LVL7 NSC 2026
S T U DY PACKS CAPS
Grade 12 Mathematics
Paper 1
Final Exam Prep
Memos included
57 exam-style questions
315 marks of practice
Full worked memos, mark by mark
Every numerical answer computer-checked
Algebra ⋅ Sequences & Series ⋅ Functions & Graphs ⋅ Finance ⋅ Differential Calculus ⋅ Probability
,LVL7 Grade 12 Mathematics ⋅ Paper 1 How the memos work
INSIDE THE PACK
How the worked memos work
Every question in this pack has a worked memo laid out like the one below: the working an examiner
expects, a tick 3 beside each line that earns a mark, and the mark allocation underneath. This is
question F1, exactly as it appears in Topic 3.
THE QUESTION
F1 Parabola and straight line from Topic 3, page 8
The sketch shows the graphs of 𝑓 (𝑥) = −𝑥 2 + 2𝑥 + 8 and y
𝑔(𝑥) = 𝑥 + 2. 𝐴 and 𝐵 are the 𝑥-intercepts of 𝑓, 𝐶 is the C
turning point of 𝑓, and 𝑓 and 𝑔 intersect at 𝐵 and 𝐷. g
(a) Determine the coordinates of 𝐴 and 𝐵. (3)
D
(b) Determine the coordinates of 𝐶. (2)
(c) Write down the range of 𝑓. (1)
(d) Determine the coordinates of 𝐷. (3) B A
O
(e) For which values of 𝑥 is 𝑓 (𝑥) > 𝑔(𝑥)? (2) x
(f) For which values of 𝑘 will 𝑓 (𝑥) = 𝑘 have two un- f
equal positive roots? (3)
[14]
THE WORKED MEMO
F1 Parabola and straight line memo page 27
(a) −𝑥 2 + 2𝑥 + 8 = 0 ⇒ 𝑥 2 − 2𝑥 − 8 = 0 ⇒ (𝑥 − 4)(𝑥 + 2) = 0 33
𝐵(−2; 0) and 𝐴(4; 0) 3 (B is the intercept shared with 𝑔, since 𝑔(−2) = 0.)
𝑏 −2
(b) 𝑥 = − = = 1; 𝑓 (1) = −1 + 2 + 8 = 9. 𝐶(1; 9) 33
2𝑎 −2
(c) 𝑦 ≤ 9 (or (−∞; 9]) 3
(d) −𝑥 2 + 2𝑥 + 8 = 𝑥 + 2 ⇒ 𝑥 2 − 𝑥 − 6 = 0 3⇒ (𝑥 − 3)(𝑥 + 2) = 0 3
𝑥 = 3: 𝑦 = 5. 𝐷(3; 5) 3
(e) −2 < 𝑥 < 3 33
(f) The horizontal line 𝑦 = 𝑘 must cut 𝑓 twice to the right of the 𝑦-axis: above the 𝑦-intercept 8 and
below the maximum 9. 3
8 < 𝑘 < 9 33
Mark allocation (a) 2 equation & factors, 1 coordinates; (b) 1 x, 1 y; (c) 1; (d) 1 equation, 1 factors, 1 Total 14
point; (e) 2; (f) 1 reasoning, 2 answer
© 2026 LVL7 Study Packs ⋅ Licensed for one learner’s personal use. Do not share or resell. 2
,LVL7 Grade 12 Mathematics ⋅ Paper 1 Contents
GRADE 12 MATHEMATICS ⋅ PAPER 1 FINAL EXAM PREP
Contents
57 315 6 100%
original exam-style marks, NSC-style topics, in exam order of questions with a worked
questions allocation memo
Topic Questions No. Marks Questions Memo
page page
1 Algebra, Equations & Inequalities A1–A11 11 46 4 19
2 Sequences & Series S1–S10 10 56 6 23
3 Functions & Graphs F1–F8 8 55 8 27
4 Finance, Growth & Decay M1–M9 9 46 11 31
5 Differential Calculus C1–C10 10 64 13 34
6 Probability & Counting P1–P9 9 48 16 38
Total 57 315
How the worked memos are marked (a real question from this pack) 2
How to use this pack, exam conventions, about this pack 41
Mark Tracker and NSC level calculator 42
What is in this pack Who it is for
• 57 original questions (no recycled • Learners: exam-style practice by topic,
past-paper questions), 315 marks (over two timed by the marks, with memos you can
full Paper 1s’ worth of practice) mark yourself against.
• All six Paper 1 topics, in the order and style • Parents: a complete, self-marking pack.
of the NSC paper Every question has a worked memo, and the
• A topic toolkit of must-know facts and Mark Tracker turns scores into an NSC
methods before each topic level.
