1.1 · 8 Marks · Quadratic equations & nature of roots
Roots of (x−4)(x+7)=0
Formulas
Δ=b²−4ac
Discriminant
Notes: Product of two brackets = 0. Unknown: x.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Roots of (x−4)(x+7)=0
Before any working — hunt first
Given
Numbers and units already in the sentence: 4 · 7 · 0. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Roots of (x−4)(x+7)=0. Circle that word — that is the missing variable.
Find
We are required to find: Roots of (x−4)(x+7)=0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write x = 4 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Roots of (x−4)(x+7)=0.
In other words — as if you have never seen this before
Zero-product: a factor is 0. Set each bracket to zero.
2. What do we already know?
Product of two brackets = 0. Unknown: x.
3. Which method — and why this one?
Zero-product: a factor is 0. Set each bracket to zero.
4. Step-by-step working
01 / 05
First bracket
x−4=0→x=4.
x=4
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
02 / 05
Second bracket
x+7=0→x=−7.
x=−7
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
4 and −7
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4 and −7
Copy onto your answer sheet — in this order
Formula: x=4
Answer: 4 and −7
6. Check it like a marker
FOIL back: x²+3x−28. At x=4: 16+12−28=0. At x=−7: 49−21−28=0.
7. A trap first-timers fall into
Swapping signs to −4 and 7 is the most common first-timer slip.
1.2 · 10 Marks · Nominal vs effective interest
Simple interest on R5 000 at 8% for 3 years
Nominal vs effective interest
Formulas
1+ieff=(1 + inom/m)m
Effective rate
Notes: P = 5000. i = 0.08. n = 3. Unknown: interest I, not A.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Simple interest on R5 000 at 8% for 3 years
In other words — as if you have never seen this before
I = Pin. The paper asked for the interest, not the accumulated amount.
2. What do we already know?
P = 5000. i = 0.08. n = 3. Unknown: interest I, not A.
3. Which method — and why this one?
I = Pin. The paper asked for the interest, not the accumulated amount.
4. Step-by-step working
01 / 05
Write I = Pin
I=5000×0.08×3.
I=Pin
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
02 / 05
Calculate
I=400×3=1 200.
I=R1 200
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
R1 200
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
R1 200
Copy onto your answer sheet — in this order
Formula: I=Pin
Answer: R1 200
6. Check it like a marker
A = P + I = 6 200. The R5 400 option is P + one year’s interest only, twice confused.
7. A trap first-timers fall into
The question says “interest”, not “amount”. Circle the asking-word.
2.1 · 10 Marks · Inclination & circles
Turning point of y = (x−3)² + 5
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
Notes: Completed square y = (x − p)² + q with p = 3, q = 5.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Turning point of y = (x−3)² + 5
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 5. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Turning point of y = (x−3)² + 5. Circle that word — that is the missing variable.
Find
We are required to find: Turning point of y = (x−3)² + 5. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write (3; 5) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Turning point of y = (x−3)² + 5.
In other words — as if you have never seen this before
TP is (p; q) = (3; 5). The square is zero at x = 3.
2. What do we already know?
Completed square y = (x − p)² + q with p = 3, q = 5.
3. Which method — and why this one?
TP is (p; q) = (3; 5). The square is zero at x = 3.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 3 · 5. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Turning point of y = (x−3)² + 5. Circle that word — that is the missing variable.
Find: We are required to find: Turning point of y = (x−3)² + 5. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write (3; 5) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Turning point of y = (x−3)² + 5.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Completed square y = (x − p)² + q with p = 3, q = 5.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
TP is (p; q) = (3; 5). The square is zero at x = 3.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Read p and q
x−3=0→x=3. Minimum y=5.
(3; 5)
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
(3; 5)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
(3; 5)
Copy onto your answer sheet — in this order
Formula: (3; 5)
Answer: (3; 5)
6. Check it like a marker
At x=3, y=0+5=5. At x=4, y=1+5=6, higher, so a minimum.
7. A trap first-timers fall into
(−3; 5) copies the minus sign from the bracket. The root of x−3 is +3.
2.2 · 12 Marks · Sine, cosine & area rules
cos θ = 12 in [0°; 360°]
Formulas
asin A=bsin B=csin C=2R
Sine rule
c²=a²+b²−2ab cos C
Cosine rule
Area=½ ab sin C
Area rule
Notes: cos θ = +12. Interval a full revolution.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: cos θ = 12 in [0°; 360°]
Before any working — hunt first
Given
Numbers and units already in the sentence: 1/ · 2 in · 0° · 360°. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: cos θ = 12 in [0°; 360°]. Circle that word — that is the missing variable.
