Grade 12 Maths · Paper 1

Calculus Grade 12

Real past paper questions and answers from 2024 to 2025, with each memo split into the ticks examiners give. Try one, mark yourself, and see exactly which skill cost you marks.

past paper questions: 80 past paper questions
skills: 20 skills
videos: 20 videos

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Question 9.1 · May/June 2025 · Paper 1 · National (DBE) · 2 marks

Given:

f (x) = x^{3} - 6 x^{2} + 9 x - 4 = (x - 4) (x - k)^{2}

9.1 Show that

k = 1

.(2)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

What's in Calculus Grade 12

Each skill is one thing a memo gives a tick for. Tap a skill for its own questions, notes and videos.

Differentiation13 past paper questions

Tangents to Graphs of Functions7 past paper questions

Stationary Points and Points of Inflection12 past paper questions

Sketching Graphs of Cubic Functions5 past paper questions

Graph Interpretation28 past paper questions

Maxima and Minima Problems4 past paper questions

Find stationary pointsFind the turning points of a curve by setting the derivative equal to 0 and solving, then say which are maximums and which are minimums.10 past paper questions · 3 videos

Rates of Change

Find a rate of changeUse the derivative to find how fast something is changing at a given moment (for example speed is the derivative of distance with respect to time).3 videos

Calculus of Motion1 past paper questions

Find a rate of changeUse the derivative to find how fast something is changing at a given moment (for example speed is the derivative of distance with respect to time).3 videos

More Calculus Grade 12 questions and answers

Try each one on paper before you open the memo.

1. Question 9.1 · November 2024 · Paper 1 · National (DBE) · 2 marks
$A (1; 9)$ and $B (\frac{5}{2}; 8)$ are the turning points of graph $f$ below. $C (0; 7)$ is the y-intercept of $f$ .
9.1 For which values of $x$ is $f$ decreasing?(2)
Show the memo (1 tick)Memo
1. ✓
  Identify interval between turning pointsDecreasing between local max at $x = 1$ and local min at $x = \frac{5}{2}$ .
  $1 < x < \frac{5}{2}$
  2 marks · State increasing and decreasing intervals
2. Question 9.1 · May/June 2024 · Paper 1 · National (DBE) · 3 marks
The graph of $f (x) = a x^{3} + b x^{2} + c x - 5$ is drawn below. $E (- 1; 0)$ and $G (5; 0)$ are the $x$ -intercepts of $f$ .
9.1 Show that $a = 1$ , $b = - 3$ and $c = - 9$ .(3)
Show the memo (3 ticks)Memo
1. ✓
  Repeated factor at the turning point E $(x + 1)^{2}$
  1 mark · Find the equation of a graph
2. ✓
  Equation in factor form $f (x) = a (x + 1)^{2} (x - 5)$
  1 mark · Find the equation of a graph
3. ✓
  Substitution of $(0; - 5)$ $- 5 = a (0 + 1)^{2} (0 - 5) ⟹ a = 1$
  $f (x) = x^{3} - 3 x^{2} - 9 x - 5$
  1 mark · Find the equation of a graph
3. Question 7.4.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks
In the diagram, the graphs of $f (x) = 2 cos x + 1$ and $g (x) = sin 2 x$ are drawn for the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
7.4 Use the graphs to determine the values of x, in the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ for which:
7.4.1 $g (x) \cdot f^{'} (x) < 0$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Analyze the conditions for the product to be negativeFor $g (x) \cdot f^{'} (x) < 0$ , the signs of $g (x)$ and $f^{'} (x)$ must be different (+ and - OR - and +).
  $f^{'} (x)$ represents the gradient of $f$ . It is positive when $f$ is increasing ( $- 18 0^{\circ} < x < 0^{\circ}$ ) and negative when $f$ is decreasing ( $0^{\circ} < x < 18 0^{\circ}$ ).
  
  Condition 1: $g (x) > 0$ and $f^{'} (x) < 0$
  $g (x)$ is positive for $(0^{\circ}; 9 0^{\circ}) \cup (- 18 0^{\circ}; - 9 0^{\circ})$ .
  $f^{'} (x)$ is negative for $(0^{\circ}; 18 0^{\circ})$ .
  Intersection: $x \in (0^{\circ}; 9 0^{\circ})$
  
  Condition 2: $g (x) < 0$ and $f^{'} (x) > 0$
  $g (x)$ is negative for $(9 0^{\circ}; 18 0^{\circ}) \cup (- 9 0^{\circ}; 0^{\circ})$ .
  $f^{'} (x)$ is positive for $(- 18 0^{\circ}; 0^{\circ})$ .
  Intersection: $x \in (- 9 0^{\circ}; 0^{\circ})$
  1 mark · Solve product inequality from graphs
2. ✓
  State final union of intervals $x \in (- 9 0^{\circ}; 0^{\circ}) \cup (0^{\circ}; 9 0^{\circ})$
  
  OR
  
  $- 9 0^{\circ} < x < 0^{\circ}$ or $0^{\circ} < x < 9 0^{\circ}$
  1 mark · Solve product inequality from graphs
4. Question 10.1 · May/June 2024 · Paper 1 · National (DBE) · 4 marks
EHGF is a rectangle. HE is produced $x^{2}$ cm to N and EH is produced $x^{2}$ cm to P. NF produced intersects PG produced at M to form an isosceles triangle MNP with NM = MP. D lies on NP where MD $⊥$ NP. NP = 4 cm and MD = 3 cm.
10.1 Show that the area of $EFG H$ is given by $A (x) = 6 x^{2} - 3 x^{4}$ .(4)
Show the memo (4 ticks)Memo
1. ✓
  Determine length of $E H$ using the base of triangle $MNP$ $E H = 4 - 2 x^{2}$
  1 mark · Set up an expression or equation
2. ✓
  Ratio for the height GH $tan P = \frac{3}{2} = \frac{G H}{x ^{2}}$
  1 mark · Calculate a side using trig
3. ✓
  Height b $b = \frac{3 x ^{2}}{2}$
  1 mark · Calculate a side using trig
4. ✓
  Multiply length by breadth and simplify $A (x) = (1, 5 x^{2}) (4 - 2 x^{2}) = 6 x^{2} - 3 x^{4}$
  1 mark · Set up an expression or equation
5. Question 8.1 · September 2024 · Paper 1 · Provincial · 4 marks
8.1 If $f (x) = - 2 x^{2} + 3 x$ , determine $f^{'} (x)$ from first principles.(4)
Show the memo (4 ticks)Memo
1. ✓
  Substitute x+h $- 2 (x + h)^{2} + 3 (x + h)$
  1 mark · Differentiate from first principles
2. ✓
  SimplificationNumerator simplifies to $- 4 x h - 2 h^{2} + 3 h$
  1 mark · Differentiate from first principles
3. ✓
  Factorise and limit $lim_{h \to 0} (- 4 x - 2 h + 3)$
  1 mark · Differentiate from first principles
4. ✓
  Answer $- 4 x + 3$
  1 mark · Differentiate from first principles