MathematicsGrade 12

Paper 1

Presented byWilbert Tenga

Mr. T

Scope for Paper 1

25 marks• Algebra

• Sequences and Series

• Functions

• Finance

• Differential Calculus

• Probability

35 marks

25 marks

15 marks

35 marks

15 marks

Algebra

Quadratic equations Example

Factorisation is key!!!!!!• Quadratic equation• Quadratic formula• Quadratic inequality• Radical (surd equations)• Exponential equations• Simultaneous equation• Fractional equations

• Nature of roots

• 𝑥! + 6𝑥 = 0 𝑜𝑟 𝑥 + 3 𝑥 + 2 = 0• 𝑥! + 3𝑥 − 5 = 0 (round off to…)• 𝑥! − 𝑥 + 12 ≤ 0• 𝑥 − 2 = 2• 3!"#$ = 9• 𝑥! − 4𝑥y + 𝑦! = 2 & 𝑦 = 3𝑥 + 2• %

"&'= !

"#%

Errors in Algebra

Quadratic equations Example• Quadratic equations

• Dividing all terms by x resulting in loss of one of the solutions.

• Expanding not required.• Expanding required.• Trying to factorise questions which

need the quadratic formula.• Not testing values and not isolating

the surd.• Representation of solutions.• Restrictions in fractional equations• Simultaneous equations • Nature of roots

• 𝑥! + 6𝑥 = 0

• 𝑥 + 3 𝑥 − 2 = 0• 𝑥 + 3 𝑥 − 2 = 1• 𝑥! + 3𝑥 − 5 = 0 (round off to…)

• 𝑥 − 2 = 2 (test values)

• 𝑥! + 6𝑥 < 0• %

"&'= !

"#%; 𝑥 ≠ 4 , 𝑥 ≠ −3

• sub linear into quadratic• 𝑏! − 4𝑎𝑐

Common MistakesQuadratic formula

• Wrong usage of formula• If b - value is negative

brackets needed.• Square root sign

• Division line

• Writing b instead of –b in the formula

Incorrect!

−(−𝑏) ± −𝑏! − 4𝑎𝑐2𝑎

−𝑏 ± 𝑏! − 4𝑎𝑐2𝑎

-b ± (!&')*!)

𝑏 ± 𝑏! − 4𝑎𝑐2𝑎

Sequence and Series

Basic Theory on sequence• Arithmetic sequence (linear)- 𝑇!= 𝑎 + 𝑛 − 1 𝑑• Quadratic sequence (parabola)- 𝑇!= 𝑎𝑛" + 𝑏𝑛 + 𝑐• Geometric sequence (exponential)- 𝑇!= 𝑎𝑟!#$

Series Sum to Infinity• Arithmetic

• 𝑆+ =+![2𝑎 + 𝑛 − 1 𝑑]

• Geometric

• 𝑆+ =)($&-")$&-

; 𝑟 ≠ 1• Sigma

• n=top-bottom+1

• Convergent series: −1 < 𝑟 < 1

• 𝑆B = CDEF ; 𝑟 ≠ 1

Misconceptions in Sequences and Series

Sequences and series Examples

• Negative common difference (brackets needed)

• 1st term in first level difference of a quadratic sequence is not 2a+b but 3a+b

• Incorrect formulas for:• Common difference written as d = 𝑇$ − 𝑇" instead of d = 𝑇" − 𝑇$Also common ratio written as

𝑟 = %!%"

instead of 𝑟 = %"%!

• 𝑇#= 𝑎 + 𝑛 − 1 𝑑• 𝑚𝑖𝑠𝑡𝑎𝑘𝑒 𝑇#= 2 + 𝑛 − 1 − 3• 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑇#= 2 + 𝑛 − 1 (−3)

• 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 3𝑎 + 𝑏

Misconceptions in Sequences and Series

Sequences and series Examples

• Incorrect formulas for:• 𝑆! =

!"[2𝑎 + 𝑛 − 1 𝑑]

• 𝑆& = '$#(

• 𝑛 = 𝑡𝑜𝑝 − 𝑏𝑜𝑡𝑡𝑜𝑚 + 1 in sigma.

• Calculating n using logs!

• Brackets! If 𝑟 = − $)

• 𝑚𝑖𝑠𝑡𝑎𝑘𝑒 𝑆! =!"[𝑎 + 𝑛 − 1 𝑑]

• 𝑚𝑖𝑠𝑡𝑎𝑘𝑒 𝑆& = '(#$

• 𝑚𝑖𝑠𝑡𝑎𝑘𝑒: Top-bottom

• 𝑆! ='($#(()$)

$#(

FunctionsTheory

Straight line Hyperbola• 𝑦 = 𝑚𝑥 + 𝑐• 𝑦 − 𝑦$ = 𝑚(𝑦 − 𝑦$)

• 𝑚 = ,"#,!-"#-!