Find
We are required to find: cos θ = 12 in [0°; 360°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write α = 60° with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives cos θ = 12 in [0°; 360°].
In other words — as if you have never seen this before
CAST: cosine is positive in Q1 and Q4. Reference angle 60°.
2. What do we already know?
cos θ = +12. Interval a full revolution.
3. Which method — and why this one?
CAST: cosine is positive in Q1 and Q4. Reference angle 60°.
4. Step-by-step working
01 / 05
Reference
cos⁻¹(1/2)=60°.
α=60°
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
02 / 05
Quadrants
Q1: 60°. Q4: 360°−60°=300°.
60° and 300°
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
60° and 300°
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
60° and 300°
Copy onto your answer sheet — in this order
Formula: α=60°
Answer: 60° and 300°
6. Check it like a marker
cos 120° = −12, the other sign. Sketch the CAST diagram.
7. A trap first-timers fall into
60° and 120° is the sine pair. Cosine does not live in Q2 when it is positive.
5 · 3 Marks · Nominal vs effective interest
R8000 depreciates at 10% p.a. reducing balance for 2 years. Value =
Nominal vs effective interest
Formulas
1+ieff=(1 + inom/m)m
Effective rate
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: R8000 depreciates at 10% p.a. reducing balance for 2 years. Value =
Before any working — hunt first
Given
Numbers and units already in the sentence: R8000 depreciates · 10% · 2 years. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Value =. Circle that word — that is the missing variable.
Find
We are required to find: Value =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 8000(0. Then substitute the given, one line at a time, to get Value =.
In other words — as if you have never seen this before
8000(0.9)²=6480.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: R8000 depreciates · 10% · 2 years. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
8000(0.9)²=6480.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: R8000 depreciates · 10% · 2 years. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Value =. Circle that word — that is the missing variable.
Find: We are required to find: Value =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 8000(0. Then substitute the given, one line at a time, to get Value =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: R8000 depreciates · 10% · 2 years. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
8000(0.9)²=6480.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
8000(0.9)²=6480.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
R6480
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
R6480
Copy onto your answer sheet — in this order
Answer: R6480
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
6 · 2 Marks · Exponents & surds
√48 simplified
Exponents & surds
Formulas
am/ⁿ=ⁿ√aᵐ
Rational exponent
√a·√b=√ab
Product of surds
Notes: √48. Unknown: simplest surd form (no square factor left under the radical).
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: √48 simplified
Before any working — hunt first
Given
Numbers and units already in the sentence: 48 simplified. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: √48 simplified. Circle that word — that is the missing variable.
Find
We are required to find: √48 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 48 = 16 × 3 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives √48 simplified.
In other words — as if you have never seen this before
Pull out the biggest square factor. 48 = 16 × 3, and 16 is 4².
2. What do we already know?
√48. Unknown: simplest surd form (no square factor left under the radical).
3. Which method — and why this one?
Pull out the biggest square factor. 48 = 16 × 3, and 16 is 4².
4. Step-by-step working
01 / 05
Factor 48
48 = 16 × 3, and 16 is a perfect square. (You could use 4 × 12, but then you would simplify again.)
48=16×3
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Take the square out
√48=√16×3=√16×√3=4√3.
√48=4√3
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
4√3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4√3
Copy onto your answer sheet — in this order
Formula: 48=16×3
Answer: 4√3
6. Check it like a marker
4√3 squared: 16 × 3 = 48. Lands back. 16√3 would square to a much bigger number.
7. A trap first-timers fall into
Stopping at √16 × √3 without taking √16. Or writing 3√4, which is not simplified (√4 is still 2).
7 · 2 Marks · Exponents & surds
√72 simplified
Exponents & surds
Formulas
am/ⁿ=ⁿ√aᵐ
Rational exponent
√a·√b=√ab
Product of surds
Notes: √72. 72=36×2.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: √72 simplified
Before any working — hunt first
Given
Numbers and units already in the sentence: 72 simplified. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: √72 simplified. Circle that word — that is the missing variable.
Find
We are required to find: √72 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 6√2 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives √72 simplified.
In other words — as if you have never seen this before
Pull perfect squares out of the radical.
2. What do we already know?
√72. 72=36×2.
