Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ddx (4x⁵) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 4x. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (4x⁵) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (4x⁵) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write ddx [k xⁿ] = k n xⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ddx (4x⁵) =.
In other words — as if you have never seen this before
Power rule in words: bring the index down to multiply, then drop the index by one.
2. What do we already know?
f(x) = 4x⁵. Unknown: the derivative f'(x).
3. Which method — and why this one?
Power rule in words: bring the index down to multiply, then drop the index by one.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 4x. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (4x⁵) =. Circle that word — that is the missing variable.
Find: We are required to find: ddx (4x⁵) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write ddx [k xⁿ] = k n xⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ddx (4x⁵) =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
f(x) = 4x⁵. Unknown: the derivative f'(x).
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Power rule in words: bring the index down to multiply, then drop the index by one.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Bring the 5 down
5 × 4 = 20, and the index becomes 4. So 20x⁴.
ddx [k xⁿ]=k n xⁿ⁻¹
Why this step: The 4 is a constant multiplier — it just tags along. You do not drop it, and you do not add it to 5.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
20x⁴
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
20x⁴
Copy onto your answer sheet — in this order
Formula: ddx [k xⁿ]=k n xⁿ⁻¹
Answer: 20x⁴
6. Check it like a marker
At x = 1, slope should be 20. A 4x⁵ curve is steep there. 4x⁴ would be four times too gentle.
7. A trap first-timers fall into
Writing 4x⁴ (forgot to multiply by 5) or x⁶ (that is integration, the reverse road).
1.2 · 12 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
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.
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.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
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.
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.
2.1 · 16 Marks
log₅ 125=
Notes: log₅ 125. In words: “5 to the what equals 125?”
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log₅ 125 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 125. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₅ 125 =. Circle that word — that is the missing variable.
Find
We are required to find: log₅ 125 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 5ˣ = 125 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log₅ 125 =.
In other words — as if you have never seen this before
A logarithm is an index. Convert to 5ˣ = 125, then notice 5³ = 125.
2. What do we already know?
log₅ 125. In words: “5 to the what equals 125?”
3. Which method — and why this one?
A logarithm is an index. Convert to 5ˣ = 125, then notice 5³ = 125.
4. Step-by-step working
01 / 05
Rewrite as an exponential
log₅ 125=x⇔5ˣ=125.
5ˣ=125
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Recognise the power
5¹=5, 5²=25, 5³=125. So x=3.
x=3
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3
Copy onto your answer sheet — in this order
Formula: 5ˣ=125
Answer: 3
6. Check it like a marker
5³=5×5×5=125. Yes.
7. A trap first-timers fall into
Writing 25 (that is 1255) or 5 (the base). A log is not a divide.
2.2 · 16 Marks · Similarity & proportion
f'(x) = 0 for f(x) = x² − 6x + 1 at
Similarity & proportion
Formulas
AA similarity ⇒ corresponding sides proportional
AA similarity
Notes: f(x) = x² − 6x + 1. Unknown: the x where the gradient is zero (stationary / turning point).
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 + 1 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 for · 6x · 1 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 + 1 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x) = 0 for f(x) = x² − 6x + 1 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write f'(x) = 2x − 6 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives f'(x) = 0 for f(x) = x² − 6x + 1 at.
In other words — as if you have never seen this before
Differentiate, then set the derivative equal to zero. That is where the tangent is flat.
2. What do we already know?
f(x) = x² − 6x + 1. Unknown: the x where the gradient is zero (stationary / turning point).
3. Which method — and why this one?
Differentiate, then set the derivative equal to zero. That is where the tangent is flat.
4. Step-by-step working
01 / 05
Differentiate
f'(x) = 2x − 6. (Power rule on x², and −6x becomes −6.)
f'(x)=2x−6
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Set equal to zero
2x − 6 = 0 → 2x = 6 → x = 3. The turning point sits at x = 3 (a minimum, because 2>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.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
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.
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
x=3
Copy onto your answer sheet — in this order
Formula: f'(x)=2x−6
Answer: x=3
6. Check it like a marker
Vertex of a parabola x = −b2a=64? Wait: a=1, b=−6, −b2a=62=3. Same answer, different road.
