2025 · practice · Mathematics · Grade 12

Grade 12 Mathematics Paper 1 — NSC style

Notes

Paper 1: algebra, calculus, finance, probability, sequences.

90 min · 137 Marks · 30 questions

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Sit it timed Formula sheet

Question breakdown

1.1 · 6 Marks
ddx (4x³ − x)=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ddx (4x³ − x) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 4x³. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (4x³ − x) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (4x³ − x) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Power rule on each term. Then substitute the given, one line at a time, to get ddx (4x³ − x) =.
In other words — as if you have never seen this before
If you have never seen this before: Power rule on each term. We will name a method in words, write the formula with no numbers, then substitute one line at a time.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 4x³. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Power rule on each term.
4. Step-by-step working
1. 01 / 08
  Step A — Read the question twice
  Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 4x³. Copy each one into a column. A name without a number is not given. Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (4x³ − x) =. Circle that word — that is the missing variable. Find: We are required to find: ddx (4x³ − x) =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Power rule on each term. Then substitute the given, one line at a time, to get ddx (4x³ − x) =.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 4x³. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  Power rule on each term.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Reason
  Power rule on each term.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
6. 06 / 08
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
7. 07 / 08
  Step F — Box the answer with its unit
  12x² − 1
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
12x² − 1
Copy onto your answer sheet — in this order
Answer: 12x² − 1
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
1.2 · 8 Marks · Arithmetic & geometric series
Geometric a=5, r=0.4. 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=5, r=0.4. S_∞ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5 · 0.4. 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_∞ =. 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: S = 5(1−0. Then substitute the given, one line at a time, to get S_∞ =.
In other words — as if you have never seen this before
If you have never seen this before: S = 5(1−0.4)=8.333… We will name a method in words, write the formula with no numbers, then substitute one line at a time.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5 · 0.4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
S=5(1−0.4)=8.333…
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 · 0.4. 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_∞ =. 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: S = 5(1−0. 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: 5 · 0.4. 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
  S=5(1−0.4)=8.333…
  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
  S=5(1−0.4)=8.333…
  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
  8.33
  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
8.33
Copy onto your answer sheet — in this order
Answer: 8.33
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…”
2.1 · 12 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: 2⁵ = 32. Then substitute the given, one line at a time, to get log₂ 32 =.
In other words — as if you have never seen this before
If you have never seen this before: 2⁵ = 32. We will name a method in words, write the formula with no numbers, then substitute one line at a time.
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?
2⁵=32.
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: 2⁵ = 32. 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
  2⁵=32.
  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…”
2.2 · 12 Marks
5C2=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 5C2 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5C · 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: 5C2 =. Circle that word — that is the missing variable.
Find
We are required to find: 5C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5!/(2!3!) = 10. Then substitute the given, one line at a time, to get 5C2 =.
In other words — as if you have never seen this before
If you have never seen this before: 5!/(2!3!) = 10. Order does not matter. We will name a method in words, write the formula with no numbers, then substitute one line at a time.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5C · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5!(2!3!)=10. Order does not matter.
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: 5C · 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: 5C2 =. Circle that word — that is the missing variable. Find: We are required to find: 5C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 5!/(2!3!) = 10. Then substitute the given, one line at a time, to get 5C2 =.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 5C · 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!(2!3!)=10. Order does not matter.
  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
  5!(2!3!)=10. Order does not matter.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
6. 06 / 08
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
7. 07 / 08
  Step F — Box the answer with its unit
  10
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
10
Copy onto your answer sheet — in this order
Answer: 10
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
3.1 · 16 Marks
f'(x)=0 for f(x)=x² − 6x + 5 at
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f'(x)=0 for f(x)=x² − 6x + 5 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 for · 6x · 5 at. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: f' = 2x − 6 = 0. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
In other words — as if you have never seen this before
If you have never seen this before: f' = 2x − 6 = 0 → x = 3. We will name a method in words, write the formula with no numbers, then substitute one line at a time.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 for · 6x · 5 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
f'=2x−6=0→x=3.
4. Step-by-step working
1. 01 / 08
  Step A — Read the question twice
  Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 0 for · 6x · 5 at. Copy each one into a column. A name without a number is not given. Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable. Find: We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: f' = 2x − 6 = 0. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 0 for · 6x · 5 at. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  f'=2x−6=0→x=3.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Reason
  f'=2x−6=0→x=3.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
