Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ddx (4x³) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 4x³. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (4x³) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (4x³) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write ddx (axⁿ)=an xⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ddx (4x³) =.
In other words — as if you have never seen this before
Power rule: ddx (axⁿ)=a n xⁿ⁻¹.
2. What do we already know?
f(x)=4x³. Unknown: f'(x).
3. Which method — and why this one?
Power rule: ddx (axⁿ)=a n xⁿ⁻¹.
4. Step-by-step working
01 / 05
Write the power rule
Bring the 3 down, multiply by 4, drop the power by 1.
ddx (axⁿ)=an xⁿ⁻¹
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Calculate
4×3 x²=12x².
12x²
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
12x²
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
12x²
Copy onto your answer sheet — in this order
Formula: ddx (axⁿ)=an xⁿ⁻¹
Answer: 12x²
6. Check it like a marker
Degree drops by 1: cubic → quadratic. Coefficient 12 is 4×3.
7. A trap first-timers fall into
Do not write 4x² (forgot to multiply by 3) or 12x³ (forgot to drop the power).
1.2 · 12 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).
2.1 · 14 Marks
log₁₀ 100=
Notes: Base 10. Argument 100. Unknown: the exponent.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log₁₀ 100 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 100. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₁₀ 100 =. Circle that word — that is the missing variable.
Find
We are required to find: log₁₀ 100 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 10ˣ = 100 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log₁₀ 100 =.
In other words — as if you have never seen this before
log₁₀ 100 is the power you raise 10 to, to get 100. 10² = 100.
2. What do we already know?
Base 10. Argument 100. Unknown: the exponent.
3. Which method — and why this one?
log₁₀ 100 is the power you raise 10 to, to get 100. 10² = 100.
4. Step-by-step working
01 / 05
Rewrite as an exponential
log₁₀ 100=x⇔10ˣ=100.
10ˣ=100
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Recognise the power
100=10², so x=2.
x=2
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
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
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
2
Copy onto your answer sheet — in this order
Formula: 10ˣ=100
Answer: 2
6. Check it like a marker
10²=100. Yes.
7. A trap first-timers fall into
log₁₀ 100 is not 10, and not 1. Count the zeros of a power of ten: 100 has two zeros.
2.2 · 16 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.
5 · 3 Marks · Remainder & factor theorem
f(x)=x³−1. f(1)=
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: f(x) = x³ − 1. f(1) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 1. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: f(1) =. Circle that word — that is the missing variable.
Find
We are required to find: f(1) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 1 − 1 = 0, so (x − 1) is a factor. Then substitute the given, one line at a time, to get f(1) =.
In other words — as if you have never seen this before
1 − 1 = 0, so (x − 1) is a factor.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
1 − 1 = 0, so (x − 1) is a factor.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 1 · 1. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: f(1) =. Circle that word — that is the missing variable.
Find: We are required to find: f(1) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 1 − 1 = 0, so (x − 1) is a factor. Then substitute the given, one line at a time, to get f(1) =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 1 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
1 − 1 = 0, so (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.
05 / 08
Working
1 − 1 = 0, so (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.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
0
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
0
Copy onto your answer sheet — in this order
Answer: 0
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
6 · 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…”
7 · 3 Marks · Arithmetic & geometric series
Geometric a=27, r=13. S∞=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
Notes: First term a = 27. Ratio r = 13. Infinite geometric series. Unknown: S_∞.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Geometric a = 27, r = 13. S_∞ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 27 · 1/ · 3. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write |r| < 1 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives S_∞ =.
In other words — as if you have never seen this before
It converges only if |r| < 1. Here |13| < 1, so S_∞ = a(1 − r).
2. What do we already know?
First term a = 27. Ratio r = 13. Infinite geometric series. Unknown: S_∞.
3. Which method — and why this one?
It converges only if |r| < 1. Here |13| < 1, so S_∞ = a(1 − r).
4. Step-by-step working
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.
8 · 4 Marks · First principles of differentiation
From first principles, derivative of f(x)=x² at x=3 is
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: From first principles, derivative of f(x)=x² at x=3 is
In other words — as if you have never seen this before
f'=2x so at 3 is 6. Or lim [ (3+h)²−9]h=6.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3 is. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
f'=2x so at 3 is 6. Or lim [ (3+h)²−9]h=6.
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: 3 is. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
f'=2x so at 3 is 6. Or lim [ (3+h)²−9]h=6.
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
f'=2x so at 3 is 6. Or lim [ (3+h)²−9]h=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.
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
6
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
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
6
Copy onto your answer sheet — in this order
Answer: 6
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
9 · 3 Marks
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…”
10 · 3 Marks
S_∞ of 2 + 4 + 8 + 16 + … is
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of 2 + 4 + 8 + 16 + … is
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 4 · 8 · 16. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 2 + 4 + 8 + 16 + … is. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 2 + 4 + 8 + 16 + … is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: r = 2, |r| ≥ 1. Then substitute the given, one line at a time, to get S_∞ of 2 + 4 + 8 + 16 + … is.
