Matric Life

SUMMARY OF SEQUENCES AND SERIES

Arithmetic Sequence (AS) (also named the linear sequence)

GENERAL TERM: T_n

a = first term T₁

d = constant diff.

Internet connection needed etc.

SUM OF TERMS: S_n

Where

i = the last term of sequence

Example

A) 2 ; 5 ; 8 ; 11 ;...
d = +3 +3 +3

T_n = 2 + (n - 1)(3)
= 2 + 3_n - 3
= 3_n - 1

B) 1 ; -4 ; -9;...
d = -5 -5

T_n = 1 + (n - 1)(-5)
= 1 - 5_n + 5
= -5_{n + 6}

Geometric Sequence (GS) (also named exponential sequence)

GENERAL TERM: T_n

a = first term T₁

r = constant ration

SUM OF TERMS: S_n

Where -1 < r < 1
(Converging series)

Example

A) 2 ; -4 ; 8 ; 16;...

r = x-2 x-2 x-2

T_n=2(-2)^n-1

NOT CONVERGING asa r < - 1

B) Internet connection needed

CONVERGING as -1 < r < 1

Quadratic Sequence (QS)

GENERAL TERM: T_n

T_n = an² + bn + c

f = 1^st difference
s = 1^nd difference

Determine a, b and c using simultaneous equations (see example)

Alternatively :

a = s / 2
b = f₁-3_a
c = T₁ - a - b

where
f₁ = first term of first difference

Example

3 ; 8 ; 16 ; 27 ;...

f: 5 8 11
s: 3 3

Setup three equations using the first three terms:

T₁ = 3:

3 = a + b + c ...(1)
T₂ = 8:
8 = 4_a + 2_b + c ...(2)
T₃ = 16:
16 = 9_a + 3_b + c ...(3)

Solving simultaneously leads
to: Internet connection needed

TYPES OF QUESTIONS YOU CAN EXPECT	STRATEGY TO ANSWER THIS TYPE OF QUESTION	EXAMPLE(S) OF THIS TYPE OF QUESTION
Identify any of the following three types of sequences: Arithmetic (AS), Geometric (GS) and Quadratic (QS)	Determine whether sequence has a constant 1_st difference (AS) constant ratio (GS) constant 2_nd difference (QS)	See examples 1 above
Determine the formula for the general term, Tⁿ, of AS, GS and QS (from Grade 11)	You need to find: a and d for an AS a and r for a GS a, b and c for a QS	See examples 1 above
Determine any specific term for a sequence e.g. T³⁰	Substitute the value of n into Tⁿ	See Text Book : Example 1, nr. 1 d and 2 d, p.8 (AS) Example 1, nr. 1 b, 3 b, p.11 (AS) Example 1, nr. 1, p. 15 (GS)
Determine the number of terms in a sequence, ??, for an AS, GS and QS or the position, n, of a specific given term or when the sum of the series is given	Substitute all known variables into the general term to get an equation with n as the only unknown. Solve for n. Substitute all known variables into the Sⁿ-formula to get an equation with n as the only unknown. Solve for n.	See Text Book: Example 1, nr.1 c, p.8 Example 1, nr.1 c, p.11 Example 1, nr. 3, p.15 Example 2, nr.3, p.20 Example 3, nr. 2, p.24
When given two sets of information, make use of simultaneous equations to solve: a and d (for an AS) a and r (for a GS)	For each set of information given, substitute the values of n and T_n or n and S_n. You then have 2 equations which you can solve simultaneously (by substitution)	See Text Book: Example 1, nr. 3, p.11 (AS) Example 1, nr.2, p.15 (AS) Example 3, nr.3, p.24 (GS)
Determine the value of a variable (x) when given a sequence in terms of x.	For AS use constant difference: T₃ - T₂ = T₂ - T₁ For GS use constant ratio:	The first three terms of an AS are given by : 8- 2_x-(x-3) = x 3 - (2x - 4) x = 5
For a series given in sigma notation: Determine the number of terms Determine the value of the series, in other words, S_n Write a given series in sigma notation.	Remember: The “counter” indicates the number of terms in the series Remember the expression next to the is the general term, T_n This will help you to determine a and d or r. Determine the general term, T_k and number of terms, n and substitute into	has n terms (counter 𝑘 runs from 1 to n) has (n+ 1) terms (counter runs from 0 to n so one term extra) has (𝑖 − 4) terms ( four terms not counted )
Determine the sum, S_n , of an AS and a GS (when the number of terms are given or not given)	In some cases you have to first determine the number of terms, _n using T_n. Substitute the values of a, n and d/r into the formula for S_n
Determine whether a GS is converging or not	Converging if −1 < r < 1
Determine S_oo for a converging GS	Substitute vales of a and r Into formula for S_oo
Determine the value of a variable (x) for which a series will converge, e.g. (2_x + 1) + (2_x + 1)2 +...	Determine 𝑟 in terms of 𝑥 and use −1 < r < 1
Apply your knowledge of sequences and series on an applied example (often involving diagram/s)	Generate a sequence of terms from the information given. Identify the type of sequence.