3. Which method — and why this one?
Pull perfect squares out of the radical.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 72 simplified. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: √72 simplified. Circle that word — that is the missing variable.
Find: We are required to find: √72 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 6√2 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives √72 simplified.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
√72. 72=36×2.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Pull perfect squares out of the radical.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Factor
√72=√36×2=6√2.
6√2
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
6√2
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
6√2
Copy onto your answer sheet — in this order
Formula: 6√2
Answer: 6√2
6. Check it like a marker
(6√2)²=36×2=72.
7. A trap first-timers fall into
12√2 would be √144×2=√288, too big.
8 · 2 Marks · Inclination & circles
Distance between (0;0) and (3;4)
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
Notes: A(0;0), B(3;4). Unknown: AB.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Distance between (0;0) and (3;4)
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 · 0 · 3 · 4. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Distance between (0;0) and (3;4). Circle that word — that is the missing variable.
Find
We are required to find: Distance between (0;0) and (3;4). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 5 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Distance between (0;0) and (3;4).
In other words — as if you have never seen this before
Distance formula is Pythagoras. 3-4-5 triangle.
2. What do we already know?
A(0;0), B(3;4). Unknown: AB.
3. Which method — and why this one?
Distance formula is Pythagoras. 3-4-5 triangle.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0 · 0 · 3 · 4. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Distance between (0;0) and (3;4). Circle that word — that is the missing variable.
Find: We are required to find: Distance between (0;0) and (3;4). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 5 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Distance between (0;0) and (3;4).
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
A(0;0), B(3;4). Unknown: AB.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Distance formula is Pythagoras. 3-4-5 triangle.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
√(3²+4²)
√9+16=√25=5.
5
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
5
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
5
Copy onto your answer sheet — in this order
Formula: 5
Answer: 5
6. Check it like a marker
A famous triple. Sketch it.
7. A trap first-timers fall into
3+4=7 adds the legs instead of squaring.
9 · 2 Marks · Dependent events & Venn
P(A or B) for mutually exclusive events
Formulas
P(A ∪ B)=P(A)+P(B)−P(A ∩ B)
Addition rule
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: P(A or B) for mutually exclusive events
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(A or B) for mutually exclusive events. Circle that word — that is the missing variable.
Find
We are required to find: P(A or B) for mutually exclusive events. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: No overlap, so add. Then substitute the given, one line at a time, to get P(A or B) for mutually exclusive events.
In other words — as if you have never seen this before
No overlap, so add.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
No overlap, so add.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(A or B) for mutually exclusive events. Circle that word — that is the missing variable.
Find: We are required to find: P(A or B) for mutually exclusive events. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: No overlap, so add. Then substitute the given, one line at a time, to get P(A or B) for mutually exclusive events.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
No overlap, so add.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
No overlap, so add.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
P(A)+P(B)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
P(A)+P(B)
Copy onto your answer sheet — in this order
Answer: P(A)+P(B)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
10 · 3 Marks · Identities, reduction & equations
cos θ = 0 in [0°; 360°]
Identities, reduction & equations
Formulas
sin²θ+cos²θ=1
Pythagorean identity
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: cos θ = 0 in [0°; 360°]
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 in · 0° · 360°. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: cos θ = 0 in [0°; 360°]. Circle that word — that is the missing variable.
Find
We are required to find: cos θ = 0 in [0°; 360°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 90° and 270°. Then substitute the given, one line at a time, to get cos θ = 0 in [0°; 360°].
In other words — as if you have never seen this before
90° and 270°
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 in · 0° · 360°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
90° and 270°
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0 in · 0° · 360°. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: cos θ = 0 in [0°; 360°]. Circle that word — that is the missing variable.
Find: We are required to find: cos θ = 0 in [0°; 360°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 90° and 270°. Then substitute the given, one line at a time, to get cos θ = 0 in [0°; 360°].
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 0 in · 0° · 360°. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
90° and 270°
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Cosine is the x-coordinate on the unit circle. x = 0 at the top (90°) and the bottom (270°).
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
90° and 270°
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
90° and 270°
Copy onto your answer sheet — in this order
Answer: 90° and 270°
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
11 · 3 Marks · Parabola, hyperbola & exponential
TP of y = −(x+3)² + 1
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: TP of y = −(x+3)² + 1
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 1. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: TP of y = −(x+3)² + 1. Circle that word — that is the missing variable.