In other words — as if you have never seen this before
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 4 · 8 · 16. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2 · 4 · 8 · 16. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 2 + 4 + 8 + 16 + … is. Circle that word — that is the missing variable.
Find: We are required to find: S_∞ of 2 + 4 + 8 + 16 + … is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: r = 2, |r| ≥ 1. Then substitute the given, one line at a time, to get S_∞ of 2 + 4 + 8 + 16 + … is.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 2 · 4 · 8 · 16. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
does not exist (diverges)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
does not exist (diverges)
Copy onto your answer sheet — in this order
Answer: does not exist (diverges)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
11 · 2 Marks · Arithmetic & geometric series
Arithmetic: a=5, d=3. T₈=
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: a=5. d=3. n=8. Unknown: Tn.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Arithmetic: a=5, d=3. T₈ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5 · 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: T₈ =. Circle that word — that is the missing variable.
Find
We are required to find: T₈ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write T₈ = 26 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives T₈ =.
In other words — as if you have never seen this before
T_n = a + (n − 1)d. Eight terms means 7 steps of d, not 8.
2. What do we already know?
a=5. d=3. n=8. Unknown: Tn.
3. Which method — and why this one?
T_n = a + (n − 1)d. Eight terms means 7 steps of d, not 8.
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 · 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: T₈ =. Circle that word — that is the missing variable.
Find: We are required to find: T₈ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write T₈ = 26 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives T₈ =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
a=5. d=3. n=8. Unknown: Tn.
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
T_n = a + (n − 1)d. Eight terms means 7 steps of d, not 8.
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
Formula
T₈=5+7×3=5+21=26.
T₈=26
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
26
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
26
Copy onto your answer sheet — in this order
Formula: T₈=26
Answer: 26
6. Check it like a marker
5,8,11,14,17,20,23,26. The eighth is 26.
7. A trap first-timers fall into
5 + 8×3 = 29 uses n instead of n−1.
12 · 2 Marks
∫ 6x² dx=
Notes: Indefinite integral of 6x².
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ∫ 6x² dx =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x². Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: ∫ 6x² dx =. Circle that word — that is the missing variable.
Find
We are required to find: ∫ 6x² dx =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 2x³ + C with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ∫ 6x² dx =.
In other words — as if you have never seen this before
Raise the index by 1, divide by the new index. Add C.
2. What do we already know?
Indefinite integral of 6x².
3. Which method — and why this one?
Raise the index by 1, divide by the new index. Add C.
4. Step-by-step working
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: 6x². Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: ∫ 6x² dx =. Circle that word — that is the missing variable.
Find: We are required to find: ∫ 6x² dx =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 2x³ + C with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives ∫ 6x² dx =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Indefinite integral of 6x².
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Raise the index by 1, divide by the new index. Add C.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Power integrate
6×x³3+C=2x³+C.
2x³+C
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
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
2x³ + C
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
2x³ + C
Copy onto your answer sheet — in this order
Formula: 2x³ + C
Answer: 2x³ + C
6. Check it like a marker
ddx (2x³) = 6x². Recovers the integrand.
7. A trap first-timers fall into
12x is the derivative of 6x², not the integral.
13 · 3 Marks · Compound angles
sin 75°=
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 75° =
Before any working — hunt first
Given
Numbers and units already in the sentence: 75°. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin 75° =. Circle that word — that is the missing variable.
Find
We are required to find: sin 75° =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: sin(45+30)=sin45cos30+cos45sin30. Then substitute the given, one line at a time, to get sin 75° =.
In other words — as if you have never seen this before
sin(45+30)=sin45cos30+cos45sin30.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 75°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
sin(45+30)=sin45cos30+cos45sin30.
4. Step-by-step working
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: 75°. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin 75° =. Circle that word — that is the missing variable.
Find: We are required to find: sin 75° =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: sin(45+30)=sin45cos30+cos45sin30. Then substitute the given, one line at a time, to get sin 75° =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 75°. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
sin(45+30)=sin45cos30+cos45sin30.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
sin(45+30)=sin45cos30+cos45sin30.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
(√6+√2)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
(√6+√2)4
Copy onto your answer sheet — in this order
Answer: (√6+√2)4
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
14 · 3 Marks
S_∞ of 9 + 3 + 1 + … =
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of 9 + 3 + 1 + … =
Before any working — hunt first
Given
Numbers and units already in the sentence: 9 · 3 · 1. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 9 + 3 + 1 + … =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 9 + 3 + 1 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: a=9, r=13. Then substitute the given, one line at a time, to get S_∞ of 9 + 3 + 1 + … =.
