2025 · trial · Mathematics · Grade 12

Grade 12 Mathematics P1 — Western Cape trial style

Notes

Answer all questions. Show a formula, substitution and a boxed answer with units. g = 9.8 m·s⁻². Calculators allowed.

90 min · 122 Marks · 30 questions

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Question breakdown

1.1 · 10 Marks
ddx (x⁴)=
Notes: f(x)=x⁴. Power 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: ddx (x⁴) =
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: ddx (x⁴) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (x⁴) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write ddx xⁿ = n xⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ddx (x⁴) =.
In other words — as if you have never seen this before
n xⁿ⁻¹. 4x³.
2. What do we already know?
f(x)=x⁴. Power rule.
3. Which method — and why this one?
n xⁿ⁻¹. 4x³.
4. Step-by-step working
1. 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.
2. 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: ddx (x⁴) =. Circle that word — that is the missing variable. Find: We are required to find: ddx (x⁴) =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: Write ddx xⁿ = n xⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ddx (x⁴) =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  f(x)=x⁴. Power rule.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  n xⁿ⁻¹. 4x³.
  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.
5. 05 / 08
  Power rule
  Bring 4 down, drop the power to 3.
  ddx xⁿ=n xⁿ⁻¹
  Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  4x³
  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.
8. 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
4x³
Copy onto your answer sheet — in this order
Formula: ddx xⁿ=n xⁿ⁻¹
Answer: 4x³
6. Check it like a marker
Integral of 4x³ is x⁴ + C, inverse check.
7. A trap first-timers fall into
x⁵5 is the integral, not the derivative.
1.2 · 12 Marks · Arithmetic & geometric series
Geometric a=5, r=2. T₄=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
Notes: First term 5, common ratio 2. Fourth term.
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: Geometric a=5, r=2. T₄ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5 · 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: T₄ =. Circle that word — that is the missing variable.
Find
We are required to find: T₄ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write Tₙ = a rⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives T₄ =.
In other words — as if you have never seen this before
Tₙ=a rⁿ⁻¹. T₄=5×2³=5×8=40.
2. What do we already know?
First term 5, common ratio 2. Fourth term.
3. Which method — and why this one?
Tₙ=a rⁿ⁻¹. T₄=5×2³=5×8=40.
4. Step-by-step working
1. 01 / 05
  Write Tₙ = arⁿ⁻¹
  The exponent is n minus 1, not n.
  Tₙ=a rⁿ⁻¹
  Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
2. 02 / 05
  Substitute
  T₄=5×2³=40.
  T₄=40
  Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
3. 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.
4. 04 / 05
  Step F — Box the answer with its unit
  40
  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.
5. 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
40
Copy onto your answer sheet — in this order
Formula: Tₙ=a rⁿ⁻¹
Answer: 40
6. Check it like a marker
Sequence: 5, 10, 20, 40. Fourth term 40.
7. A trap first-timers fall into
2⁴ = 16 would give T₅ if you use rⁿ instead of rⁿ⁻¹.
2.1 · 14 Marks
log_a a=
Notes: Logarithm of the base to itself.
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: log_a a =
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: log_a a =. Circle that word — that is the missing variable.
Find
We are required to find: log_a a =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write a¹ = a with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log_a a =.
In other words — as if you have never seen this before
log_a a=1 because a¹=a.
2. What do we already know?
Logarithm of the base to itself.
3. Which method — and why this one?
log_a a=1 because a¹=a.
4. Step-by-step working
1. 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.
2. 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: log_a a =. Circle that word — that is the missing variable. Find: We are required to find: log_a a =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: Write a¹ = a with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log_a 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).
3. 03 / 08
  Step B — Write the given in a column
  Logarithm of the base to itself.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  log_a a=1 because a¹=a.
  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.
5. 05 / 08
  Convert
  log_a a=x⇔aˣ=a⇔x=1.
  a¹=a
  Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  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.
8. 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
Copy onto your answer sheet — in this order
Formula: a¹=a
Answer: 1
6. Check it like a marker
log_a 1 = 0, different identity. log_a a = 1.
7. A trap first-timers fall into
0 is log of 1, not log of a.
2.2 · 14 Marks · Cubic functions & graphs
A cubic can have at most how many x-intercepts?
Formulas
f'(x)=0⇒stationary points
Turning / stationary
f''(x) > 0⇒local min
Second-derivative test
Notes: Degree 3 polynomial. x-intercepts are real roots.