Find
We are required to find: TP of y = −(x+3)² + 1. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: (−3; 1). Then substitute the given, one line at a time, to get TP of y = −(x+3)² + 1.
In other words — as if you have never seen this before
(−3; 1)
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
(−3; 1)
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 3 · 1. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: TP of y = −(x+3)² + 1. Circle that word — that is the missing variable.
Find: We are required to find: TP of y = −(x+3)² + 1. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: (−3; 1). Then substitute the given, one line at a time, to get TP of y = −(x+3)² + 1.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
(−3; 1)
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
a(x−(−3))²+1.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
(−3; 1)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
(−3; 1)
Copy onto your answer sheet — in this order
Answer: (−3; 1)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
12 · 4 Marks · Parabola, hyperbola & exponential
Range of y = (x−1)² + 4
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Range of y = (x−1)² + 4
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 4. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Range of y = (x−1)² + 4. Circle that word — that is the missing variable.
Find
We are required to find: Range of y = (x−1)² + 4. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Parabola opens up, minimum 4. Then substitute the given, one line at a time, to get Range of y = (x−1)² + 4.
In other words — as if you have never seen this before
Parabola opens up, minimum 4.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Parabola opens up, minimum 4.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 1 · 4. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Range of y = (x−1)² + 4. Circle that word — that is the missing variable.
Find: We are required to find: Range of y = (x−1)² + 4. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Parabola opens up, minimum 4. Then substitute the given, one line at a time, to get Range of y = (x−1)² + 4.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 1 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Parabola opens up, minimum 4.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Parabola opens up, minimum 4.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
y ≥ 4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
y ≥ 4
Copy onto your answer sheet — in this order
Answer: y ≥ 4
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
13 · 2 Marks
Solve x²=49
Notes: x squared equals 49.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Solve x² = 49
Before any working — hunt first
Given
Numbers and units already in the sentence: 49. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Solve x² = 49. Circle that word — that is the missing variable.
Find
We are required to find: Solve x² = 49. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write x = ±7 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Solve x² = 49.
In other words — as if you have never seen this before
Square root both sides. Remember the ± . 7² and (−7)² both equal 49.
2. What do we already know?
x squared equals 49.
3. Which method — and why this one?
Square root both sides. Remember the ± . 7² and (−7)² both equal 49.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 49. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Solve x² = 49. Circle that word — that is the missing variable.
Find: We are required to find: Solve x² = 49. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write x = ±7 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Solve x² = 49.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
x squared equals 49.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Square root both sides. Remember the ± . 7² and (−7)² both equal 49.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Square root
x=±√49=±7.
x=±7
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
x=±7
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
x=±7
Copy onto your answer sheet — in this order
Formula: x=±7
Answer: x=±7
6. Check it like a marker
7×7=49 and (−7)×(−7)=49.
7. A trap first-timers fall into
x = 7 only loses a mark. The paper wants both roots.
14 · 2 Marks · Inclination & circles
Midpoint of (2; 4) and (6; 8)
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
Notes: A(2;4), B(6;8). Unknown: midpoint M.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Midpoint of (2; 4) and (6; 8)
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 4 · 6 · 8. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Midpoint of (2; 4) and (6; 8). Circle that word — that is the missing variable.
Find
We are required to find: Midpoint of (2; 4) and (6; 8). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write (4; 6) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Midpoint of (2; 4) and (6; 8).
In other words — as if you have never seen this before
Average the x’s, average the y’s.
2. What do we already know?
A(2;4), B(6;8). Unknown: midpoint M.
3. Which method — and why this one?
Average the x’s, average the y’s.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2 · 4 · 6 · 8. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Midpoint of (2; 4) and (6; 8). Circle that word — that is the missing variable.
Find: We are required to find: Midpoint of (2; 4) and (6; 8). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write (4; 6) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Midpoint of (2; 4) and (6; 8).
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
A(2;4), B(6;8). Unknown: midpoint M.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Average the x’s, average the y’s.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Averages
x=(2+6)2=4. y=(4+8)2=6.
(4; 6)
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
(4; 6)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
(4; 6)
Copy onto your answer sheet — in this order
Formula: (4; 6)
Answer: (4; 6)
6. Check it like a marker
M is halfway: from 2 to 6 is +4, half is +2 → 4.
7. A trap first-timers fall into
(8;12) added without dividing by 2.
