2025 · practice · Mathematics · Grade 12

Grade 12 Mathematics — KZN calculus & series

Notes

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

90 min · 107 Marks · 30 questions

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Question breakdown

1.1 · 8 Marks · Area, sine and cosine rules
ddx (x⁴)=
Formulas
Area=½ ab sin C
Area rule
c²=a²+b²−2ab cos C
Cosine rule
Notes: Power x⁴. Unknown: first derivative.
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 4x³ 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
Power rule: bring the power down, drop the index by 1.
2. What do we already know?
Power x⁴. Unknown: first derivative.
3. Which method — and why this one?
Power rule: bring the power down, drop the index by 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: 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 4x³ 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
  Power x⁴. Unknown: first derivative.
  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
  Power rule: bring the power down, drop the index by 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
  Power rule
  n x^ⁿ⁻¹=4 x³.
  4x³
  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
  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: 4x³
Answer: 4x³
6. Check it like a marker
Integral of 4x³ is x⁴ + C. Inverse operations.
7. A trap first-timers fall into
x⁵5 is the integral, not the derivative. Read ddx.
1.2 · 10 Marks
S_∞ of 16 + 8 + 4 + …
Notes: a=16. r=816=12. |r| < 1 so S_∞ exists.
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 16 + 8 + 4 + …
Before any working — hunt first
Given
Numbers and units already in the sentence: 16 · 8 · 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: S_∞ of 16 + 8 + 4 + …. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 16 + 8 + 4 + …. 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 16 + 8 + 4 + ….
In other words — as if you have never seen this before
S_∞ = a(1 − r) for |r| < 1.
2. What do we already know?
a=16. r=816=12. |r| < 1 so S_∞ exists.
3. Which method — and why this one?
S_∞ = a(1 − r) for |r| < 1.
4. Step-by-step working
1. 01 / 05
  Name r
  r=12. |r| < 1, so it 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
  Formula
  S=16(1 − 1/2)=16(1/2)=32.
  S_∞=32
  Why this step: Write the formula before numbers. The memo awards a method mark 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
  32
  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
32
Copy onto your answer sheet — in this order
Formula: r=12
Answer: 32
6. Check it like a marker
16+8+4+2+1+… approaches 32, never 28.
7. A trap first-timers fall into
Adding the first three terms only (28) is not S_∞.
2.1 · 10 Marks · First principles of differentiation
f(x)=x²−6x. Stationary x=
Formulas
f'(x)=limh→0[f(x+h) − f(x)]h
Definition of the derivative
Notes: f(x)=x²−6x. Unknown: x where f'=0.
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)=x²−6x. Stationary x =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x. 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: Stationary x =. Circle that word — that is the missing variable.
Find
We are required to find: Stationary x =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write f' = 2x−6 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Stationary x =.
In other words — as if you have never seen this before
Stationary ⇒ f'(x) = 0. Differentiate first.
2. What do we already know?
f(x)=x²−6x. Unknown: x where f'=0.
3. Which method — and why this one?
Stationary ⇒ f'(x) = 0. Differentiate first.
4. Step-by-step working
1. 01 / 05
  Differentiate
  f'=2x−6.
  f'=2x−6
  Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
2. 02 / 05
  Set to zero
  2x−6=0→x=3.
  x=3
  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
  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: f'=2x−6
Answer: 3
6. Check it like a marker
f''=2 > 0, a minimum. f(3)=9−18=−9.
7. A trap first-timers fall into
x = 6 is where f(x)=0 (a root), not where the gradient is zero.
2.2 · 12 Marks
log₁₀ 100=
Notes: log base 10 of 100.
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 2 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
log_b a=x means bx=a. 10²=100.
2. What do we already know?
log base 10 of 100.
3. Which method — and why this one?
log_b a=x means bx=a. 10²=100.
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: 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 2 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log₁₀ 100 =.
  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
  log base 10 of 100.
  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_b a=x means bx=a. 10²=100.
  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
  Rewrite as a power
  10x=100=10²→x=2.