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: A cubic can have at most how many x-intercepts?
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: how many x-intercepts. Circle that word — that is the missing variable.
Find
We are required to find: how many x-intercepts. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write at most 3 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives how many x-intercepts.
In other words — as if you have never seen this before
A degree n polynomial has at most n real roots.
2. What do we already know?
Degree 3 polynomial. x-intercepts are real roots.
3. Which method — and why this one?
A degree n polynomial has at most n real roots.
4. Step-by-step working
1. 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.
2. 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: how many x-intercepts. Circle that word — that is the missing variable. Find: We are required to find: how many x-intercepts. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: Write at most 3 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives how many x-intercepts.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Degree 3 polynomial. x-intercepts are real roots.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  A degree n polynomial has at most n real roots.
  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.
5. 05 / 08
  Use the degree
  Cubic → at most 3 real roots → at most 3 x-intercepts.
  at most 3
  Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
6. 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.
7. 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.
8. 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
Formula: at most 3
Answer: 3
6. Check it like a marker
f(x)=x³ − x = x(x−1)(x+1) has three. f(x)=x³ + 1 has one. Maximum is three.
7. A trap first-timers fall into
It can have 1 or 3 real roots (and 2 with a turning-point root), but never 4.
5 · 3 Marks · Remainder & factor theorem
Remainder when x³ − 2x + 1 is divided by (x − 1) is
Formulas
f(a)=0⇔(x − a) | f(x)
Factor 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: Remainder when x³ − 2x + 1 is divided by (x − 1) is
In other words — as if you have never seen this before
Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2x · 1 is · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
4. Step-by-step working
1. 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.
2. 02 / 07
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 2x · 1 is · 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.
3. 03 / 07
  Step C — Name the method in words
  Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
  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.
4. 04 / 07
  Working
  Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
  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.
5. 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.
6. 06 / 07
  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.
7. 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
0
Copy onto your answer sheet — in this order
Answer: 0
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
∫ 6x² dx=
Notes: Indefinite integral of 6x².
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: ∫ 6x² dx =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x². 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: ∫ 6x² dx =. Circle that word — that is the missing variable.
Find
We are required to find: ∫ 6x² dx =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 2x³ + C with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ∫ 6x² dx =.
In other words — as if you have never seen this before
Raise the index by 1, divide by the new index. Add C.
2. What do we already know?
Indefinite integral of 6x².
3. Which method — and why this one?
Raise the index by 1, divide by the new index. Add C.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 6x². 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: ∫ 6x² dx =. Circle that word — that is the missing variable. Find: We are required to find: ∫ 6x² dx =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: Write 2x³ + C with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ∫ 6x² dx =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Indefinite integral of 6x².
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  Raise the index by 1, divide by the new index. Add C.
  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.
5. 05 / 08
  Power integrate
  6×x³3+C=2x³+C.
  2x³+C
  Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  2x³ + C
  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.
8. 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
2x³ + C
Copy onto your answer sheet — in this order
Formula: 2x³ + C
Answer: 2x³ + C
6. Check it like a marker
ddx (2x³) = 6x². Recovers the integrand.
7. A trap first-timers fall into
12x is the derivative of 6x², not the integral.
7 · 3 Marks
S_∞ of 8 + 4 + 2 + … =
Notes: a = 8, next is 4, then 2. Geometric with r = 12.
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: S_∞ of 8 + 4 + 2 + … =
Before any working — hunt first
Given
Numbers and units already in the sentence: 8 · 4 · 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: S_∞ of 8 + 4 + 2 + … =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 8 + 4 + 2 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write r = 12 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives S_∞ of 8 + 4 + 2 + … =.
In other words — as if you have never seen this before
|r|<1 so S_∞=a(1−r).
2. What do we already know?
a = 8, next is 4, then 2. Geometric with r = 12.
3. Which method — and why this one?
|r|<1 so S_∞=a(1−r).
4. Step-by-step working
1. 01 / 05
  Find r
  r=48=1/2. |12|<1, converges.
  r=12
  Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
2. 02 / 05
  Substitute
  S_∞=8(1−1/2)=8(1/2)=16.
  S_∞=16
  Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
3. 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.
4. 04 / 05
  Step F — Box the answer with its unit
  16
  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.
