Grade 12 Maths Paper 1: Every Topic You Need to Know [CAPS 2026]

Paper 1 is the longer and generally harder of the two NSC Maths papers. It covers pure algebra and the four big applied topics: functions and graphs, financial mathematics, differential calculus, and probability. Together these account for 150 marks — and your performance here will largely determine your final maths symbol.

This guide covers every topic in CAPS Paper 1 with the key formulas, what the examiners actually test, and the mistakes to avoid.

Paper 1 at a Glance

| Topic | Approximate marks | |---|---| | Algebra, equations, and inequalities | 25 | | Patterns, sequences, and series | 25 | | Functions and graphs | 35 | | Financial mathematics | 15 | | Differential calculus | 35 | | Probability | 15 | | Total | 150 |

These allocations shift slightly from year to year, but they give you a reliable sense of where to invest your time. Calculus and functions together are worth roughly half the paper.

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Topic 1: Algebra, Equations, and Inequalities

This section tests whether you can manipulate expressions fluently and solve a range of equation types. It sounds like revision, but the Paper 1 versions are harder than anything in Grade 11.

What's tested

• Simplifying expressions with surds and exponents
• Solving quadratic equations (factoring, quadratic formula, completing the square)
• Simultaneous equations (one linear, one quadratic)
• Nature of roots using the discriminant
• Inequalities (quadratic and rational)

Key formulas

Quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Discriminant and nature of roots:

$\Delta = b^2 - 4ac$

| $\Delta$ | Nature of roots | |---|---| | $\Delta > 0$ | Two real, unequal roots | | $\Delta = 0$ | Two real, equal roots | | $\Delta < 0$ | No real roots | | $\Delta > 0$ and a perfect square | Rational roots |

Exam tips

The most common error in this section is handling surds incorrectly — particularly $\sqrt{a^2 + b^2} \neq a + b$. The square root of a sum is not the sum of the square roots. Write this on a sticky note if you need to.

For inequalities, always sketch a number line or parabola to determine the correct direction of the inequality. Sign errors from skipping this step cost hundreds of students marks every year.

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Topic 2: Patterns, Sequences, and Series

Sequences require you to work fluently with both arithmetic and geometric progressions — their general terms, sums, and limits.

Arithmetic sequences

General term:

$T_n = a + (n-1)d$

Sum of first $n$ terms:

$S_n = \frac{n}{2}[2a + (n-1)d] \qquad \text{or} \qquad S_n = \frac{n}{2}(a + l)$

where $a$ is the first term, $d$ is the common difference, and $l$ is the last term.

Geometric sequences

General term:

$T_n = ar^{n-1}$

Sum of first $n$ terms:

$S_n = \frac{a(r^n - 1)}{r - 1} \qquad (r \neq 1)$

Sum to infinity (only valid when $|r| < 1$):

$S_\infty = \frac{a}{1 - r}$

What's tested

The examiners love to combine both types in one question — for example, giving you a sequence where alternate terms form an arithmetic progression and the others a geometric one. Slowing down to identify the pattern before applying any formula will save you from using the wrong one.

Sigma notation ($\sum$) appears regularly. Remember that $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$ and practise converting sigma expressions to standard form before summing.

Exam tips

Write down $a$, $d$ or $r$, and what you're solving for before touching a formula. Students who skip this step often substitute into the wrong formula or use $n$ and $T_n$ the wrong way around.

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Topic 3: Functions and Graphs

Functions carries the most marks of any single section in Paper 1. You need to work fluently with five function families and understand how transformations affect each one.

Function families and their forms

| Function | Standard form | Key features | |---|---|---| | Quadratic (parabola) | $f(x) = a(x-p)^2 + q$ | Vertex at $(p, q)$, axis $x = p$ | | Hyperbola | $f(x) = \frac{a}{x-p} + q$ | Asymptotes $x = p$, $y = q$ | | Exponential | $f(x) = ab^{x-p} + q$ | Asymptote $y = q$ | | Logarithm | $f(x) = \log_b(x)$ | Inverse of exponential | | Straight line | $f(x) = mx + c$ | Gradient $m$, $y$-intercept $c$ |

Transformations

For any function $f(x)$:

• $f(x) + k$ shifts up by $k$ units
• $f(x-h)$ shifts right by $h$ units
• $-f(x)$ reflects in the $x$-axis
• $f(-x)$ reflects in the $y$-axis

What's tested

Questions typically give you a sketch with some information and ask you to determine the equation, or give the equation and ask for features (intercepts, asymptotes, turning points, domain, range). Intersections between functions are heavily tested — usually set the two expressions equal and solve.

The inverse function appears in almost every exam. Remember:

• The inverse of $y = b^x$ is $y = \log_b x$
• To find the inverse, swap $x$ and $y$ and solve for $y$
• The graph of the inverse is a reflection of the original in the line $y = x$

Exam tips

Always state the domain and range when asked — examiners award marks for these explicitly. The range of $f(x) = \frac{a}{x-p} + q$ is $y \in \mathbb{R},\ y \neq q$, not just "$y \neq q$". Be precise.

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Topic 4: Financial Mathematics

Finance is one of the most practically useful sections in the syllabus and one of the most reliably scoreable — the formula structure is predictable and the calculations are methodical.