7. A trap first-timers fall into
Picking x = 6, which is −b, or x = 1, a root-ish guess. Stationary is where f' is zero, not where f is zero.
3.1 · 18 Marks · Remainder & factor theorem
6C2=
Remainder & factor theorem
Formulas
f(a)=0⇔(x − a) | f(x)
Factor theorem
Notes: n = 6, r = 2. Combinations: order does not matter.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 6C2 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6C · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 6C2 =. Circle that word — that is the missing variable.
Find
We are required to find: 6C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write nCr = n! / (r!(n−r)!) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 6C2 =.
In other words — as if you have never seen this before
nCr = n! / (r! (n−r)!). For 6C2 this is (6×5) / 2, because most of the factorial cancels.
2. What do we already know?
n = 6, r = 2. Combinations: order does not matter.
3. Which method — and why this one?
nCr = n! / (r! (n−r)!). For 6C2 this is (6×5) / 2, because most of the factorial cancels.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Cancel and divide
(6 × 5) / 2 = 30 / 2 = 15. There are 15 ways to pick 2 people out of 6 if order does not count.
6C2=15
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
15
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
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
15
Copy onto your answer sheet — in this order
Formula: nCr=n!(r!(n−r)!)
Answer: 15
6. Check it like a marker
6P2 = 6×5 = 30 (permutations, order counts). Combinations are half of that here because 2! = 2. 15 = 302.
7. A trap first-timers fall into
Writing 30 (that is 6P2) or 720 (that is 6!). Combinations divide by r! because swapping the two chosen people is the same pair.
6 · 3 Marks · Remainder & factor theorem
Remainder when x³ − 2x + 1 is divided by (x − 1) is
Remainder & factor theorem
Formulas
f(a)=0⇔(x − a) | f(x)
Factor theorem
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Remainder when x³ − 2x + 1 is divided by (x − 1) is
In other words — as if you have never seen this before
Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2x · 1 is · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 2x · 1 is · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
Remainder theorem: f(1)=1−2+1=0. (x−1) is a factor.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
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.
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…”
7 · 2 Marks
2ˣ=16. x=
Notes: Exponential: base 2, equals 16.
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: 2ˣ = 16. x =
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 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: x =. Circle that word — that is the missing variable.
Find
We are required to find: x =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 16 = 2⁴ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives x =.
In other words — as if you have never seen this before
Write 16 as a power of 2. 16 = 2⁴, so x = 4.
2. What do we already know?
Exponential: base 2, equals 16.
3. Which method — and why this one?
Write 16 as a power of 2. 16 = 2⁴, so x = 4.
4. Step-by-step working
01 / 05
Same base
16=2⁴. So 2ˣ=2⁴.
16=2⁴
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Equate exponents
x=4.
x=4
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4
Copy onto your answer sheet — in this order
Formula: 16=2⁴
Answer: 4
6. Check it like a marker
2⁴=16. Yes. 2⁸=256, far too big.
7. A trap first-timers fall into
Do not write x = 162=8. You cannot divide when the unknown is an exponent.
8 · 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
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 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).
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.
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.
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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3
Copy onto your answer sheet — in this order
Answer: 3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
9 · 2 Marks · Arithmetic & geometric series
S_∞ of a geometric series exists only when
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of a geometric series exists only when
In other words — as if you have never seen this before
Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
Terms must shrink. |r| ≥ 1 ⇒ the series diverges.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
|r| < 1
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
|r| < 1
Copy onto your answer sheet — in this order
Answer: |r| < 1
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
10 · 3 Marks
S_∞ of 8 + 4 + 2 + … =
Notes: a = 8, next is 4, then 2. Geometric with r = 12.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of 8 + 4 + 2 + … =
Before any working — hunt first
Given
Numbers and units already in the sentence: 8 · 4 · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 8 + 4 + 2 + … =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 8 + 4 + 2 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write r = 12 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives S_∞ of 8 + 4 + 2 + … =.
In other words — as if you have never seen this before
|r|<1 so S∞=a(1−r).
2. What do we already know?
a = 8, next is 4, then 2. Geometric with r = 12.
3. Which method — and why this one?
|r|<1 so S∞=a(1−r).
4. Step-by-step working
01 / 05
Find r
r=48=1/2. |12|<1, converges.
r=12
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
02 / 05
Substitute
S∞=8(1−1/2)=8(1/2)=16.