5. 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
16
Copy onto your answer sheet — in this order
Formula: r=12
Answer: 16
6. Check it like a marker
Partial sums 8, 12, 14, 15, 15.5… climbing to 16.
7. A trap first-timers fall into
Do not use the arithmetic sum formula. Do not write 8(1+1/2).
8 · 3 Marks
S_∞ of 9 + 3 + 1 + … =
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: S_∞ of 9 + 3 + 1 + … =
Before any working — hunt first
Given
Numbers and units already in the sentence: 9 · 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: S_∞ of 9 + 3 + 1 + … =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 9 + 3 + 1 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: a=9, r=13. Then substitute the given, one line at a time, to get S_∞ of 9 + 3 + 1 + … =.
In other words — as if you have never seen this before
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 9 · 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 9 · 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: S_∞ of 9 + 3 + 1 + … =. Circle that word — that is the missing variable. Find: We are required to find: S_∞ of 9 + 3 + 1 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: a=9, r=13. Then substitute the given, one line at a time, to get S_∞ of 9 + 3 + 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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 9 · 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.
4. 04 / 08
  Step C — Name the method in words
  a=9, r=13. S=a(1−r)=9(2/3)=13.5.
  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.
5. 05 / 08
  Working
  a=9, r=13. S=a(1−r)=9(2/3)=13.5.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  13.5
  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.
8. 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
13.5
Copy onto your answer sheet — in this order
Answer: 13.5
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…”
9 · 4 Marks
f'(x)=0 for f(x)=x² − 6x + 5 at
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: f'(x)=0 for f(x)=x² − 6x + 5 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 for · 6x · 5 at. 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: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. 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. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
In other words — as if you have never seen this before
x=3
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 for · 6x · 5 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
x=3
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 0 for · 6x · 5 at. 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: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable. Find: We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. 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. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 0 for · 6x · 5 at. 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.
4. 04 / 08
  Step C — Name the method in words
  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.
5. 05 / 08
  Reason
  f'=2x−6=0→x=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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  x=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.
8. 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=3
Copy onto your answer sheet — in this order
Answer: x=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…”
10 · 2 Marks
ddx (4x³ − 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: ddx (4x³ − x) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 4x³. 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: ddx (4x³ − x) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (4x³ − x) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 12x² − 1. Then substitute the given, one line at a time, to get ddx (4x³ − x) =.
In other words — as if you have never seen this before
12x² − 1
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 4x³. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
12x² − 1
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 4x³. 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: ddx (4x³ − x) =. Circle that word — that is the missing variable. Find: We are required to find: ddx (4x³ − x) =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 12x² − 1. Then substitute the given, one line at a time, to get ddx (4x³ − x) =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 4x³. 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.
4. 04 / 08
  Step C — Name the method in words
  12x² − 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.
5. 05 / 08
  Reason
  Power rule on each term.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  12x² − 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.
8. 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
12x² − 1
Copy onto your answer sheet — in this order
Answer: 12x² − 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…”
11 · 2 Marks
log₁₀ 1000=
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: log₁₀ 1000 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 1000. 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: log₁₀ 1000 =. Circle that word — that is the missing variable.
Find
We are required to find: log₁₀ 1000 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 10³ = 1000. Then substitute the given, one line at a time, to get log₁₀ 1000 =.
In other words — as if you have never seen this before
10³=1000.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1000. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
10³=1000.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 1000. 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: log₁₀ 1000 =. Circle that word — that is the missing variable. Find: We are required to find: log₁₀ 1000 =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 10³ = 1000. Then substitute the given, one line at a time, to get log₁₀ 1000 =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 1000. 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.
4. 04 / 08
  Step C — Name the method in words
  10³=1000.
  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.
5. 05 / 08
  Working
  10³=1000.
  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.
6. 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.
7. 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.
8. 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…”
12 · 2 Marks · Compound angles
sin 15° in surd form uses
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
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: sin 15° in surd form uses
Before any working — hunt first
Given
Numbers and units already in the sentence: 15°. 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: sin 15° in surd form uses. Circle that word — that is the missing variable.
Find
We are required to find: sin 15° in surd form uses. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: sin(45−30). Then substitute the given, one line at a time, to get sin 15° in surd form uses.
In other words — as if you have never seen this before
sin(45−30)
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 15°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
sin(45−30)
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 15°. 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: sin 15° in surd form uses. Circle that word — that is the missing variable. Find: We are required to find: sin 15° in surd form uses. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: sin(45−30). Then substitute the given, one line at a time, to get sin 15° in surd form uses.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 15°. 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.