In other words — as if you have never seen this before
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 9 · 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
4. Step-by-step working
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: 9 · 3 · 1. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 9 + 3 + 1 + … =. Circle that word — that is the missing variable.
Find: We are required to find: S_∞ of 9 + 3 + 1 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: a=9, r=13. Then substitute the given, one line at a time, to get S_∞ of 9 + 3 + 1 + … =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 9 · 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
13.5
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
13.5
Copy onto your answer sheet — in this order
Answer: 13.5
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
15 · 4 Marks
Which formula is future value of an ordinary annuity?
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Which formula is future value of an ordinary annuity?
In other words — as if you have never seen this before
Money going in grows to F. Loans use P.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Money going in grows to F. Loans use P.
4. Step-by-step working
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
Money going in grows to F. Loans use P.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
Money going in grows to F. Loans use P.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
F=x[((1+i)ⁿ−1)/i]
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
F=x[((1+i)ⁿ−1)/i]
Copy onto your answer sheet — in this order
Answer: F=x[((1+i)ⁿ−1)/i]
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
16 · 2 Marks
Present value of an annuity is used for
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: Present value of an annuity is used for
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: Present value of an annuity is used for. Circle that word — that is the missing variable.
Find
We are required to find: Present value of an annuity is used for. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: P formula discounts future payments. Then substitute the given, one line at a time, to get Present value of an annuity is used for.
In other words — as if you have never seen this before
P formula discounts future payments.
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?
P formula discounts future payments.
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: Present value of an annuity is used for. Circle that word — that is the missing variable.
Find: We are required to find: Present value of an annuity is used for. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: P formula discounts future payments. Then substitute the given, one line at a time, to get Present value of an annuity is used for.
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
P formula discounts future payments.
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
P formula discounts future payments.
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
loanshow much you still owe
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
loanshow much you still owe
Copy onto your answer sheet — in this order
Answer: loanshow much you still owe
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 · 4 Marks
Stationary point of f(x)=x³−3x+1 is at x =
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Stationary point of f(x)=x³−3x+1 is at x =
In other words — as if you have never seen this before
f'=3x²−3=0→x²=1→x=±1.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3x · 1 is. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
f'=3x²−3=0→x²=1→x=±1.
4. Step-by-step working
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: 3x · 1 is. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
f'=3x²−3=0→x²=1→x=±1.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
f'=3x²−3=0→x²=1→x=±1.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
±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
±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…”
18 · 2 Marks · Arithmetic & geometric series
Geometric a=5, r=0.4. S∞=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Geometric a=5, r=0.4. S_∞ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5 · 0.4. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 8. Then substitute the given, one line at a time, to get S_∞ =.
In other words — as if you have never seen this before
8.33
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5 · 0.4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
8.33
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 · 0.4. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ =. Circle that word — that is the missing variable.
Find: We are required to find: S_∞ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 8. 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: 5 · 0.4. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
8.33
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
S=5(1−0.4)=8.333…
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
8.33
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
8.33
Copy onto your answer sheet — in this order
Answer: 8.33
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
19 · 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…”
20 · 3 Marks · Area, sine and cosine rules
If ΔABC ∼ ΔDEF with ratio 2:3, area ratio is
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: If ΔABC ∼ ΔDEF with ratio 2:3, area ratio is
In other words — as if you have never seen this before
Areas scale with the square of corresponding sides.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 3. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Areas scale with the square of corresponding sides.
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: 2 · 3. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
Areas scale with the square of corresponding sides.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
Areas scale with the square of corresponding sides.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
4:9
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
4:9
Copy onto your answer sheet — in this order
Answer: 4:9
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
21 · 2 Marks · 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…”
22 · 2 Marks
r=−0.9 means
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: r = −0.9 means
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.9 means. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 9 means. Circle that word — that is the missing variable.
Find
We are required to find: 9 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Close to −1: strong negative. Then substitute the given, one line at a time, to get 9 means.
In other words — as if you have never seen this before
Close to −1: strong negative.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0.9 means. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Close to −1: strong negative.
4. Step-by-step working
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: 0.9 means. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: 9 means. Circle that word — that is the missing variable.
Find: We are required to find: 9 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Close to −1: strong negative. Then substitute the given, one line at a time, to get 9 means.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 0.9 means. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Close to −1: strong negative.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Close to −1: strong negative.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
strong negative linear correlation
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
strong negative linear correlation
Copy onto your answer sheet — in this order
Answer: strong negative linear correlation
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
23 · 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.
24 · 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…”
25 · 3 Marks
5C2=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 5C2 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5C · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 5C2 =. Circle that word — that is the missing variable.
Find
We are required to find: 5C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 10. Then substitute the given, one line at a time, to get 5C2 =.
In other words — as if you have never seen this before
10
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5C · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
10
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: 5C · 2. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: 5C2 =. Circle that word — that is the missing variable.