• A worked memo for every question, with • Tutors: original questions with NSC-style
the tick-by-tick mark allocation mark allocations, and question and memo
• Sketches for the graph questions, drawn to page references for every item.
scale
• Every numerical answer re-checked by
computer before publication (page 41)
• A Mark Tracker that converts your score
into an NSC level
© 2026 LVL7 Study Packs ⋅ Licensed for one learner’s personal use. Do not share or resell. 3
,PART A ⋅ TOPIC 1
Algebra, Equations & Inequalities
11 questions ⋅ 46 marks ⋅ about 55 minutes
Questions A1–A11 ⋅ worked memo for this topic starts on page 19
TOPIC TOOLKIT
−𝑏 ± √𝑏 2 − 4𝑎𝑐
• Quadratic formula: 𝑥 = . Always write the equation in standard form 𝑎𝑥 2 +𝑏𝑥+𝑐 =
2𝑎
0 first.
• Nature of roots from Δ = 𝑏 2 − 4𝑎𝑐: Δ < 0 non-real; Δ = 0 real & equal; Δ > 0 real & unequal; Δ a
perfect square ⇒ rational.
• Quadratic inequalities: get 0 on one side, find the critical values, then use a sketch or number
line.
• Surd equations: isolate the root, square both sides, and always check for extraneous answers.
• Exponents: same base ⇒ equate exponents. Factor out the smallest power: 3𝑥+2 − 3𝑥 = 3𝑥 (9 − 1).
A1 Factorising memo p. 19
Solve for 𝑥: 𝑥(𝑥 − 3) = 10 (3)
A2 Quadratic formula memo p. 19
Solve for 𝑥, correct to TWO decimal places: 2𝑥 2 − 5𝑥 − 1 = 0 (4)
A3 Quadratic inequality memo p. 19
Solve for 𝑥: (𝑥 + 2)(3 − 𝑥) < 0 (3)
A4 Surd equation memo p. 20
Solve for 𝑥: √𝑥 + 7 = 𝑥 − 5 (5)
A5 Simultaneous equations memo p. 20
Solve for 𝑥 and 𝑦 simultaneously:
𝑦 =𝑥 −1 and 𝑥 2 − 2𝑥𝑦 + 3𝑦 2 = 9
(6)
A6 Exponential equation memo p. 20
Solve for 𝑥: 3𝑥+2 − 3𝑥 = 72 (4)
© 2026 LVL7 Study Packs ⋅ Licensed for one learner’s personal use. Do not share or resell. 4
S T U DY PACKS CAPS
Grade 12 Mathematics
Paper 1
Final Exam Prep
Memos included
57 exam-style questions
315 marks of practice
Full worked memos, mark by mark
Every numerical answer computer-checked
Algebra ⋅ Sequences & Series ⋅ Functions & Graphs ⋅ Finance ⋅ Differential Calculus ⋅ Probability
,LVL7 Grade 12 Mathematics ⋅ Paper 1 How the memos work
INSIDE THE PACK
How the worked memos work
Every question in this pack has a worked memo laid out like the one below: the working an examiner
expects, a tick 3 beside each line that earns a mark, and the mark allocation underneath. This is
question F1, exactly as it appears in Topic 3.
THE QUESTION
F1 Parabola and straight line from Topic 3, page 8
The sketch shows the graphs of 𝑓 (𝑥) = −𝑥 2 + 2𝑥 + 8 and y
𝑔(𝑥) = 𝑥 + 2. 𝐴 and 𝐵 are the 𝑥-intercepts of 𝑓, 𝐶 is the C
turning point of 𝑓, and 𝑓 and 𝑔 intersect at 𝐵 and 𝐷. g
(a) Determine the coordinates of 𝐴 and 𝐵. (3)
D
(b) Determine the coordinates of 𝐶. (2)
(c) Write down the range of 𝑓. (1)
(d) Determine the coordinates of 𝐷. (3) B A
O
(e) For which values of 𝑥 is 𝑓 (𝑥) > 𝑔(𝑥)? (2) x
(f) For which values of 𝑘 will 𝑓 (𝑥) = 𝑘 have two un- f
equal positive roots? (3)
[14]
THE WORKED MEMO
F1 Parabola and straight line memo page 27
(a) −𝑥 2 + 2𝑥 + 8 = 0 ⇒ 𝑥 2 − 2𝑥 − 8 = 0 ⇒ (𝑥 − 4)(𝑥 + 2) = 0 33
𝐵(−2; 0) and 𝐴(4; 0) 3 (B is the intercept shared with 𝑔, since 𝑔(−2) = 0.)