4. 04 / 08
  Step C — Name the method in words
  sin(45−30)
  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.
5. 05 / 08
  Reason
  Compound 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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  sin(45−30)
  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.
8. 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
sin(45−30)
Copy onto your answer sheet — in this order
Answer: sin(45−30)
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 · 4 Marks
Stationary point of f(x)=x³−3x+1 is at 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: Stationary point of f(x)=x³−3x+1 is at x =
In other words — as if you have never seen this before
f'=3x²−3=0→x²=1→x=±1.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3x · 1 is. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
f'=3x²−3=0→x²=1→x=±1.
4. Step-by-step working
1. 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.
2. 02 / 07
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 3x · 1 is. 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.
3. 03 / 07
  Step C — Name the method in words
  f'=3x²−3=0→x²=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.
4. 04 / 07
  Working
  f'=3x²−3=0→x²=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.
5. 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.
6. 06 / 07
  Step F — Box the answer with its unit
  ±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.
7. 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
±1
Copy onto your answer sheet — in this order
Answer: ±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…”
14 · 4 Marks
Predicting y outside the given x-range is
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: Predicting y outside the given x-range is
In other words — as if you have never seen this before
Outside the data cloud.
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?
Outside the data cloud.
4. Step-by-step working
1. 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.
2. 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.
3. 03 / 07
  Step C — Name the method in words
  Outside the data cloud.
  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.
4. 04 / 07
  Working
  Outside the data cloud.
  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.
5. 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.
6. 06 / 07
  Step F — Box the answer with its unit
  extrapolation
  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.
7. 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
extrapolation
Copy onto your answer sheet — in this order
Answer: extrapolation
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…”
15 · 2 Marks · Arithmetic & geometric series
Arithmetic: a=3, d=4. T₅=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
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: Arithmetic: a=3, d=4. T₅ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 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: T₅ =. Circle that word — that is the missing variable.
Find
We are required to find: T₅ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: T₅ = 3 + 4×4 = 19. Then substitute the given, one line at a time, to get T₅ =.
In other words — as if you have never seen this before
T₅=3+4×4=19.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
T₅=3+4×4=19.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 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: T₅ =. Circle that word — that is the missing variable. Find: We are required to find: T₅ =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: T₅ = 3 + 4×4 = 19. Then substitute the given, one line at a time, to get T₅ =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 3 · 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.
4. 04 / 08
  Step C — Name the method in words
  T₅=3+4×4=19.
  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.
5. 05 / 08
  Working
  T₅=3+4×4=19.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  19
  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.
8. 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
19
Copy onto your answer sheet — in this order
Answer: 19
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 · Arithmetic & geometric series
Geometric a=8, r=12. S_∞=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
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: Geometric a = 8, r = 12. S_∞ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 8 · 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: S_∞ =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 16. Then substitute the given, one line at a time, to get S_∞ =.
In other words — as if you have never seen this before
16
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 8 · 1/ · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
16
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 8 · 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: S_∞ =. Circle that word — that is the missing variable. Find: We are required to find: S_∞ =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 16. Then substitute the given, one line at a time, to get S_∞ =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 8 · 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.
4. 04 / 08
  Step C — Name the method in words
  16
  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.
5. 05 / 08
  Working
  |r| < 1 so it converges. S_∞ = a(1 − r) = 8(1/2) = 16. Do not add the first two terms and stop.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  16
  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.
8. 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
16
Copy onto your answer sheet — in this order
Answer: 16
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 · 3 Marks
r=−0.95 means
Notes: Pearson’s r = −0.95 for a bivariate data set.
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: r = −0.95 means
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.95 means. 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: 95 means. Circle that word — that is the missing variable.
Find
We are required to find: 95 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write |r| = 0.95 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 95 means.
In other words — as if you have never seen this before
r near −1: strong negative linear. r does not prove causation.
2. What do we already know?
Pearson’s r = −0.95 for a bivariate data set.
3. Which method — and why this one?
r near −1: strong negative linear. r does not prove causation.
4. Step-by-step working
1. 01 / 05
  Read the sign
  Negative: as x increases, y tends to decrease.
  Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
2. 02 / 05
  Read the size
  |r|=0.95 is close to 1, so the linear pattern is strong.
  |r|=0.95
  Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
3. 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.