15 · 2 Marks · Proportion & midpoint theorem
Angle in a semicircle is
Proportion & midpoint theorem
Formulas
DE ∥ BC, AD=DB⇒AE=EC
Midpoint theorem
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Angle in a semicircle is
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Angle in a semicircle is. Circle that word — that is the missing variable.
Find
We are required to find: Angle in a semicircle is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Theorem: angle in a semicircle is a right angle. Then substitute the given, one line at a time, to get Angle in a semicircle is.
In other words — as if you have never seen this before
Theorem: angle in a semicircle is a right angle.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Theorem: angle in a semicircle is a right angle.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Angle in a semicircle is. Circle that word — that is the missing variable.
Find: We are required to find: Angle in a semicircle is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Theorem: angle in a semicircle is a right angle. Then substitute the given, one line at a time, to get Angle in a semicircle is.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Theorem: angle in a semicircle is a right angle.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Theorem: angle in a semicircle is a right angle.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
90°
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
90°
Copy onto your answer sheet — in this order
Answer: 90°
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
16 · 2 Marks · Exponents & surds
√50 simplified is
Exponents & surds
Formulas
am/ⁿ=ⁿ√aᵐ
Rational exponent
√a·√b=√ab
Product of surds
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: √50 simplified is
Before any working — hunt first
Given
Numbers and units already in the sentence: 50 simplified. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: √50 simplified is. Circle that word — that is the missing variable.
Find
We are required to find: √50 simplified is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: √50 = √(25×2) = 5√2. Then substitute the given, one line at a time, to get √50 simplified is.
In other words — as if you have never seen this before
√50=√25×2=5√2.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 50 simplified. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
√50=√25×2=5√2.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 50 simplified. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: √50 simplified is. Circle that word — that is the missing variable.
Find: We are required to find: √50 simplified is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: √50 = √(25×2) = 5√2. Then substitute the given, one line at a time, to get √50 simplified is.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 50 simplified. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
√50=√25×2=5√2.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
√50=√25×2=5√2.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
5√2
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
5√2
Copy onto your answer sheet — in this order
Answer: 5√2
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
17 · 2 Marks · Quadratic equations & nature of roots
If second differences are constant, the pattern is
Formulas
Δ=b²−4ac
Discriminant
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: If second differences are constant, the pattern is
In other words — as if you have never seen this before
Constant second difference ⇒ quadratic.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Constant second difference ⇒ quadratic.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
Constant second difference ⇒ quadratic.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
Constant second difference ⇒ quadratic.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
quadratic
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
quadratic
Copy onto your answer sheet — in this order
Answer: quadratic
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
18 · 3 Marks · Quadratic equations & nature of roots
First differences of a quadratic with a=2. Second difference =
Formulas
Δ=b²−4ac
Discriminant
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: First differences of a quadratic with a=2. Second difference =
Before any working — hunt first
Given
Numbers and units already in the sentence: 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Second difference =. Circle that word — that is the missing variable.
Find
We are required to find: Second difference =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 2a = second difference = 4. Then substitute the given, one line at a time, to get Second difference =.
In other words — as if you have never seen this before
2a = second difference = 4.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
2a = second difference = 4.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Second difference =. Circle that word — that is the missing variable.
Find: We are required to find: Second difference =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 2a = second difference = 4. Then substitute the given, one line at a time, to get Second difference =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 2. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
2a = second difference = 4.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
2a = second difference = 4.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4
Copy onto your answer sheet — in this order
Answer: 4
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
19 · 3 Marks · Nominal vs effective interest
12% p.a. compounded monthly. Monthly interest rate i =
Nominal vs effective interest
Formulas
1+ieff=(1 + inom/m)m
Effective rate
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 12% p.a. compounded monthly. Monthly interest rate i =
Before any working — hunt first
Given
Numbers and units already in the sentence: 12%. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Monthly interest rate i =. Circle that word — that is the missing variable.
Find
We are required to find: Monthly interest rate i =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: i = 0. Then substitute the given, one line at a time, to get Monthly interest rate i =.
In other words — as if you have never seen this before
i=0.1212=0.01.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 12%. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
i=0.1212=0.01.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 12%. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Monthly interest rate i =. Circle that word — that is the missing variable.