  2
  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
  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.
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
2
Copy onto your answer sheet — in this order
Formula: 2
Answer: 2
6. Check it like a marker
10² is 100. Count the zeros: 2.
7. A trap first-timers fall into
Writing 10 treats log as a linear scale.
5 · 3 Marks · Arithmetic & geometric series
Geometric a=27, r=13. 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
Notes: First term a = 27. Ratio r = 13. Infinite geometric series. Unknown: S_∞.
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 = 27, r = 13. S_∞ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 27 · 1/ · 3. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 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
Write |r| < 1 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives S_∞ =.
In other words — as if you have never seen this before
It converges only if |r| < 1. Here |13| < 1, so S_∞ = a(1 − r).
2. What do we already know?
First term a = 27. Ratio r = 13. Infinite geometric series. Unknown: S_∞.
3. Which method — and why this one?
It converges only if |r| < 1. Here |13| < 1, so S_∞ = a(1 − r).
4. Step-by-step working
1. 01 / 05
  Test |r|
  |13| = 13 which is less than 1. The terms shrink (27, 9, 3, 1, …) so a finite sum exists.
  |r| < 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.
2. 02 / 05
  Apply S_∞
  S_∞=27(1 − 1/3)=27(2/3)=27×32=40.5.
  S_∞=a(1−r)=40.5
  Why this step: Dividing by a fraction is multiplying by the flip. 27 ÷ (2/3) = 27 × 3/2.
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
  40.5
  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.
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
40.5
Copy onto your answer sheet — in this order
Formula: |r| < 1
Answer: 40.5
6. Check it like a marker
27+9+3+1+13+… is clearly more than 27+9=36 and less than 27+27=54. 40.5 sits in the middle.
7. A trap first-timers fall into
Writing ∞ because “it never stops.” It never stops, but the leftover becomes tiny. That is what |r|<1 means.
6 · 2 Marks
r=−0.9 means
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 = −0.9 means
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.9 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: 9 means. Circle that word — that is the missing variable.
Find
We are required to find: 9 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Close to −1: strong negative. Then substitute the given, one line at a time, to get 9 means.
In other words — as if you have never seen this before
Close to −1: strong negative.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0.9 means. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Close to −1: strong negative.
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: 0.9 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: 9 means. Circle that word — that is the missing variable. Find: We are required to find: 9 means. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Close to −1: strong negative. Then substitute the given, one line at a time, to get 9 means.
  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: 0.9 means. 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
  Close to −1: strong negative.
  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
  Close to −1: strong negative.
  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
  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.
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
strong negative linear correlation
Copy onto your answer sheet — in this order
Answer: strong negative linear correlation
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…”
7 · 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…”
8 · 3 Marks · Compound angles
sin 75°=
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 75° =
Before any working — hunt first
Given
Numbers and units already in the sentence: 75°. 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 75° =. Circle that word — that is the missing variable.
Find
We are required to find: sin 75° =. 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)=sin45cos30+cos45sin30. Then substitute the given, one line at a time, to get sin 75° =.
In other words — as if you have never seen this before
sin(45+30)=sin45cos30+cos45sin30.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 75°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
sin(45+30)=sin45cos30+cos45sin30.
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: 75°. 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 75° =. Circle that word — that is the missing variable. Find: We are required to find: sin 75° =. 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)=sin45cos30+cos45sin30. Then substitute the given, one line at a time, to get sin 75° =.
  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: 75°. 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)=sin45cos30+cos45sin30.
  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
  sin(45+30)=sin45cos30+cos45sin30.
  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
  (√6+√2)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
(√6+√2)4
Copy onto your answer sheet — in this order
Answer: (√6+√2)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…”
9 · 3 Marks · Regression & correlation
r=0.2 means
Formulas
ŷ=a+bx
Least-squares line
−1 ≤ r ≤ 1
Pearson’s r
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 = 0.2 means
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.2 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: 2 means. Circle that word — that is the missing variable.
Find
We are required to find: 2 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: weak positive correlation. Then substitute the given, one line at a time, to get 2 means.