4. 04 / 05
  Step F — Box the answer with its unit
  strong negative linear correlation
  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.
5. 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
strong negative linear correlation
Copy onto your answer sheet — in this order
Formula: |r|=0.95
Answer: strong negative linear correlation
6. Check it like a marker
Scatter would slope down, tightly around a line.
7. A trap first-timers fall into
Correlation is not causation. Do not say “x causes y to fall”.
18 · 3 Marks
Angle of elevation is measured from the
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 of elevation is measured from the
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 of elevation is measured from the. Circle that word — that is the missing variable.
Find
We are required to find: Angle of elevation is measured from the. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Elevation from horizontal. Then substitute the given, one line at a time, to get Angle of elevation is measured from the.
In other words — as if you have never seen this before
Elevation from horizontal.
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?
Elevation from horizontal.
4. Step-by-step working
1. 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.
2. 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 of elevation is measured from the. Circle that word — that is the missing variable. Find: We are required to find: Angle of elevation is measured from the. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Elevation from horizontal. Then substitute the given, one line at a time, to get Angle of elevation is measured from the.
  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).
3. 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.
4. 04 / 08
  Step C — Name the method in words
  Elevation from horizontal.
  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.
5. 05 / 08
  Working
  Elevation from horizontal.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  horizontal up to the line of sight
  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.
8. 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
horizontal up to the line of sight
Copy onto your answer sheet — in this order
Answer: horizontal up to the line of sight
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 · 4 Marks
Solve 2ˣ=8
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 2ˣ = 8
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 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: Solve 2ˣ = 8. Circle that word — that is the missing variable.
Find
We are required to find: Solve 2ˣ = 8. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 8=2³. Then substitute the given, one line at a time, to get Solve 2ˣ = 8.
In other words — as if you have never seen this before
8=2³.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 8. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
8=2³.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 2 · 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: Solve 2ˣ = 8. Circle that word — that is the missing variable. Find: We are required to find: Solve 2ˣ = 8. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 8=2³. Then substitute the given, one line at a time, to get Solve 2ˣ = 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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 2 · 8. 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.
4. 04 / 08
  Step C — Name the method in words
  8=2³.
  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.
5. 05 / 08
  Working
  8=2³.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  x=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.
8. 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=3
Copy onto your answer sheet — in this order
Answer: x=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…”
20 · 3 Marks
S_∞ of 2 + 4 + 8 + 16 + … is
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: S_∞ of 2 + 4 + 8 + 16 + … is
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 4 · 8 · 16. 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: S_∞ of 2 + 4 + 8 + 16 + … is. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 2 + 4 + 8 + 16 + … is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: r = 2, |r| ≥ 1. Then substitute the given, one line at a time, to get S_∞ of 2 + 4 + 8 + 16 + … is.
In other words — as if you have never seen this before
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 4 · 8 · 16. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 2 · 4 · 8 · 16. 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: S_∞ of 2 + 4 + 8 + 16 + … is. Circle that word — that is the missing variable. Find: We are required to find: S_∞ of 2 + 4 + 8 + 16 + … is. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: r = 2, |r| ≥ 1. Then substitute the given, one line at a time, to get S_∞ of 2 + 4 + 8 + 16 + … 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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 2 · 4 · 8 · 16. 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.
4. 04 / 08
  Step C — Name the method in words
  r=2, |r| ≥ 1. Do not write 2(1−2)=−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.
5. 05 / 08
  Working
  r=2, |r| ≥ 1. Do not write 2(1−2)=−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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  does not exist (diverges)
  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.
8. 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
does not exist (diverges)
Copy onto your answer sheet — in this order
Answer: does not exist (diverges)
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
Stationary points of f(x) = x³ − 12x + 1 at 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: Stationary points of f(x) = x³ − 12x + 1 at x =
In other words — as if you have never seen this before
±2
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 12x · 1 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
±2
4. Step-by-step working
1. 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.
2. 02 / 07
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 12x · 1 at. 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.
3. 03 / 07
  Step C — Name the method in words
  ±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.
4. 04 / 07
  Working
  f'=3x²−12=0→x²=4→x=±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.
5. 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.
6. 06 / 07
  Step F — Box the answer with its unit
  ±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.
7. 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
±2
Copy onto your answer sheet — in this order
Answer: ±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…”
22 · 2 Marks
If f(x) = x + 4, f⁻¹(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: If f(x) = x + 4, f⁻¹(x) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 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: If f(x) = x + 4, f⁻¹(x) =. Circle that word — that is the missing variable.