𝑏 −2
(b) 𝑥 = − = = 1; 𝑓 (1) = −1 + 2 + 8 = 9. 𝐶(1; 9) 33
2𝑎 −2
(c) 𝑦 ≤ 9 (or (−∞; 9]) 3
(d) −𝑥 2 + 2𝑥 + 8 = 𝑥 + 2 ⇒ 𝑥 2 − 𝑥 − 6 = 0 3⇒ (𝑥 − 3)(𝑥 + 2) = 0 3
𝑥 = 3: 𝑦 = 5. 𝐷(3; 5) 3
(e) −2 < 𝑥 < 3 33
(f) The horizontal line 𝑦 = 𝑘 must cut 𝑓 twice to the right of the 𝑦-axis: above the 𝑦-intercept 8 and
below the maximum 9. 3
8 < 𝑘 < 9 33
Mark allocation (a) 2 equation & factors, 1 coordinates; (b) 1 x, 1 y; (c) 1; (d) 1 equation, 1 factors, 1 Total 14
point; (e) 2; (f) 1 reasoning, 2 answer
© 2026 LVL7 Study Packs ⋅ Licensed for one learner’s personal use. Do not share or resell. 2
,LVL7 Grade 12 Mathematics ⋅ Paper 1 Contents
GRADE 12 MATHEMATICS ⋅ PAPER 1 FINAL EXAM PREP
Contents
57 315 6 100%
original exam-style marks, NSC-style topics, in exam order of questions with a worked
questions allocation memo
Topic Questions No. Marks Questions Memo
page page
1 Algebra, Equations & Inequalities A1–A11 11 46 4 19
2 Sequences & Series S1–S10 10 56 6 23
3 Functions & Graphs F1–F8 8 55 8 27
4 Finance, Growth & Decay M1–M9 9 46 11 31
5 Differential Calculus C1–C10 10 64 13 34
6 Probability & Counting P1–P9 9 48 16 38
Total 57 315
How the worked memos are marked (a real question from this pack) 2
How to use this pack, exam conventions, about this pack 41
Mark Tracker and NSC level calculator 42
What is in this pack Who it is for
• 57 original questions (no recycled • Learners: exam-style practice by topic,
past-paper questions), 315 marks (over two timed by the marks, with memos you can
full Paper 1s’ worth of practice) mark yourself against.
• All six Paper 1 topics, in the order and style • Parents: a complete, self-marking pack.
of the NSC paper Every question has a worked memo, and the
• A topic toolkit of must-know facts and Mark Tracker turns scores into an NSC
methods before each topic level.
• A worked memo for every question, with • Tutors: original questions with NSC-style
the tick-by-tick mark allocation mark allocations, and question and memo
• Sketches for the graph questions, drawn to page references for every item.
scale
• Every numerical answer re-checked by
computer before publication (page 41)
• A Mark Tracker that converts your score
into an NSC level
© 2026 LVL7 Study Packs ⋅ Licensed for one learner’s personal use. Do not share or resell. 3
,PART A ⋅ TOPIC 1
Algebra, Equations & Inequalities
11 questions ⋅ 46 marks ⋅ about 55 minutes
Questions A1–A11 ⋅ worked memo for this topic starts on page 19
TOPIC TOOLKIT
−𝑏 ± √𝑏 2 − 4𝑎𝑐
• Quadratic formula: 𝑥 = . Always write the equation in standard form 𝑎𝑥 2 +𝑏𝑥+𝑐 =
2𝑎
0 first.
• Nature of roots from Δ = 𝑏 2 − 4𝑎𝑐: Δ < 0 non-real; Δ = 0 real & equal; Δ > 0 real & unequal; Δ a
perfect square ⇒ rational.
• Quadratic inequalities: get 0 on one side, find the critical values, then use a sketch or number
line.
• Surd equations: isolate the root, square both sides, and always check for extraneous answers.
• Exponents: same base ⇒ equate exponents. Factor out the smallest power: 3𝑥+2 − 3𝑥 = 3𝑥 (9 − 1).
A1 Factorising memo p. 19
Solve for 𝑥: 𝑥(𝑥 − 3) = 10 (3)
A2 Quadratic formula memo p. 19
Solve for 𝑥, correct to TWO decimal places: 2𝑥 2 − 5𝑥 − 1 = 0 (4)
A3 Quadratic inequality memo p. 19
Solve for 𝑥: (𝑥 + 2)(3 − 𝑥) < 0 (3)
A4 Surd equation memo p. 20
Solve for 𝑥: √𝑥 + 7 = 𝑥 − 5 (5)
A5 Simultaneous equations memo p. 20
Solve for 𝑥 and 𝑦 simultaneously:
𝑦 =𝑥 −1 and 𝑥 2 − 2𝑥𝑦 + 3𝑦 2 = 9
(6)
A6 Exponential equation memo p. 20
Solve for 𝑥: 3𝑥+2 − 3𝑥 = 72 (4)
© 2026 LVL7 Study Packs ⋅ Licensed for one learner’s personal use. Do not share or resell. 4