In other words — as if you have never seen this before
weak positive correlation
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0.2 means. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
weak positive correlation
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: 0.2 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: 2 means. Circle that word — that is the missing variable. Find: We are required to find: 2 means. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: weak positive correlation. Then substitute the given, one line at a time, to get 2 means.
  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: 0.2 means. 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
  weak positive correlation
  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
  Close to 0, positive.
  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
  weak positive 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.
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
weak positive correlation
Copy onto your answer sheet — in this order
Answer: weak positive correlation
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
10 · 3 Marks
6C2=
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: 6C2 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6C · 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: 6C2 =. Circle that word — that is the missing variable.
Find
We are required to find: 6C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 15. Then substitute the given, one line at a time, to get 6C2 =.
In other words — as if you have never seen this before
15
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6C · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
15
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: 6C · 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: 6C2 =. Circle that word — that is the missing variable. Find: We are required to find: 6C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 15. Then substitute the given, one line at a time, to get 6C2 =.
  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: 6C · 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
  15
  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
  6! / (2! 4!) = (6 × 5) / 2 = 15. Combinations do not care about order — that is why we divide by 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
  15
  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
15
Copy onto your answer sheet — in this order
Answer: 15
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 · 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…”
13 · 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…”
14 · 2 Marks · Circles & tangents
A tangent is perpendicular to the
Formulas
(x − a)²+(y − b)²=r²
Circle
m_radius×m_tangent=−1
Tangent ⟂ radius
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 tangent is perpendicular to 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: A tangent is perpendicular to the. Circle that word — that is the missing variable.
Find
We are required to find: A tangent is perpendicular to the. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Classic circle theorem in analytical form. Then substitute the given, one line at a time, to get A tangent is perpendicular to the.
In other words — as if you have never seen this before
Classic circle theorem in analytical form.
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?
Classic circle theorem in analytical form.
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: A tangent is perpendicular to the. Circle that word — that is the missing variable. Find: We are required to find: A tangent is perpendicular to the. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Classic circle theorem in analytical form. Then substitute the given, one line at a time, to get A tangent is perpendicular to 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
  Classic circle theorem in analytical form.
  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
  Classic circle theorem in analytical form.
  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
  radius at the point of contact
  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
radius at the point of contact
Copy onto your answer sheet — in this order
Answer: radius at the point of contact
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
S_∞ of a geometric series exists only when
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: S_∞ of a geometric series exists only when
In other words — as if you have never seen this before
Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
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?
Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
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
  Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
  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
  Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
  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
  |r| < 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
|r| < 1
Copy onto your answer sheet — in this order
Answer: |r| < 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…”
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 · 2 Marks · Arithmetic & geometric series
Geometric first term 3, 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
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 first term 3, r=2. T₄ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 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
What we need to calculate: T₄ = 3×2³ = 24. Then substitute the given, one line at a time, to get T₄ =.
In other words — as if you have never seen this before
T₄=3×2³=24.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
T₄=3×2³=24.
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 · 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: What we need to calculate: T₄ = 3×2³ = 24. Then substitute the given, one line at a time, to get T₄ =.
  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 · 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
  T₄=3×2³=24.
  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
  T₄=3×2³=24.
  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
  24
  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
24
Copy onto your answer sheet — in this order
Answer: 24
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
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…”
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 · First principles of differentiation
f(x)=x². From first principles f'(x)=
Formulas
f'(x)=limh→0[f(x+h) − f(x)]h
Definition of the derivative
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)=x². From first principles f'(x) =
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: From first principles f'(x) =. Circle that word — that is the missing variable.
Find
We are required to find: From first principles f'(x) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: [2xh + h²]h. Then substitute the given, one line at a time, to get From first principles f'(x) =.
In other words — as if you have never seen this before
[2xh + h²]h→2x as h→0.
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?
[2xh + h²]h→2x as h→0.