Find
We are required to find: If f(x) = x + 4, f⁻¹(x) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Swap and subtract 4. Then substitute the given, one line at a time, to get If f(x) = x + 4, f⁻¹(x) =.
In other words — as if you have never seen this before
Swap and subtract 4.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Swap and subtract 4.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 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: If f(x) = x + 4, f⁻¹(x) =. Circle that word — that is the missing variable. Find: We are required to find: If f(x) = x + 4, f⁻¹(x) =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Swap and subtract 4. Then substitute the given, one line at a time, to get If f(x) = x + 4, f⁻¹(x) =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 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.
4. 04 / 08
  Step C — Name the method in words
  Swap and subtract 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.
5. 05 / 08
  Working
  Swap and subtract 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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  x − 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.
8. 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 − 4
Copy onto your answer sheet — in this order
Answer: x − 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…”
23 · 3 Marks · Compound angles
tan θ = m. If m is undefined the line is
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
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: tan θ = m. If m is undefined the line 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: If m is undefined the line is. Circle that word — that is the missing variable.
Find
We are required to find: If m is undefined the line is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: vertical. Then substitute the given, one line at a time, to get If m is undefined the line is.
In other words — as if you have never seen this before
vertical
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?
vertical
4. Step-by-step working
1. 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.
2. 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: If m is undefined the line is. Circle that word — that is the missing variable. Find: We are required to find: If m is undefined the line is. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: vertical. Then substitute the given, one line at a time, to get If m is undefined the line 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).
3. 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.
4. 04 / 08
  Step C — Name the method in words
  vertical
  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.
5. 05 / 08
  Reason
  No defined gradient.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  vertical
  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.
8. 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
vertical
Copy onto your answer sheet — in this order
Answer: vertical
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
5P2=
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: 5P2 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5P · 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: 5P2 =. Circle that word — that is the missing variable.
Find
We are required to find: 5P2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5×4 = 20. Then substitute the given, one line at a time, to get 5P2 =.
In other words — as if you have never seen this before
5×4=20.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5P · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5×4=20.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 5P · 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: 5P2 =. Circle that word — that is the missing variable. Find: We are required to find: 5P2 =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 5×4 = 20. Then substitute the given, one line at a time, to get 5P2 =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 5P · 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.
4. 04 / 08
  Step C — Name the method in words
  5×4=20.
  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.
5. 05 / 08
  Working
  5×4=20.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  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.
8. 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
20
Copy onto your answer sheet — in this order
Answer: 20
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…”
25 · 4 Marks
log₅ 125=
Notes: log₅ 125. In words: “5 to the what equals 125?”
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: log₅ 125 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 125. 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: log₅ 125 =. Circle that word — that is the missing variable.
Find
We are required to find: log₅ 125 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 5ˣ = 125 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log₅ 125 =.
In other words — as if you have never seen this before
A logarithm is an index. Convert to 5ˣ = 125, then notice 5³ = 125.
2. What do we already know?
log₅ 125. In words: “5 to the what equals 125?”
3. Which method — and why this one?
A logarithm is an index. Convert to 5ˣ = 125, then notice 5³ = 125.
4. Step-by-step working
1. 01 / 05
  Rewrite as an exponential
  log₅ 125=x⇔5ˣ=125.
  5ˣ=125
  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.
2. 02 / 05
  Recognise the power
  5¹=5, 5²=25, 5³=125. So x=3.
  x=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.
3. 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.
4. 04 / 05
  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.
5. 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
3
Copy onto your answer sheet — in this order
Formula: 5ˣ=125
Answer: 3
6. Check it like a marker
5³=5×5×5=125. Yes.
7. A trap first-timers fall into
Writing 25 (that is 1255) or 5 (the base). A log is not a divide.
26 · 3 Marks
log₁₀ 100=
Notes: Base 10. Argument 100. Unknown: the exponent.
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: log₁₀ 100 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 100. 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: log₁₀ 100 =. Circle that word — that is the missing variable.
Find
We are required to find: log₁₀ 100 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 10ˣ = 100 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log₁₀ 100 =.
In other words — as if you have never seen this before
log₁₀ 100 is the power you raise 10 to, to get 100. 10² = 100.
2. What do we already know?
Base 10. Argument 100. Unknown: the exponent.