Find: We are required to find: 5C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 10. Then substitute the given, one line at a time, to get 5C2 =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 5C · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
10
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
5!(2!3!)=10. Order does not matter.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
10
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
10
Copy onto your answer sheet — in this order
Answer: 10
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
26 · 4 Marks · Area, sine and cosine rules
Area of Δ with a=6, b=10, included angle 90° is
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: Area of Δ with a=6, b=10, included angle 90° is
In other words — as if you have never seen this before
½ab sin90=½×6×10×1=30.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6 · 10 · 90°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
½ab sin90=½×6×10×1=30.
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: 6 · 10 · 90°. 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
½ab sin90=½×6×10×1=30.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
½ab sin90=½×6×10×1=30.
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
30
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
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
30
Copy onto your answer sheet — in this order
Answer: 30
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
27 · 3 Marks · Similarity & proportion
AA similarity requires
Similarity & proportion
Formulas
AA similarity ⇒ corresponding sides proportional
AA similarity
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: AA similarity requires
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: AA similarity requires. Circle that word — that is the missing variable.
Find
We are required to find: AA similarity requires. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: two equal corresponding angles. Then substitute the given, one line at a time, to get AA similarity requires.
In other words — as if you have never seen this before
two equal corresponding angles
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
two equal corresponding angles
4. Step-by-step working
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: AA similarity requires. Circle that word — that is the missing variable.
Find: We are required to find: AA similarity requires. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: two equal corresponding angles. Then substitute the given, one line at a time, to get AA similarity requires.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
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
two equal corresponding angles
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
AA is sufficient.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
two equal corresponding angles
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
two equal corresponding angles
Copy onto your answer sheet — in this order
Answer: two equal corresponding angles
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
28 · 4 Marks · Compound angles
A vertical pole 8 m, observer 15 m away. Angle of elevation tan⁻¹(8/15) ≈
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: A vertical pole 8 m, observer 15 m away. Angle of elevation tan⁻¹(8/15) ≈
Before any working — hunt first
Given
Numbers and units already in the sentence: 8 m · 15 m · 8/ · 15. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Angle of elevation tan⁻¹(8/15) ≈. Circle that word — that is the missing variable.
Find
We are required to find: Angle of elevation tan⁻¹(8/15) ≈. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: tan θ = oppositeadjacent=815. Then substitute the given, one line at a time, to get Angle of elevation tan⁻¹(8/15) ≈.
In other words — as if you have never seen this before
tan θ = oppositeadjacent=815.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 8 m · 15 m · 8/ · 15. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
tan θ = oppositeadjacent=815.
4. Step-by-step working
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 m · 15 m · 8/ · 15. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Angle of elevation tan⁻¹(8/15) ≈. Circle that word — that is the missing variable.
Find: We are required to find: Angle of elevation tan⁻¹(8/15) ≈. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: tan θ = oppositeadjacent=815. Then substitute the given, one line at a time, to get Angle of elevation tan⁻¹(8/15) ≈.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 8 m · 15 m · 8/ · 15. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
tan θ = oppositeadjacent=815.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
tan θ = oppositeadjacent=815.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
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
28.1°
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
28.1°
Copy onto your answer sheet — in this order
Answer: 28.1°
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
29 · 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…”
30 · 3 Marks · Arithmetic & geometric series
Geometric a=5, r=2. T₄=
Formulas
Tₙ=a rⁿ⁻¹
Geometric nth term — exponent is n minus 1
Sₙ=n2 [2a + (n−1)d]
Arithmetic sum
Sₙ=a(rⁿ − 1)(r − 1), r ≠ 1
Geometric sum
Notes: First term 5, common ratio 2. Fourth term.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Geometric a=5, r=2. T₄ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5 · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: T₄ =. Circle that word — that is the missing variable.
Find
We are required to find: T₄ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write Tₙ = a rⁿ⁻¹ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives T₄ =.
In other words — as if you have never seen this before
Tₙ=a rⁿ⁻¹. T₄=5×2³=5×8=40.
2. What do we already know?
First term 5, common ratio 2. Fourth term.
3. Which method — and why this one?
Tₙ=a rⁿ⁻¹. T₄=5×2³=5×8=40.
4. Step-by-step working
01 / 05
Write Tₙ = arⁿ⁻¹
The exponent is n minus 1, not n.
Tₙ=a rⁿ⁻¹
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Substitute
T₄=5×2³=40.
T₄=40
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
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
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
Copy onto your answer sheet — in this order
Formula: Tₙ=a rⁿ⁻¹
Answer: 40
6. Check it like a marker
Sequence: 5, 10, 20, 40. Fourth term 40.
7. A trap first-timers fall into
2⁴ = 16 would give T₅ if you use rⁿ instead of rⁿ⁻¹.