Find: We are required to find: Monthly interest rate i =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: i = 0. Then substitute the given, one line at a time, to get Monthly interest rate i =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 12%. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
i=0.1212=0.01.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
i=0.1212=0.01.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
0.01
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
0.01
Copy onto your answer sheet — in this order
Answer: 0.01
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
20 · 3 Marks · Sine, cosine & area rules
In △ABC, A = 30°, a = 5, B = 45°. b =
Formulas
asin A=bsin B=csin C=2R
Sine rule
c²=a²+b²−2ab cos C
Cosine rule
Area=½ ab sin C
Area rule
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: In △ABC, A = 30°, a = 5, B = 45°. b =
Before any working — hunt first
Given
Numbers and units already in the sentence: 30° · 5 · 45°. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: b =. Circle that word — that is the missing variable.
Find
We are required to find: b =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: bsin45=5sin30. Then substitute the given, one line at a time, to get b =.
In other words — as if you have never seen this before
bsin45=5sin30→b=5×(√2/2)(1/2)=5√2.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 30° · 5 · 45°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
bsin45=5sin30→b=5×(√2/2)(1/2)=5√2.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 30° · 5 · 45°. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: b =. Circle that word — that is the missing variable.
Find: We are required to find: b =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: bsin45=5sin30. Then substitute the given, one line at a time, to get b =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 30° · 5 · 45°. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
bsin45=5sin30→b=5×(√2/2)(1/2)=5√2.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
bsin45=5sin30→b=5×(√2/2)(1/2)=5√2.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
5√2
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
5√2
Copy onto your answer sheet — in this order
Answer: 5√2
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
21 · 2 Marks · Nominal vs effective interest
12% p.a. compounded monthly. i per period =
Nominal vs effective interest
Formulas
1+ieff=(1 + inom/m)m
Effective rate
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 12% p.a. compounded monthly. i per period =
Before any working — hunt first
Given
Numbers and units already in the sentence: 12%. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: i per period =. Circle that word — that is the missing variable.
Find
We are required to find: i per period =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 0. Then substitute the given, one line at a time, to get i per period =.
In other words — as if you have never seen this before
0.01
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 12%. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
0.01
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 12%. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: i per period =. Circle that word — that is the missing variable.
Find: We are required to find: i per period =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 0. Then substitute the given, one line at a time, to get i per period =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 12%. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
0.01
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
i = 0.1212=0.01. Nominal rate divided by number of periods per year.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
0.01
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
0.01
Copy onto your answer sheet — in this order
Answer: 0.01
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
22 · 4 Marks · Sine, cosine & area rules
Cosine rule is used when you know
Formulas
asin A=bsin B=csin C=2R
Sine rule
c²=a²+b²−2ab cos C
Cosine rule
Area=½ ab sin C
Area rule
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Cosine rule is used when you know
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Cosine rule is used when you know. Circle that word — that is the missing variable.
Find
We are required to find: Cosine rule is used when you know. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Need two sides and included angle, or three sides. Then substitute the given, one line at a time, to get Cosine rule is used when you know.
In other words — as if you have never seen this before
Need two sides and included angle, or three sides.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Need two sides and included angle, or three sides.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Cosine rule is used when you know. Circle that word — that is the missing variable.
Find: We are required to find: Cosine rule is used when you know. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Need two sides and included angle, or three sides. Then substitute the given, one line at a time, to get Cosine rule is used when you know.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Need two sides and included angle, or three sides.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Need two sides and included angle, or three sides.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
SSS or SAS
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
SSS or SAS
Copy onto your answer sheet — in this order
Answer: SSS or SAS
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
23 · 2 Marks · Quadratic equations & nature of roots
Roots of (x−3)(x+5)=0 are
Formulas
Δ=b²−4ac
Discriminant
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Roots of (x−3)(x+5)=0 are
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 5 · 0 are. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Roots of (x−3)(x+5)=0 are. Circle that word — that is the missing variable.
Find
We are required to find: Roots of (x−3)(x+5)=0 are. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: x−3=0 or x+5=0. Then substitute the given, one line at a time, to get Roots of (x−3)(x+5)=0 are.
In other words — as if you have never seen this before
x−3=0 or x+5=0.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3 · 5 · 0 are. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
x−3=0 or x+5=0.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 3 · 5 · 0 are. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Roots of (x−3)(x+5)=0 are. Circle that word — that is the missing variable.