Core formulas

Simple interest (growth and decay):

$A = P(1 + in) \qquad \text{and} \qquad A = P(1 - in)$

Compound interest (growth and decay):

$A = P(1 + i)^n \qquad \text{and} \qquad A = P(1 - i)^n$

Effective vs nominal interest rate (compounded $m$ times per year):

$(1 + i_{\text{eff}}) = \left(1 + \frac{i_{\text{nom}}}{m}\right)^m$

Future value of an annuity (regular payments in):

$F = \frac{x\left[(1+i)^n - 1\right]}{i}$

Present value of an annuity (loan repayment):

$P = \frac{x\left[1-(1+i)^{-n}\right]}{i}$

where $x$ is the payment per period, $i$ is the interest rate per period, and $n$ is the number of periods.

What's tested

The NSC examiners consistently test:

1. Compound growth over multiple periods with a mid-term change in rate
2. Sinking funds (future value annuity to replace an asset)
3. Loan amortisation (present value annuity with outstanding balance calculation)
4. Nominal vs effective rate conversions

Exam tips

The most common mistake is using the annual rate instead of the periodic rate. If interest is compounded monthly at 12% per year, then $i = 0.12 / 12 = 0.01$ per month and $n$ is the number of months, not years. Always convert before substituting.

Write your formula, substitute, and show every step. Finance questions carry multiple method marks — you can lose the answer but still get 4 out of 6 if your working is clear.

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Topic 5: Differential Calculus

Calculus is the topic students fear most, but it's also one of the most predictable in terms of what gets asked. Master the three sub-sections and you can reliably pick up 25+ marks here.

First principles

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

First principles questions appear almost every year and are worth 5–6 marks. You must expand $f(x+h)$, subtract $f(x)$, simplify, and then take the limit.

Differentiation rules

Power rule:

$\frac{d}{dx}\left[x^n\right] = nx^{n-1}$

Scalar multiple, sum, and difference rules follow directly:

$\frac{d}{dx}\left[af(x) \pm bg(x)\right] = af'(x) \pm bg'(x)$

Before differentiating, always simplify to the form $ax^n$ — expand brackets, split fractions, convert roots to exponents. Trying to differentiate $\frac{x^2 + 3x}{x}$ without simplifying first is a common source of errors.

Cubic graphs and optimisation

A cubic function $f(x) = ax^3 + bx^2 + cx + d$ has:

• Stationary points where $f'(x) = 0$
• Inflection point where $f''(x) = 0$
• Increasing where $f'(x) > 0$, decreasing where $f'(x) < 0$

| Condition | Meaning | |---|---| | $f'(x) = 0$ | Stationary point (local max or min) | | $f''(x) > 0$ at stationary point | Local minimum | | $f''(x) < 0$ at stationary point | Local maximum | | $f''(x) = 0$ | Point of inflection |

Optimisation questions give a real-world scenario (volume of a box, cost of a container) and ask you to maximise or minimise a quantity. Always:

1. Write the constraint equation
2. Express the objective function in one variable
3. Differentiate and set equal to zero
4. Verify it's a maximum or minimum using the second derivative

Exam tips

The most common error is forgetting to rewrite a function before differentiating. $\sqrt{x} = x^{1/2}$, $\frac{1}{x^2} = x^{-2}$, and $\frac{3}{x} = 3x^{-1}$. If you're differentiating something that looks like a fraction or a root, rewrite it first, every time.

Also note: the chain rule, product rule, and quotient rule are not in the CAPS Grade 12 syllabus. If you think you need them, you've probably not simplified enough.

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Topic 6: Probability

Probability in Paper 1 focuses on counting principles and formal probability rules — not just intuition.

Core rules

Complement:

$P(A') = 1 - P(A)$

Addition rule:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

For mutually exclusive events: $P(A \cap B) = 0$, so:

$P(A \cup B) = P(A) + P(B)$

For independent events:

$P(A \cap B) = P(A) \times P(B)$

Counting principles

Fundamental counting principle: if there are $m$ ways to do one thing and $n$ ways to do another, there are $m \times n$ ways to do both.

Permutations (arrangements where order matters):

${}^nP_r = \frac{n!}{(n-r)!}$

Combinations (selections where order does not matter):

${}^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}$

What's tested

Exam questions typically involve:

• Venn diagrams with three overlapping sets
• Tree diagrams for dependent or independent events
• Counting arrangements with restrictions (e.g. "letters must not be repeated", "vowels must be together")

Exam tips

For Venn diagram questions, always fill in the intersection first, then work outward. Students who start with the outer regions almost always make errors in the overlap.

For counting questions, read the restriction carefully before choosing between permutations and combinations. "How many ways can 5 people sit in a row" → permutations. "How many 3-person committees can be chosen from 10" → combinations.

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How to Prepare for Paper 1

Prioritise by marks. Calculus (35) and functions (35) are worth two-thirds of the paper. If your time is limited, these two sections give you the best return.

Do past papers under timed conditions. The NSC examiners reuse structures and question types year after year. Someone who has worked through the last five years of Paper 1 under exam conditions will recognise the shape of most questions they see on the day.

Learn to show working. Paper 1 carries method marks throughout. A wrong answer with clear, correct working often scores 3 out of 4. A wrong answer with no working scores 0.

Fix your algebra first. Calculus and function questions that go wrong usually go wrong because of an algebra error in the middle — a sign flip, a wrong expansion, or a divided-by-zero. Solid basic algebra is the foundation everything else sits on.

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