TYPES OF RELATIONS BETWEEN TWO VARIABLES

TYPE	DESCRIPTION	PROPERTIES	TYPICAL EXAMPLES
NON-FUNCTIONS	One-to-many	• One 𝑥-value in domain has MORE THAN ONE 𝑦-value • Does NOT pass vertical line test	• Inverse of a parabola
FUNCTIONS	One-to-one	Each 𝑥-value has a unique 𝑦- value • No 𝑥- or 𝑦-value appear more than once in domain or range • Passes VERTICAL line test	• Straight line graph and its inverse • Hyperbola and its inverse • Exponential graph and its inverse, the logarithmic function
	Many-to- one	• No 𝑥-value appears more than once in domain • More than one 𝑥-value maps onto the same 𝑦-value • Passes VERTICAL line test	• Parabola • Graph of the cubic function • Trigonometric graphs

REVISION OF THE PARABOLA

EQUATION IN TURNING POINT FORM

DETERMINE THE EQUATION OF A PARABOLA

REVISION OF THE HYPERBOLA

REVISION OF THE EXPONENTIAL GRAPH

EXAMPLES OF SYMMETRICAL EXPONENTIAL GRAPHS

INTERSECTS OF TWO GRAPHS

To determine the coordinates of the point where two graphs INTERSECT: Use SIMULTANEOUS EQUATIONS

EXAMPLE

Determine the coordinates of the points of intersection of f(x) = 3x + 6 and g(x) = −2_x² + 3_x + 14

Equate the two equations and solve for _x:

3_x + 6 = −2_x² + 3_x + 14 2_x² − 8 = 0 x² − 4 = 0 (x − 2)(x + 2) = 0 x = 2 or x = −2

Substitute _x-values back into one of equations (choose the easier one):

If x = 2 then y = 3(2) + 6 = 12 So one point of intersection is (2; 12).

If x = −2 then y = 3(−2) + 6 = 0 The other point of intersection is (−2; 0) which is also the x-intercept of both graphs.

THE INVERSE OF A FUNCTION

The inverse of a function, f, is denoted by f⁻¹.
f⁻¹ is a reflection of f in the line y = x
To determine the equation of f⁻¹, swop x and y in the equation of f
The x-intercept of f is the y-intercept of f⁻¹

Exercise

Definition of logarithm

If log_b x = y, then b^y = x

Note that:

log_a a = 1 (a ≠ 0)
log_a 1 = 0
log_a = log₁₀

USING LOGARITHMS TO SOLVE EQUATIONS

We know that equations involving exponents can be solved using exponential laws:

2^x = 128 2^x = 2⁷(prime factories) ∴ x = 7

But, what if we cannot use prime factors?

2^x = 13
log2^x = log13
xlog2 = log13
x = log13/log2 = 3,7

THE INVERSE OF THE EXPONENTIAL GRAPH

Exercise 2

1 Make use of the definition of the logarithm to solve for x:

HIRE PURCHASE AGREEMENTS

A = P(1 + in)

Example:Simple Interest

Kelvin buys computer equipment on hire purchase for R20 000. He has to put down 10% deposit and repays the amount monthly over 3 years. The interest rate is 15% p.a.

Deposit = 10% of R20 000 = R2 000.
He has to repay 𝐴 = 18000(1 + 0,15 × 1) = R26 100 in total.
36 monthly payments of R26 100÷ 36 =R725 each.

INFLATION / INCREASE IN PRICE OR VALUE

A = P(1 + i)ⁿ

n = number of years

NOMINAL AND EFFECTIVE INTEREST RATES

EXAMPLE:

What is the effective rate if the nominal rate is 18% p.a. compounded quarterly?
In other words:
Which rate compounded annually will give me the same return as 18% compounded quarterly?

FUTURE VALUE ANNUITIES

OUTSTANDING BALANCE OF LOAN

Exercise 3

1 Determine through calculation which of the following investments is the best, if R15 000 is invested for 5 years at:

a 10,6% p.a. simple interest
b 9,6% p.a., interest compounded quarterly.