Find: We are required to find: Roots of (x−3)(x+5)=0 are. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: x−3=0 or x+5=0. Then substitute the given, one line at a time, to get Roots of (x−3)(x+5)=0 are.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 3 · 5 · 0 are. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
x−3=0 or x+5=0.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
x−3=0 or x+5=0.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
3 and −5
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3 and −5
Copy onto your answer sheet — in this order
Answer: 3 and −5
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
24 · 2 Marks · Quadratic equations & nature of roots
Δ for x² − 4x + 4 = 0
Formulas
Δ=b²−4ac
Discriminant
Notes: a = 1, b = −4, c = 4. Unknown: discriminant Δ = b² − 4ac.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Δ for x² − 4x + 4 = 0
Before any working — hunt first
Given
Numbers and units already in the sentence: 4x · 4 · 0. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Δ for x² − 4x + 4 = 0. Circle that word — that is the missing variable.
Find
We are required to find: Δ for x² − 4x + 4 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write Δ = b² − 4ac with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Δ for x² − 4x + 4 = 0.
In other words — as if you have never seen this before
Δ tells the nature of roots. Zero means one real rational root (a repeated root) — a perfect square.
2. What do we already know?
a = 1, b = −4, c = 4. Unknown: discriminant Δ = b² − 4ac.
3. Which method — and why this one?
Δ tells the nature of roots. Zero means one real rational root (a repeated root) — a perfect square.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 4x · 4 · 0. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Δ for x² − 4x + 4 = 0. Circle that word — that is the missing variable.
Find: We are required to find: Δ for x² − 4x + 4 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write Δ = b² − 4ac with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Δ for x² − 4x + 4 = 0.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
a = 1, b = −4, c = 4. Unknown: discriminant Δ = b² − 4ac.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Δ tells the nature of roots. Zero means one real rational root (a repeated root) — a perfect square.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Write Δ
Δ=b²−4ac=(−4)²−4(1)(4)=16−16=0.
Δ=b²−4ac
Why this step: The minus on b is squared away. (−4)² is +16, not −16.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
0
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
0
Copy onto your answer sheet — in this order
Formula: Δ=b²−4ac
Answer: 0
6. Check it like a marker
x² − 4x + 4 = (x − 2)². A repeated root x = 2, which is what Δ = 0 promised.
7. A trap first-timers fall into
Writing 16 (that is b², you forgot to subtract 4ac) or −16 (you squared −4 into −16).
25 · 2 Marks · Quadratic equations & nature of roots
Quadratic pattern. Second difference constantly 6. a =
Formulas
Δ=b²−4ac
Discriminant
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Quadratic pattern. Second difference constantly 6. a =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: a =. Circle that word — that is the missing variable.
Find
We are required to find: a =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 2a = second difference, a = 3. Then substitute the given, one line at a time, to get a =.
In other words — as if you have never seen this before
2a = second difference, a = 3.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
2a = second difference, a = 3.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 6. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: a =. Circle that word — that is the missing variable.
Find: We are required to find: a =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 2a = second difference, a = 3. Then substitute the given, one line at a time, to get a =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 6. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
2a = second difference, a = 3.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
2a = second difference, a = 3.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3
Copy onto your answer sheet — in this order
Answer: 3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
26 · 3 Marks · Inclination & circles
y=(x−1)²+2. Turning point?
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: y = (x−1)² + 2. Turning point?
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Turning point. Circle that word — that is the missing variable.
Find
We are required to find: Turning point. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: (1; 2). Then substitute the given, one line at a time, to get Turning point.
In other words — as if you have never seen this before
(1; 2)
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
(1; 2)
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 1 · 2. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Turning point. Circle that word — that is the missing variable.
Find: We are required to find: Turning point. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: (1; 2). Then substitute the given, one line at a time, to get Turning point.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 1 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
(1; 2)
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
TP (p; q) with opposite sign on p.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
(1; 2)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
(1; 2)
Copy onto your answer sheet — in this order
Answer: (1; 2)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
27 · 2 Marks · Quadratic equations & nature of roots
x² − 1 < 0 has solution
Formulas
Δ=b²−4ac
Discriminant
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: x² − 1 < 0 has solution
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 0 has. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: x² − 1 < 0 has solution. Circle that word — that is the missing variable.
Find
We are required to find: x² − 1 < 0 has solution. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Negative between the roots of (x−1)(x+1). Then substitute the given, one line at a time, to get x² − 1 < 0 has solution.
In other words — as if you have never seen this before
Negative between the roots of (x−1)(x+1).
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 0 has. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Negative between the roots of (x−1)(x+1).
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 1 · 0 has. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: x² − 1 < 0 has solution. Circle that word — that is the missing variable.