3. Which method — and why this one?
log₁₀ 100 is the power you raise 10 to, to get 100. 10² = 100.
4. Step-by-step working
1. 01 / 05
  Rewrite as an exponential
  log₁₀ 100=x⇔10ˣ=100.
  10ˣ=100
  Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
2. 02 / 05
  Recognise the power
  100=10², so x=2.
  x=2
  Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
3. 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.
4. 04 / 05
  Step F — Box the answer with its unit
  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.
5. 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
2
Copy onto your answer sheet — in this order
Formula: 10ˣ=100
Answer: 2
6. Check it like a marker
10²=100. Yes.
7. A trap first-timers fall into
log₁₀ 100 is not 10, and not 1. Count the zeros of a power of ten: 100 has two zeros.
27 · 3 Marks · Compound angles
ddx not on Paper 2. Which topic is Paper 2?
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
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: ddx not on Paper 2. Which topic is Paper 2?
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 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: Which topic is Paper 2. Circle that word — that is the missing variable.
Find
We are required to find: Which topic is Paper 2. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Trigonometry. Then substitute the given, one line at a time, to get Which topic is Paper 2.
In other words — as if you have never seen this before
Trigonometry
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Trigonometry
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 2 · 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: Which topic is Paper 2. Circle that word — that is the missing variable. Find: We are required to find: Which topic is Paper 2. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Trigonometry. Then substitute the given, one line at a time, to get Which topic is Paper 2.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 2 · 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.
4. 04 / 08
  Step C — Name the method in words
  Trigonometry
  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.
5. 05 / 08
  Reason
  Calculus and finance sit on Paper 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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  Trigonometry
  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.
8. 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
Trigonometry
Copy onto your answer sheet — in this order
Answer: Trigonometry
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 · 3 Marks
log₂ 32=
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: log₂ 32 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 32. 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: log₂ 32 =. Circle that word — that is the missing variable.
Find
We are required to find: log₂ 32 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5. Then substitute the given, one line at a time, to get log₂ 32 =.
In other words — as if you have never seen this before
5
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 32. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 32. 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: log₂ 32 =. Circle that word — that is the missing variable. Find: We are required to find: log₂ 32 =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 5. Then substitute the given, one line at a time, to get log₂ 32 =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 32. 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.
4. 04 / 08
  Step C — Name the method in words
  5
  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.
5. 05 / 08
  Reason
  2⁵=32.
  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.
6. 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.
7. 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.
8. 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
Answer: 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…”
29 · 2 Marks
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: 5! =
Before any working — hunt first
Given
Numbers and units already in the sentence: 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: =. Circle that word — that is the missing variable.
Find
We are required to find: =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5×4×3×2×1=120. Then substitute the given, one line at a time, to get =.
In other words — as if you have never seen this before
5×4×3×2×1=120.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5×4×3×2×1=120.
4. Step-by-step working
1. 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.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 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: =. Circle that word — that is the missing variable. Find: We are required to find: =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 5×4×3×2×1=120. Then substitute the given, one line at a time, to get =.
  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).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 5. 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.
4. 04 / 08
  Step C — Name the method in words
  5×4×3×2×1=120.
  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.
5. 05 / 08
  Working
  5×4×3×2×1=120.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  120
  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.
8. 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
120
Copy onto your answer sheet — in this order
Answer: 120
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…”
30 · 2 Marks
⁵C₂=
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: ⁵C₂ =
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: ⁵C₂ =. Circle that word — that is the missing variable.
Find
We are required to find: ⁵C₂ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5!/(2!3!) = 10. Then substitute the given, one line at a time, to get ⁵C₂ =.
In other words — as if you have never seen this before
5!(2!3!)=10.
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?
5!(2!3!)=10.
4. Step-by-step working
1. 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.
2. 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: ⁵C₂ =. Circle that word — that is the missing variable. Find: We are required to find: ⁵C₂ =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 5!/(2!3!) = 10. Then substitute the given, one line at a time, to get ⁵C₂ =.
  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).
3. 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.
4. 04 / 08
  Step C — Name the method in words
  5!(2!3!)=10.
  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.
5. 05 / 08
  Working
  5!(2!3!)=10.
  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.
6. 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.
7. 07 / 08
  Step F — Box the answer with its unit
  10
  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.
8. 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
10
Copy onto your answer sheet — in this order
Answer: 10
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…”