2 An amount of money is now invested at 8,5% p.a compounded monthly to grow to R95 000 in 5 years.

a Is 8,5% called the effective or nominal interest rate?
b Calculate the amount that must be invested now.
c Calculate the interest earned on this investment.

3 Shirley wants to buy a flat screen TV. The TV that she wants currently costs R8 000.

a The TV will increase in cost according to the rate of inflation, which is 6% per annum. How much will the TV cost in two years’ time?
b For two years Shirley puts R2 000 into her savings account at the beginning of every six month period (starting immediately). Interest on her savings is paid at 7% per annum, compounded six-monthly. Will she have enough to pay for the TV in two years’ time? Show all your calculations.

4 Calculate:

a the effective interest rate to 2 dec. places if the nominal interest rate is 7,85% p.a., compounded monthly.
b the nominal interest rate if interest on an investment is compounded quarterly, using an effective interest rate of 9,25% p.a.

5 Equipment with a value(new) of R350 000 depreciated to R179 200 after 3 years, based on the reducing balance method. Determine the annual rate of depreciation.

6 R20 000 is deposited into a new savings account at 9,75% p.a., compounded quarterly. After18 months, R10 000 more is deposited. After a further 3 months, the interest rate changes to 9,95% p.a., compounded monthly. Determine the balance in the account 3 years after the account was opened

7 A company recently bought new equipment to the value of R900 000 which has to be replaced in 5 years’ time. The value of the equipment depreciates at 15% per year according to the reduced-balance method. After 5 years the equipment can be sold second hand at the reduced value. The inflation rate on the equipment is 18% per year.

a The company wants to establish a sinking fund to replace the equipment in 5 years’ time. Calculate what the value of the sinking fund should be to replace the equipment.
b Calculate the quarterly amount that the company has to pay into the sinking fund to be able to replace the equipment in 5 years’ time. The company makes the first payment immediately and the last payment at the end of the 5 year period. The interest rate for the sinking fund is 8% per year compounded quarterly.

8 Goods to the value of R1 500 is bought on hire purchase and repaid in 24 monthly payments of R85. Calculate the annual interest rate that applied for the hire purchase agreement.

9 Peter makes a loan to buy a house. He pays back the loan over a period of 20 years in monthly payments of R6 500. Peter qualifies for an interest rate of 12% per years compounded monthly. He makes his first payment one month after the loan was granted.

a Calculate the amount Peter borrowed.
b Calculate the amount that Peter still owes on his house after he has been paying back the loan for 8 years.

10 Megan’s father wants to make provision for her studies. He starts paying R1000 on a monthly base into an investment on her 12th birthday. He makes the last payment on her 18th birthday. She needs the money 5 months after her 18th birthday. The interest rate on the investment is 10% per annum compounded monthly. Calculate the amount Megan has available for her studies.

11 Stephan starts investing R300 into an investment monthly, starting one month from now. He earns interest of 9% per annum compounded monthly. For how long must he make these monthly investments so that the total value of his investment is R48 000? Give your answer as follows: …. years and …. Months

12 Carl purchases sound equipment to the value of R15 000 on hire purchase. The dealer expects him to put down a 10% deposit. The interest rate is 12% per annum and he has to repay the money monthly over 4 years. It is compulsory for him to insure the equipment through the dealer at a premium of R30 per month. Calculate the total amount Carl has to pay the dealer monthly.

13 Tony borrows money to the value of R400 000. He has to pay back the money in 16 quarterly payments, but only has to make his first payment one year from now. The interest rate is 8% per annum compounded quarterly. Calculate the quarterly payment Tony has to pay.

REVISION OF TRIGONOMETRY

YOU HAVE TO KNOW IN WHICH QUADRANT AN ANGLES LIES AND WHICH RATIO (AND ITS INVERSE) IS POSITIVE THERE:

REDUCTION FORMULAE

Exercise 3

REVISION ON THE USE OF THE sinus −, cosinus − and the area − FORMULAE

TIPS FOR SOLVING PROBLEMS IN THREE DIMENSIONS

Where there are 3 triangles, start with the ∆ with the most information and work via the 2nd ∆ to the 3rd ∆ which contains the unknown to be calculated.
Indicate all RIGHT angles – remember they don’t always look like 90° angles
Shade the horizontal plane in the diagram (e.g. floor, ground)
Be on the lookout for reductions like cos(90^o − a) = sina and sin(180^o − a) = sina to simplify expressions
Use compound and double angle formulae to convert to single angles
When writing out the solution – always indicate in which ∆ you are working