• Parallel & perpendicular line

• Sketching

• 𝑦 = )"&/

+ 𝑞• Asymptotes: 𝑥 = 𝑝 & 𝑦 = 𝑞

not 𝑝 = ⋯or q = ⋯• Axis of symmetry:

𝑦 = 𝑥 + 𝑐 𝑜𝑟 𝑦 = −𝑥 + 𝑐• Sketching

Parabola Exponential• 𝑦 = 𝑎𝑥! + 𝑏𝑥 +c• 𝑦 = 𝑎(𝑥 − 𝑝)! + 𝑞• Inverse 𝑓&$• Reflections • Shifts (horizontal or vertical)• Sketching• Interpretation

• 𝑦 = 𝑎. 𝑏"#/ + 𝑞• Asymptote• 𝑓&$ leading to logarithms• Sketch• Domain• Range

FunctionsErrors

Straight line Hyperbola• Gradient treated as change in

x/change in y• 𝑚 = ,"#,!

-"#-!• Tangents

• Formula.• Sketch curves in wrong quadrants• Not labelling key points• Shifts • Reflections• Curves crossing asymptote• Failure to determine equation

having given a sketch.

Parabola Exponential• Interpretative questions – bracket

0;∞ 𝑎𝑛𝑑 0; 2• Determining equation in the form𝑦 = 𝑎(𝑥 − 𝑥$)(𝑥 − 𝑥!)

• Integration with tangent• Determining the inverse of a

parabola.

• Wrong manipulation from exponential to logarithm

• Graphs which cross asymptote• Domain or range of log function

FinanceTheory

Grade 10-11 Grade 12

Appreciation• Simple (Hire purchase) and

compound interest (Inflation).

Depreciation• Straight line and reducing

balance method• Effective to nominal interest

and vice versa• Timelines

Annuities• Future value (savings and

investments)• Present value (loans)• Outstanding balance• Sinking funds

FinanceErrorsMistake Notes

Calculation of n(Possible remedy use timelines!)

Calculation of i

Wrong formula!

• Immediate payment resulting in an additional payment for investments.

• Immediate payment acts as a deposit for loans.

• log equation in solving for n

• Not dividing by the correct compound period e.g monthly, quarterly and yearly.

• n for PV is negative!• Effective to nominal

FinanceErrors continued

Mistake Notes

Wrong formula!• Present value

n must be negative.

• n written as 12n instead of saying:𝑛 = 𝑦𝑒𝑎𝑟𝑠 ×𝑐𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝑖𝑛𝑔 𝑝𝑒𝑟𝑖𝑜𝑑

• i not divided by the compounding period

• Pv=- $#($./)$

/WRONG!!

• Pv=- $#($./)%$

/CORRECT!

• For compounded monthly:• 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 = 0

$!

CalculusTheory

Content Context

• First principles• Differential rules • Cubic function

• Turning point• Point of inflection• concavity• Sketching using notation• Using 𝑓1 𝑥 sketch to find 𝑓 𝑥

original graph.• Rate of change• Tangents

• Optimisation• Area and volume of various

shapes.

• 𝑓1 𝑥 = 0• 𝑓1′ 𝑥 = 0• 𝑓11 " < 0 𝑜𝑟 𝑓1′ 𝑥 > 0

• Surface area and volumes of various shapes.

CalculusCommon errors

Content Error• First principles

• Notational mistake

• Surd to exponent for

example ! 𝑥H

• Inability to factorise a cubic function.

• Sketching 𝑓 𝑥 from 𝑓1 𝑥 .• Using Chain or quotient rule.

• 𝑥!" instead of 𝑥

"!

• Various methods

• Time wasted!!!!

ProbabilityNotes

Content Context

• Venn diagram• Tree diagram

• Compound events• Independent and dependent

event• Probability rule• Mutually exclusive events• Contingency table

• Counting principles

• 2 or 3 rings • Independent and dependent event

• P(A and B)=P(A).P(B)

• P(A or B)=P(A)+P(B)-P(A and B)

• P(A or B)=P(A)+P(B)

• Combinations• Pin numbers

ProbabilityErrorsContent Mistake

• Tree diagram

• Venn diagram

• Counting methods

• Contingency table

• Adding consecutive events instead of multiplying

• Sum of probability not equating 1.

• Assigning correct values in correct parts of the ring.

• Concept of at ‘least’

• Restrictions• Repetitions allowed and not

allowed.

• Applying conditional probability were not required

General tips

• Rest well and try to relax!

• Treat this exam like any other.

• Know your calculator and formulas before the exam!

• Read instructions and strategize the sequence through

which you will answer the questions.

General tips

• Start with the easiest questions.

• Time management is key! You need to set aside

revision time.

• Read and highlight the main highlights of questions

and annotate the given diagrams.

• After the exam focus on the next exam!

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