL CHAPTERS

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MERCY!!!!!

WHERE TO START MATHS AND SCIENCE TUTORING

“Where to Start Maths and Science tutoring” is aiming at assisting learners with

understanding of basic skills for Maths and Sciences and also changes the stigma of learners

towards Maths and Science subjects, we also help Schools around KZN and even to other

provinces.

WTS VISITING SCHOOL PROGRAM

DAYS : FRIDAYS, SATURDAYS & SUNDAYS

SUBJECTS : MATHS, MATHS LIT AND PHYSCS

TIME : ANY TIME AND EVEN CROSSNIGHTS

BOOK US ON : 0826727928

WTS PRIVATE CLASSES

PLACE : RICHARDS BAY ARBORETUM

GRADES : 8 TO 12

WEEKENDS

LEARNERS FROM FAR PLACES

TIME : 17:00 TO 21:00

SUBJECTS : MATHS & SCIENCES

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WEEKDAYS

MONDAY TO FRIDAY

TIME : 17:30 TO 20:20

SUBJECTS : MATHS & SCIENCES

ACKNOWLEDGEMENTS

TEACHING APROACH

Mr AS MABASO @ MANGOSUTHU UNIVERSITY

TEAM TEACHING

MR MJIYAKHO @ KWANOTSHELA SCHOOL [VRYHEID]

MR MAGWAZA @ NOMBUSO SCHOOL [PORT SHEPSTONE]

UNIZULU TUTORS

TYPING

MISS SP NTSHALINTSHALI

EDDITING

ADVISOR MRS EN PHAKATHI

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