Exercise 4

SYNTHETIC DIVISION

Exercise 5

3 Show that x − 3 is a factor of f(x) = _x3 − _x2 − 5_x − 3 and hence solve f(x) = 0.

4 Show that 2_x − 1 is a factor of _g(x) = 4_x3 − 8_x2 − x + 2 and hence solve _g(x) = 0

NB: NOTE THE DIFFERENCE BETWEEN THE FOLLOWING:

f(4) is the y-VALUE of the function at x = 4
f'(4) is the GRADIENT of the function at x = 4
As well as the gradient of the TANGENT at x = 4

x − INTERCEPTS/ROOTS AND SHAPE

EQUATION OF TANGENT TO GRAPH AT A SPECIFIC POINT

SPECIAL APPLICATIONS OF DERIVATIVES

USING FIRST DERIVATIVE TO DETERMINE MINIMUM OR MAXIMUM

Exercise 6

REVISION OF CONCEPTS FROM PREVIOUS GRADES

THE CIRCLE : (x − a)² + (y − b)² = r² (centre-radius form)

Exercise 7

REVISION of geometry From previous years

PROPERTIES OF SPECIAL QUADRILATERALS

HOW TO PROVE THAT A QUADRILATERAL IS A PARALLELOGRAM

Prove any ONE of the following (most often by congruency):

Prove that both pairs of opposite sides are parallel
Prove that both pairs of opposite sides are equal
Prove that both pairs of opposite angles are equal
Prove that the diagonals bisect each other

HOW TO PROVE THAT A PARALLLELOGRAM IS A RHOMBUS

Prove ONE of the following:

Prove that the diagonals bisect each other perpendicularly
Prove that any two adjacent sides are equal in length

THREE WAYS TO PROVE THAT A QUADRILATERAL IS A CYCLIC QUADRILATERAL

Prove that:

one pair of opposite angles are supplementary
the exterior angle is equal to the opposite interior angle
two angles subtended by a line segment at two other vertices of the quadrilateral, are equal.

GRADE 12 GEOMETRY

Exercise 8

OGIVE

THE OGIVE CAN BE USED TO DETERMINE THE MEDIAN AND QUARTILES

MEASURES OF DISPERSION AROUND THE MEAN

USING A TABLE TO CALCULATE VARIANCE AND STANDARD DEVIATION

USING CASIO fx-82ZA PLUS CALCULATOR TO CALCULATE STANDARD DEVIATION

SCATTER DIAGRAMS (SCATTER PLOTS) FOR BIVARIATE DATA

Scatter diagrams are used to graphically determine whether there is an association between two variables.

By investigation one can determine which of the following curves (regression functions) would best fit the diagram:

CORRELATION

The strength of the relationship between the two variables represented in a scatter diagram, depends on how close the points lie to the line of best fit. The closer the points lie to this line, the stronger the relationship or correlation.

Correlation (tendency of the graph) can be described in terms of the general distribution of data points, as follows:

CORRELATION COEFFICIENT

The correlation between two variables can also be described in terms of a number, called the correlation coefficient. The correlation coefficient, r, indicates the strength and the direction of the correlation between two variables. This number can be anything between −1 and 1

Example

Refer to the previous example again. For the given data set r = 0,958 which means that there is a strong positive relationship between the two variables.

Exercise 9

4 Five number 4; 8; 10; x and y have a mean of 10 and a standard deviation of 4. Find 𝑥 and y.

5 The standard deviation of five numbers is 7,5. Each number is increased by 2. What will the standard deviation of the new set of numbers be? Explain your answer

SUMMARY OF THEORY ON PROBABILITY

FACTORIAL NOTATION

The product 5 × 4 × 3 × 2 × 1 can be written as 5!
.^.. n! = n × (h − 1) × (n − 2) × … × 3 × 2 × 1

LETTER ARRANGEMENTS

When making new words from the letters in a given word , one has to distinguish between:

Exercise 9

1 How many different 074- cell phone numbers are possible if the digits may not repeat?

2 How many different 082- cell phone numbers are possible if the digits may only be integers?

3 What is the probability that you will draw a queen of diamonds from a pack cards?

4 How many different arrangements can be made with the letters of the word TSITSIKAMMA, if:

a repeating letters are regarded as different letters
b repeating letters are regarded as identical.

5 Four different English books, three different German books and two different Afrikaans books are randomly arranged on a shelf. Calculate the number of arrangements if:

a the English books have to be kept together
b all books of the same language have to be kept together
c the order of the books does not matter.

6 In how many different ways can a chairman and a vice-chairman be chosen from a committee of 12 people?

7 The letters of the word MATHEMATICS have to be rearranged. Calculate the probability that the “word” formed will not start and end with the same letter.

8 In how many different ways can the letters of the word MATHEMATICS rearranged so that