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: From first principles f'(x) =. Circle that word — that is the missing variable. Find: We are required to find: From first principles f'(x) =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: [2xh + h²]h. Then substitute the given, one line at a time, to get From first principles f'(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: 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
  [2xh + h²]h→2x as h→0.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Working
  [2xh + h²]h→2x as h→0.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
  2x
  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
2x
Copy onto your answer sheet — in this order
Answer: 2x
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 · First principles of differentiation
The definition of f'(x) uses
Formulas
f'(x)=limh→0[f(x+h) − f(x)]h
Definition of the derivative
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: The definition of f'(x) uses
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: The definition of f'(x) uses. Circle that word — that is the missing variable.
Find
We are required to find: The definition of f'(x) uses. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: First principles. Then substitute the given, one line at a time, to get The definition of f'(x) uses.
In other words — as if you have never seen this before
First principles.
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?
First principles.
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: The definition of f'(x) uses. Circle that word — that is the missing variable. Find: We are required to find: The definition of f'(x) uses. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: First principles. Then substitute the given, one line at a time, to get The definition of f'(x) 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: 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
  First principles.
  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
  First principles.
  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
  limh→0[f(x+h)−f(x)]h
  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
limh→0[f(x+h)−f(x)]h
Copy onto your answer sheet — in this order
Answer: limh→0 [f(x+h)−f(x)]h
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
23 · 2 Marks · Area, sine and cosine rules
ddx (4x⁵)=
Formulas
Area=½ ab sin C
Area rule
c²=a²+b²−2ab cos C
Cosine rule
Notes: f(x) = 4x⁵. Unknown: the derivative f'(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 (4x⁵) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 4x. 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 (4x⁵) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (4x⁵) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write ddx [k xⁿ] = k 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 (4x⁵) =.
In other words — as if you have never seen this before
Power rule in words: bring the index down to multiply, then drop the index by one.
2. What do we already know?
f(x) = 4x⁵. Unknown: the derivative f'(x).
3. Which method — and why this one?
Power rule in words: bring the index down to multiply, then drop the index by one.
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: 4x. 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 (4x⁵) =. Circle that word — that is the missing variable. Find: We are required to find: ddx (4x⁵) =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: Write ddx [k xⁿ] = k 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 (4x⁵) =.
  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) = 4x⁵. Unknown: the derivative f'(x).
  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
  Power rule in words: bring the index down to multiply, then drop the index by one.
  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
  Bring the 5 down
  5 × 4 = 20, and the index becomes 4. So 20x⁴.
  ddx [k xⁿ]=k n xⁿ⁻¹
  Why this step: The 4 is a constant multiplier — it just tags along. You do not drop it, and you do not add it to 5.
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
  20x⁴
  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
20x⁴
Copy onto your answer sheet — in this order
Formula: ddx [k xⁿ]=k n xⁿ⁻¹
Answer: 20x⁴
6. Check it like a marker
At x = 1, slope should be 20. A 4x⁵ curve is steep there. 4x⁴ would be four times too gentle.
7. A trap first-timers fall into
Writing 4x⁴ (forgot to multiply by 5) or x⁶ (that is integration, the reverse road).
24 · 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…”
25 · 2 Marks · Area, sine and cosine rules
ddx (x³)=
Formulas
Area=½ ab sin C
Area rule
c²=a²+b²−2ab cos C
Cosine 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 (x³) =
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 (x³) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (x³) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Power rule: n x^{n−1}. Then substitute the given, one line at a time, to get ddx (x³) =.
In other words — as if you have never seen this before
Power rule: n x^ⁿ⁻¹.
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?
Power rule: n x^ⁿ⁻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: 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 (x³) =. Circle that word — that is the missing variable. Find: We are required to find: ddx (x³) =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Power rule: n x^{n−1}. Then substitute the given, one line at a time, to get ddx (x³) =.
  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
  Power rule: n x^ⁿ⁻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
  Working
  Power rule: n x^ⁿ⁻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
  3x²
  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
3x²
Copy onto your answer sheet — in this order
Answer: 3x²
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
26 · 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…”
27 · 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…”
28 · 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…”
29 · 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…”
30 · 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.