Find: We are required to find: x² − 1 < 0 has solution. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Negative between the roots of (x−1)(x+1). Then substitute the given, one line at a time, to get x² − 1 < 0 has solution.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 1 · 0 has. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Negative between the roots of (x−1)(x+1).
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Negative between the roots of (x−1)(x+1).
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
−1 < x < 1
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
−1 < x < 1
Copy onto your answer sheet — in this order
Answer: −1 < x < 1
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
28 · 2 Marks · Parabola, hyperbola & exponential
Asymptotes of y = 1(x−2) + 3
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Asymptotes of y = 1(x−2) + 3
Before any working — hunt first
Given
Numbers and units already in the sentence: 1/ · 2 · 3. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Asymptotes of y = 1(x−2) + 3. Circle that word — that is the missing variable.
Find
We are required to find: Asymptotes of y = 1(x−2) + 3. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Vertical x=−p=2, horizontal y=q=3. Then substitute the given, one line at a time, to get Asymptotes of y = 1(x−2) + 3.
In other words — as if you have never seen this before
Vertical x=−p=2, horizontal y=q=3.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1/ · 2 · 3. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Vertical x=−p=2, horizontal y=q=3.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 1/ · 2 · 3. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Asymptotes of y = 1(x−2) + 3. Circle that word — that is the missing variable.
Find: We are required to find: Asymptotes of y = 1(x−2) + 3. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Vertical x=−p=2, horizontal y=q=3. Then substitute the given, one line at a time, to get Asymptotes of y = 1(x−2) + 3.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 1/ · 2 · 3. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Vertical x=−p=2, horizontal y=q=3.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Vertical x=−p=2, horizontal y=q=3.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
x=2 and y=3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
x=2 and y=3
Copy onto your answer sheet — in this order
Answer: x=2 and y=3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Independent P(A)=0.4, P(B)=0.5. P(A and B) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.4 · 0.5. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(A and B) =. Circle that word — that is the missing variable.
Find
We are required to find: P(A and B) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write P(A ∩ B) = P(A)P(B) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(A and B) =.
In other words — as if you have never seen this before
Independent means “one does not change the other.” Then P(A and B) = P(A) × P(B). Adding is “or” for mutually exclusive events.
Independent means “one does not change the other.” Then P(A and B) = P(A) × P(B). Adding is “or” for mutually exclusive events.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0.4 · 0.5. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(A and B) =. Circle that word — that is the missing variable.
Find: We are required to find: P(A and B) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write P(A ∩ B) = P(A)P(B) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(A and B) =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Independent means “one does not change the other.” Then P(A and B) = P(A) × P(B). Adding is “or” for mutually exclusive events.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Multiply
P(A ∩ B)=0.4×0.5=0.20.
P(A ∩ B)=P(A)P(B)
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
0.20
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
0.20
Copy onto your answer sheet — in this order
Formula: P(A ∩ B)=P(A)P(B)
Answer: 0.20
6. Check it like a marker
0.20 is smaller than both 0.4 and 0.5, as an “and” must be. 0.90 would be the “or” if they were mutually exclusive — they are not, and that was not asked.
7. A trap first-timers fall into
Adding 0.4+0.5=0.90. That is P(A or B) for mutually exclusive events, a different sentence.
30 · 2 Marks · Quadratic equations & nature of roots
Nature of roots of x² + 6x + 9 = 0
Formulas
Δ=b²−4ac
Discriminant
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Nature of roots of x² + 6x + 9 = 0
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x · 9 · 0. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Nature of roots of x² + 6x + 9 = 0. Circle that word — that is the missing variable.
Find
We are required to find: Nature of roots of x² + 6x + 9 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: real, rational, equal. Then substitute the given, one line at a time, to get Nature of roots of x² + 6x + 9 = 0.
In other words — as if you have never seen this before
real, rational, equal
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6x · 9 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
real, rational, equal
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 6x · 9 · 0. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Nature of roots of x² + 6x + 9 = 0. Circle that word — that is the missing variable.
Find: We are required to find: Nature of roots of x² + 6x + 9 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: real, rational, equal. Then substitute the given, one line at a time, to get Nature of roots of x² + 6x + 9 = 0.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 6x · 9 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
real, rational, equal
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Δ = 36 − 36 = 0. Equal real rational roots; it is a perfect square (x+3)².
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
real, rational, equal
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
real, rational, equal
Copy onto your answer sheet — in this order
Answer: real, rational, equal
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”