6. 06 / 08
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
7. 07 / 08
  Step F — Box the answer with its unit
  x=3
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
x=3
Copy onto your answer sheet — in this order
Answer: x=3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
3.2 · 16 Marks · Compound angles
sin(A+B) expands to
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(A+B) expands to
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: sin(A+B) expands to. Circle that word — that is the missing variable.
Find
We are required to find: sin(A+B) expands to. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Compound-angle sine. Then substitute the given, one line at a time, to get sin(A+B) expands to.
In other words — as if you have never seen this before
If you have never seen this before: Compound-angle sine. We will name a method in words, write the formula with no numbers, then substitute one line at a time.
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?
Compound-angle sine.
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: sin(A+B) expands to. Circle that word — that is the missing variable. Find: We are required to find: sin(A+B) expands to. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Compound-angle sine. Then substitute the given, one line at a time, to get sin(A+B) expands to.
  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
  Compound-angle sine.
  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 sine.
  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
  sinA cosB + cosA sinB
  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
sinA cosB + cosA sinB
Copy onto your answer sheet — in this order
Answer: sinA cosB + cosA sinB
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 · 3 Marks · Arithmetic & geometric series
Geometric a=5, r=2. T₄=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
Notes: First term 5, common ratio 2. Fourth term.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Geometric a=5, r=2. T₄ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5 · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: T₄ =. Circle that word — that is the missing variable.
Find
We are required to find: T₄ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write Tₙ = a rⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives T₄ =.
In other words — as if you have never seen this before
Tₙ=a rⁿ⁻¹. T₄=5×2³=5×8=40.
2. What do we already know?
First term 5, common ratio 2. Fourth term.
3. Which method — and why this one?
Tₙ=a rⁿ⁻¹. T₄=5×2³=5×8=40.
4. Step-by-step working
1. 01 / 05
  Write Tₙ = arⁿ⁻¹
  The exponent is n minus 1, not n.
  Tₙ=a rⁿ⁻¹
  Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
2. 02 / 05
  Substitute
  T₄=5×2³=40.
  T₄=40
  Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
3. 03 / 05
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
4. 04 / 05
  Step F — Box the answer with its unit
  40
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
5. 05 / 05
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
40
Copy onto your answer sheet — in this order
Formula: Tₙ=a rⁿ⁻¹
Answer: 40
6. Check it like a marker
Sequence: 5, 10, 20, 40. Fourth term 40.
7. A trap first-timers fall into
2⁴ = 16 would give T₅ if you use rⁿ instead of rⁿ⁻¹.
8 · 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…”
9 · 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…”
10 · 4 Marks · Annuities & outstanding balance
Outstanding balance after k payments uses P on the remaining n−k periods
Formulas
F=x[((1+i)ⁿ − 1)/i]
Future value of an annuity
P=x[1 − (1+i)⁻ⁿ]i
Present value of an annuity
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: Outstanding balance after k payments uses P on the remaining n−k periods
In other words — as if you have never seen this before
Present-value of remaining instalments.
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?
Present-value of remaining instalments.
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
  Present-value of remaining instalments.
  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
  Present-value of remaining instalments.
  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
  true
  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
true
Copy onto your answer sheet — in this order
Answer: true
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 · 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…”
12 · 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…”
13 · 3 Marks · Similarity & proportion
AA similarity requires
Formulas
AA similarity ⇒ corresponding sides proportional
AA similarity
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: AA similarity requires
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: AA similarity requires. Circle that word — that is the missing variable.
Find
We are required to find: AA similarity requires. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: two equal corresponding angles. Then substitute the given, one line at a time, to get AA similarity requires.
In other words — as if you have never seen this before
two equal corresponding angles
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?
two equal corresponding angles
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: AA similarity requires. Circle that word — that is the missing variable. Find: We are required to find: AA similarity requires. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: two equal corresponding angles. Then substitute the given, one line at a time, to get AA similarity requires.
  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
  two equal corresponding angles
  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
  AA is sufficient.
  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
  two equal corresponding angles
  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
two equal corresponding angles
Copy onto your answer sheet — in this order
Answer: two equal corresponding angles
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
∫ 6x² dx=
Notes: Indefinite integral of 6x².
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ∫ 6x² dx =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x². Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: ∫ 6x² dx =. Circle that word — that is the missing variable.
Find
We are required to find: ∫ 6x² dx =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 2x³ + C with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ∫ 6x² dx =.
In other words — as if you have never seen this before
Raise the index by 1, divide by the new index. Add C.
2. What do we already know?
Indefinite integral of 6x².
3. Which method — and why this one?
Raise the index by 1, divide by the new index. Add C.
4. Step-by-step working
1. 01 / 08
  Step A — Read the question twice
  Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 6x². Copy each one into a column. A name without a number is not given. Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: ∫ 6x² dx =. Circle that word — that is the missing variable. Find: We are required to find: ∫ 6x² dx =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: Write 2x³ + C with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ∫ 6x² dx =.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Indefinite integral of 6x².
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  Raise the index by 1, divide by the new index. Add C.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Power integrate
  6×x³3+C=2x³+C.
  2x³+C
  Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
6. 06 / 08
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
7. 07 / 08
  Step F — Box the answer with its unit
  2x³ + C
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
2x³ + C
Copy onto your answer sheet — in this order
Formula: 2x³ + C
Answer: 2x³ + C
6. Check it like a marker
ddx (2x³) = 6x². Recovers the integrand.
7. A trap first-timers fall into
12x is the derivative of 6x², not the integral.
15 · 4 Marks
Which formula is future value of an ordinary annuity?
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: Which formula is future value of an ordinary annuity?
In other words — as if you have never seen this before
Money going in grows to F. Loans use P.
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?
Money going in grows to F. Loans use P.
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
  Money going in grows to F. Loans use P.
  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
  Money going in grows to F. Loans use P.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
  F=x[((1+i)ⁿ−1)/i]
  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
F=x[((1+i)ⁿ−1)/i]
Copy onto your answer sheet — in this order
Answer: F=x[((1+i)ⁿ−1)/i]
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
S_∞ of 16 + 8 + 4 + … =
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
What we need to calculate: 32. Then substitute the given, one line at a time, to get S_∞ of 16 + 8 + 4 + … =.
In other words — as if you have never seen this before
32
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 16 · 8 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
32
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: 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: What we need to calculate: 32. Then substitute the given, one line at a time, to get S_∞ of 16 + 8 + 4 + … =.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 16 · 8 · 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
  32
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Working
  a=16, r=12. S=16(1−0.5)=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
  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.
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
32
Copy onto your answer sheet — in this order
Answer: 32
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
17 · 2 Marks · 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…”
18 · 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…”
19 · 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…”
20 · 3 Marks
f(x)=x³−3x². f'(x)=0 at
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f(x)=x³−3x². f'(x)=0 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 3x² · 0 at. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x)=0 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: f'=3x²−6x=3x(x−2). Then substitute the given, one line at a time, to get f'(x)=0 at.
In other words — as if you have never seen this before
f'=3x²−6x=3x(x−2).
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3x² · 0 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
f'=3x²−6x=3x(x−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: 3x² · 0 at. Copy each one into a column. A name without a number is not given. Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 at. Circle that word — that is the missing variable. Find: We are required to find: f'(x)=0 at. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: f'=3x²−6x=3x(x−2). Then substitute the given, one line at a time, to get f'(x)=0 at.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 3x² · 0 at. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  f'=3x²−6x=3x(x−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
  f'=3x²−6x=3x(x−2).
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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=0 and x=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
x=0 and x=2
Copy onto your answer sheet — in this order
Answer: x=0 and x=2
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
21 · 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.
22 · 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”.
23 · 2 Marks · Circles & tangents
Gradient of radius to (4; 0) from (0; 0) is
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: Gradient of radius to (4; 0) from (0; 0) is
In other words — as if you have never seen this before
0
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 4 · 0 · 0 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
0
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: 4 · 0 · 0 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
3. 03 / 07
  Step C — Name the method in words
  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.
4. 04 / 07
  Reason
  Horizontal radius, m=0. Tangent then vertical.
  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…”
24 · 4 Marks · Compound angles
A vertical pole 8 m, observer 15 m away. Angle of elevation tan⁻¹(8/15) ≈
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: A vertical pole 8 m, observer 15 m away. Angle of elevation tan⁻¹(8/15) ≈
Before any working — hunt first
Given
Numbers and units already in the sentence: 8 m · 15 m · 8/ · 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: Angle of elevation tan⁻¹(8/15) ≈. Circle that word — that is the missing variable.
Find
We are required to find: Angle of elevation tan⁻¹(8/15) ≈. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: tan θ = oppositeadjacent=815. Then substitute the given, one line at a time, to get Angle of elevation tan⁻¹(8/15) ≈.
In other words — as if you have never seen this before
tan θ = oppositeadjacent=815.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 8 m · 15 m · 8/ · 15. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
tan θ = oppositeadjacent=815.
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 m · 15 m · 8/ · 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: Angle of elevation tan⁻¹(8/15) ≈. Circle that word — that is the missing variable. Find: We are required to find: Angle of elevation tan⁻¹(8/15) ≈. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: tan θ = oppositeadjacent=815. Then substitute the given, one line at a time, to get Angle of elevation tan⁻¹(8/15) ≈.
  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 m · 15 m · 8/ · 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
  tan θ = oppositeadjacent=815.
  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
  tan θ = oppositeadjacent=815.
  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
  28.1°
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