S∞=16
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
16
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
16
Copy onto your answer sheet — in this order
Formula: r=12
Answer: 16
6. Check it like a marker
Partial sums 8, 12, 14, 15, 15.5… climbing to 16.
7. A trap first-timers fall into
Do not use the arithmetic sum formula. Do not write 8(1+1/2).
11 · 2 Marks · First principles of differentiation
The definition of f'(x) uses
Formulas
f'(x)=limh→0[f(x+h) − f(x)]h
Definition of the derivative
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: The definition of f'(x) uses
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: The definition of f'(x) uses. Circle that word — that is the missing variable.
Find
We are required to find: The definition of f'(x) uses. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: First principles. Then substitute the given, one line at a time, to get The definition of f'(x) uses.
In other words — as if you have never seen this before
First principles.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
First principles.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: The definition of f'(x) uses. Circle that word — that is the missing variable.
Find: We are required to find: The definition of f'(x) uses. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: First principles. Then substitute the given, one line at a time, to get The definition of f'(x) uses.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
First principles.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
First principles.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
limh→0[f(x+h)−f(x)]h
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
limh→0[f(x+h)−f(x)]h
Copy onto your answer sheet — in this order
Answer: limh→0 [f(x+h)−f(x)]h
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
12 · 4 Marks
f'(x)=0 for f(x)=x² − 6x + 5 at
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f'(x)=0 for f(x)=x² − 6x + 5 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 for · 6x · 5 at. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: x = 3. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
In other words — as if you have never seen this before
x=3
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 for · 6x · 5 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
x=3
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0 for · 6x · 5 at. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable.
Find: We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: x = 3. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
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.
04 / 08
Step C — Name the method in words
x=3
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
x=3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
x=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…”
13 · 3 Marks · Compound angles
sin(A − B) expands to
Compound angles
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: sinA cosB − cosA sinB. 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
sinA cosB − cosA sinB
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?
sinA cosB − cosA sinB
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: 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: sinA cosB − cosA sinB. 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).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
sinA cosB − cosA sinB
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Compound-angle sine with a minus: the minus stays in the middle. Plus would be sin(A+B).
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
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.
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…”
14 · 2 Marks · First principles of differentiation
f(x)=x²−6x. Stationary x=
Formulas
f'(x)=limh→0[f(x+h) − f(x)]h
Definition of the derivative
Notes: f(x)=x²−6x. Unknown: x where f'=0.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f(x)=x²−6x. Stationary x =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Stationary x =. Circle that word — that is the missing variable.
Find
We are required to find: Stationary x =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write f' = 2x−6 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Stationary x =.
In other words — as if you have never seen this before
Stationary ⇒ f'(x) = 0. Differentiate first.
2. What do we already know?
f(x)=x²−6x. Unknown: x where f'=0.
3. Which method — and why this one?
Stationary ⇒ f'(x) = 0. Differentiate first.
4. Step-by-step working
01 / 05
Differentiate
f'=2x−6.
f'=2x−6
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
02 / 05
Set to zero
2x−6=0→x=3.
x=3
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3
Copy onto your answer sheet — in this order
Formula: f'=2x−6
Answer: 3
6. Check it like a marker
f''=2 > 0, a minimum. f(3)=9−18=−9.
7. A trap first-timers fall into
x = 6 is where f(x)=0 (a root), not where the gradient is zero.
15 · 4 Marks
Stationary points of f(x)=x² − 4x: f'(x)=0 at
Notes: f(x)=x² − 4x. Stationary means the derivative is zero.
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)=x² − 4x: f'(x)=0 at
In other words — as if you have never seen this before
Differentiate, set equal to zero, solve for x.
2. What do we already know?
f(x)=x² − 4x. Stationary means the derivative is zero.
3. Which method — and why this one?
Differentiate, set equal to zero, solve for x.
4. Step-by-step working
01 / 05
Differentiate
f'(x)=2x−4.
f'=2x−4
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Set to zero
2x−4=0→2x=4→x=2.
x=2
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
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.
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
x=2
Copy onto your answer sheet — in this order
Formula: f'=2x−4
Answer: x=2
6. Check it like a marker
Second derivative f''=2>0 so it is a minimum at x=2. f(2)=4−8=−4.