28.1°
Copy onto your answer sheet — in this order
Answer: 28.1°
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…”
25 · 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…”
26 · 4 Marks
Solve 2ˣ=8
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Solve 2ˣ = 8
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 8. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Solve 2ˣ = 8. Circle that word — that is the missing variable.
Find
We are required to find: Solve 2ˣ = 8. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 8=2³. Then substitute the given, one line at a time, to get Solve 2ˣ = 8.
In other words — as if you have never seen this before
8=2³.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 8. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
8=2³.
4. Step-by-step working
1. 01 / 08
  Step A — Read the question twice
  Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 2 · 8. Copy each one into a column. A name without a number is not given. Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Solve 2ˣ = 8. Circle that word — that is the missing variable. Find: We are required to find: Solve 2ˣ = 8. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: 8=2³. Then substitute the given, one line at a time, to get Solve 2ˣ = 8.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 2 · 8. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  8=2³.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Working
  8=2³.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
6. 06 / 08
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
7. 07 / 08
  Step F — Box the answer with its unit
  x=3
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
x=3
Copy onto your answer sheet — in this order
Answer: x=3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
27 · 2 Marks · Arithmetic & geometric series
Arithmetic: a=3, d=4. T₅=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Arithmetic: a=3, d=4. T₅ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 4. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: T₅ =. Circle that word — that is the missing variable.
Find
We are required to find: T₅ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: T₅ = 3 + 4×4 = 19. Then substitute the given, one line at a time, to get T₅ =.
In other words — as if you have never seen this before
T₅=3+4×4=19.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
T₅=3+4×4=19.
4. Step-by-step working
1. 01 / 08
  Step A — Read the question twice
  Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 3 · 4. Copy each one into a column. A name without a number is not given. Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: T₅ =. Circle that word — that is the missing variable. Find: We are required to find: T₅ =. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: T₅ = 3 + 4×4 = 19. Then substitute the given, one line at a time, to get T₅ =.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 3 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  T₅=3+4×4=19.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Working
  T₅=3+4×4=19.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
6. 06 / 08
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
7. 07 / 08
  Step F — Box the answer with its unit
  19
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
19
Copy onto your answer sheet — in this order
Answer: 19
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
28 · 2 Marks
Stationary points of f(x) = x³ − 12x + 1 at x =
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Stationary points of f(x) = x³ − 12x + 1 at x =
In other words — as if you have never seen this before
±2
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 12x · 1 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
±2
4. Step-by-step working
1. 01 / 07
  Step A — Read the question twice
  Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
2. 02 / 07
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 12x · 1 at. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
3. 03 / 07
  Step C — Name the method in words
  ±2
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
4. 04 / 07
  Working
  f'=3x²−12=0→x²=4→x=±2.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
5. 05 / 07
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
6. 06 / 07
  Step F — Box the answer with its unit
  ±2
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
7. 07 / 07
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
±2
Copy onto your answer sheet — in this order
Answer: ±2
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
29 · 3 Marks · Compound angles
ddx not on Paper 2. Which topic is Paper 2?
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ddx not on Paper 2. Which topic is Paper 2?
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Which topic is Paper 2. Circle that word — that is the missing variable.
Find
We are required to find: Which topic is Paper 2. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Trigonometry. Then substitute the given, one line at a time, to get Which topic is Paper 2.
In other words — as if you have never seen this before
Trigonometry
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Trigonometry
4. Step-by-step working
1. 01 / 08
  Step A — Read the question twice
  Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
2. 02 / 08
  Step A2 — Hunt: Given, Missing, Find, Calculate
  Given: Numbers and units already in the sentence: 2 · 2. Copy each one into a column. A name without a number is not given. Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Which topic is Paper 2. Circle that word — that is the missing variable. Find: We are required to find: Which topic is Paper 2. Write that as a symbol on the left, with a question mark, before you touch a formula. Calculate: What we need to calculate: Trigonometry. Then substitute the given, one line at a time, to get Which topic is Paper 2.
  Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
3. 03 / 08
  Step B — Write the given in a column
  Copy these onto the left of your page, one per line, with units: 2 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
  Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
4. 04 / 08
  Step C — Name the method in words
  Trigonometry
  Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
5. 05 / 08
  Reason
  Calculus and finance sit on Paper 1.
  Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
6. 06 / 08
  Step G — Sense-check the size
  Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
  Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
7. 07 / 08
  Step F — Box the answer with its unit
  Trigonometry
  Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
8. 08 / 08
  Step H — What the marker is looking for
  On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
  Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
Trigonometry
Copy onto your answer sheet — in this order
Answer: Trigonometry
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
30 · 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…”