7. A trap first-timers fall into
Do not solve f(x)=0 — those are x-intercepts (0 and 4), not stationary points.
16 · 2 Marks · Arithmetic & geometric series
Geometric a=8, r=12. S∞=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Geometric a = 8, r = 12. S_∞ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 8 · 1/ · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 16. Then substitute the given, one line at a time, to get S_∞ =.
In other words — as if you have never seen this before
16
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 8 · 1/ · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
16
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 8 · 1/ · 2. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ =. Circle that word — that is the missing variable.
Find: We are required to find: S_∞ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 16. Then substitute the given, one line at a time, to get S_∞ =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 8 · 1/ · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
16
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
|r| < 1 so it converges. S_∞ = a(1 − r) = 8(1/2) = 16. Do not add the first two terms and stop.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
16
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
16
Copy onto your answer sheet — in this order
Answer: 16
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
17 · 2 Marks
S_∞ of 16 + 8 + 4 + …
Notes: a=16. r=816=12. |r| < 1 so S∞ exists.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of 16 + 8 + 4 + …
Before any working — hunt first
Given
Numbers and units already in the sentence: 16 · 8 · 4. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 16 + 8 + 4 + …. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 16 + 8 + 4 + …. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write r = 12 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives S_∞ of 16 + 8 + 4 + ….
In other words — as if you have never seen this before
S_∞ = a(1 − r) for |r| < 1.
2. What do we already know?
a=16. r=816=12. |r| < 1 so S∞ exists.
3. Which method — and why this one?
S_∞ = a(1 − r) for |r| < 1.
4. Step-by-step working
01 / 05
Name r
r=12. |r| < 1, so it converges.
r=12
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
02 / 05
Formula
S=16(1 − 1/2)=16(1/2)=32.
S∞=32
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
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.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
32
Copy onto your answer sheet — in this order
Formula: r=12
Answer: 32
6. Check it like a marker
16+8+4+2+1+… approaches 32, never 28.
7. A trap first-timers fall into
Adding the first three terms only (28) is not S_∞.
18 · 2 Marks
loga a=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log_a a =
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: log_a a =. Circle that word — that is the missing variable.
Find
We are required to find: log_a a =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: a¹ = a. Then substitute the given, one line at a time, to get log_a a =.
In other words — as if you have never seen this before
a¹=a.
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?
a¹=a.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: log_a a =. Circle that word — that is the missing variable.
Find: We are required to find: log_a a =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: a¹ = a. Then substitute the given, one line at a time, to get log_a a =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
a¹=a.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
a¹=a.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
1
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
1
Copy onto your answer sheet — in this order
Answer: 1
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
19 · 2 Marks · Area, sine and cosine rules
ddx (x³)=
Formulas
Area=½ ab sin C
Area rule
c²=a²+b²−2ab cos C
Cosine rule
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ddx (x³) =
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (x³) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (x³) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Power rule: n x^{n−1}. Then substitute the given, one line at a time, to get ddx (x³) =.
In other words — as if you have never seen this before
Power rule: n xⁿ⁻¹.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Power rule: n xⁿ⁻¹.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (x³) =. Circle that word — that is the missing variable.
Find: We are required to find: ddx (x³) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Power rule: n x^{n−1}. Then substitute the given, one line at a time, to get ddx (x³) =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Power rule: n xⁿ⁻¹.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Power rule: n xⁿ⁻¹.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
3x²
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3x²
Copy onto your answer sheet — in this order
Answer: 3x²
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
20 · 2 Marks · Cubic functions & graphs
A cubic can have at most how many x-intercepts?
Cubic functions & graphs
Formulas
f'(x)=0⇒stationary points
Turning / stationary
f''(x) > 0⇒local min
Second-derivative test
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: A cubic can have at most how many x-intercepts?
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: how many x-intercepts. Circle that word — that is the missing variable.
Find
We are required to find: how many x-intercepts. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Degree 3 polynomial. Then substitute the given, one line at a time, to get how many x-intercepts.
In other words — as if you have never seen this before
Degree 3 polynomial.
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?
Degree 3 polynomial.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: how many x-intercepts. Circle that word — that is the missing variable.
Find: We are required to find: how many x-intercepts. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Degree 3 polynomial. Then substitute the given, one line at a time, to get how many x-intercepts.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Degree 3 polynomial.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Degree 3 polynomial.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3
Copy onto your answer sheet — in this order
Answer: 3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
21 · 3 Marks · Compound angles
ddx not on Paper 2. Which topic is Paper 2?
Compound angles
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
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2 · 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).
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.
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.
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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
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.
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…”
22 · 3 Marks
log₂ 32=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log₂ 32 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 32. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₂ 32 =. Circle that word — that is the missing variable.
Find
We are required to find: log₂ 32 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5. Then substitute the given, one line at a time, to get log₂ 32 =.
In other words — as if you have never seen this before
5
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 32. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 32. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₂ 32 =. Circle that word — that is the missing variable.
Find: We are required to find: log₂ 32 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 5. Then substitute the given, one line at a time, to get log₂ 32 =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
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.
04 / 08
Step C — Name the method in words
5
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
5
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
5
Copy onto your answer sheet — in this order
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…”
23 · 4 Marks · Annuities & outstanding balance
Outstanding balance after k payments uses P on the remaining n−k periods
Annuities & outstanding balance
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
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
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.
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.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
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.
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…”
24 · 2 Marks
5!=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 5! =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: =. Circle that word — that is the missing variable.
Find
We are required to find: =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5×4×3×2×1=120. Then substitute the given, one line at a time, to get =.
In other words — as if you have never seen this before
5×4×3×2×1=120.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5×4×3×2×1=120.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 5. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: =. Circle that word — that is the missing variable.
Find: We are required to find: =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 5×4×3×2×1=120. Then substitute the given, one line at a time, to get =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 5. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
5×4×3×2×1=120.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
5×4×3×2×1=120.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
120
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
120
Copy onto your answer sheet — in this order
Answer: 120
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
25 · 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
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 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).
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.
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.
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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
x=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.
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…”
26 · 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
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 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).
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.
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.
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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
x − 4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
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…”
27 · 2 Marks · Circles & tangents
Gradient of radius to (4; 0) from (0; 0) is
Formulas
(x − a)²+(y − b)²=r²
Circle
mradius×mtangent=−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
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 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.
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.
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.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
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.
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…”
28 · 2 Marks
ddx (5x⁴)=
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 (5x⁴) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5x. 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 (5x⁴) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (5x⁴) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5×4 x³ = 20x³. Then substitute the given, one line at a time, to get ddx (5x⁴) =.
In other words — as if you have never seen this before
5×4 x³=20x³.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5x. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5×4 x³=20x³.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 5x. 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 (5x⁴) =. Circle that word — that is the missing variable.
Find: We are required to find: ddx (5x⁴) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 5×4 x³ = 20x³. Then substitute the given, one line at a time, to get ddx (5x⁴) =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 5x. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
5×4 x³=20x³.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
5×4 x³=20x³.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
20x³
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
20x³
Copy onto your answer sheet — in this order
Answer: 20x³
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 · 4 Marks · Compound angles
sin(A+B) expands to
Compound angles
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: sinA cosB + cosA sinB. 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
sinA cosB + cosA sinB
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?
sinA cosB + cosA sinB
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: 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: sinA cosB + cosA sinB. 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).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
sinA cosB + cosA sinB
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
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.
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…”
30 · 3 Marks
f'(x) = 0 for f(x) = x² − 8x + 3 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 − 8x + 3 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 for · 8x · 3 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 − 8x + 3 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x) = 0 for f(x) = x² − 8x + 3 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: x = 4. Then substitute the given, one line at a time, to get f'(x) = 0 for f(x) = x² − 8x + 3 at.
In other words — as if you have never seen this before
x=4
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 for · 8x · 3 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
x=4
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0 for · 8x · 3 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 − 8x + 3 at. Circle that word — that is the missing variable.
Find: We are required to find: f'(x) = 0 for f(x) = x² − 8x + 3 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: x = 4. Then substitute the given, one line at a time, to get f'(x) = 0 for f(x) = x² − 8x + 3 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).
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 · 8x · 3 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.
04 / 08
Step C — Name the method in words
x=4
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
f' = 2x − 8 = 0 → x = 4. Stationary point sits where the gradient is zero.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
x=4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
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…”