Matric maths, one small step at a time.

Solving an equation means finding the number that makes both sides equal. In Question 1 you meet the same six disguises every year.

• ✗ Dividing \(8x^2\gt2x\) by \(x\) to get \(x\gt\frac14\). ✓ Factorise instead. You keep the \(x\lt0\) part.
• ✗ Forgetting to check surd answers. ✓ Substitute every answer back into the original equation.
• ✗ Writing \(2^x=-\frac12\) as an answer. ✓ \(2^x\) is never negative, so write “no solution” for that factor.
• ✗ Rounding \(\sqrt{41}\) to 6,4 halfway through. ✓ Keep the full value in your calculator; round only the final answer.
• ✗ Giving \(x\) without \(y\) in simultaneous equations. ✓ Every \(x\) needs its partner \(y\).

A pattern is a recipe for the next number. Your first job is always to identify the recipe by testing the gaps between terms.

A sequence is the list of terms. A series is the sum of the terms. \(T_n\) is one term; \(S_n\) is the total of the first \(n\) terms.

• ✗ Counting \(\sum_{p=21}^{117}\) as 96 terms. ✓ \(117-21+1=97\) terms.
• ✗ Using \(S_\infty\) when \(r=2\). ✓ Only when \(-1\lt r\lt1\). State the condition for the mark.
• ✗ \(r=\frac{T_1}{T_2}\). ✓ \(r=\frac{T_2}{T_1}\): a term divided by the one before it.
• ✗ Answering \(n=-19{,}5\) for a position. ✓ \(n\) is a counting number. Reject negatives and fractions, and say why.

A function is a machine: put in \(x\), get out exactly one \(y\). A graph is just a picture of every input and output at once.

Each family has its own shape, and the numbers in the equation move that shape around. Play with the sliders below and watch what each letter does.

• ✗ \(y=\frac{3}{x+2}\) has asymptote \(x=2\). ✓ Set the denominator to zero: \(x=-2\).
• ✗ Including the asymptote in an interval. ✓ Use a round bracket or a strict sign at an asymptote: \(-\frac12\le x\lt1\).
• ✗ Vertical distance \(=g-f\) when \(f\) is on top. ✓ Always top minus bottom.
• ✗ Range of a sad parabola written as \(y\ge q\). ✓ Its highest point is \(q\), so \(y\le q\).

If \(f\) turns 2 into 9, then \(f^{-1}\) turns 9 back into 2. Every point \((a;b)\) becomes \((b;a)\), so the graph is mirrored in the line \(y=x\).

The exponential graph \(y=b^x\) and the log graph \(y=\log_bx\) are inverses of each other. A log is simply the question “what power?”: \(\log_28=3\) because \(2^3=8\).

• ✗ \(f^{-1}(x)=\frac{1}{f(x)}\). ✓ The inverse undoes \(f\). It is not the reciprocal.
• ✗ Shifting 5 right as \(x+5\). ✓ Right is \(x-5\): replace \(x\) with \((x-5)\).
• ✗ Log graph domain “all real numbers”. ✓ You can only take the log of a positive number: \(x\gt0\).

Compound interest is interest on interest. Every period, the balance is multiplied by \((1+i)\). Everything else in this topic is bookkeeping around that one idea.

• ✗ Using 6,8 instead of 0,068, or forgetting to divide by 12. ✓ Write \(i\) as a fraction in your working: \(\frac{0{,}068}{12}\).
• ✗ Counting 23 deposits from January 2020 to October 2025. ✓ Use a timeline. Count both ends.
• ✗ Rounding \(n=129{,}42\) down to 129 months. ✓ The 0,42 is a real (smaller) payment, so it counts as a month.
• ✗ Rounding \(i\) to 0,0057. ✓ Keep \(\frac{0{,}068}{12}\) exact in the calculator. Early rounding changes the rand answer.

Imagine walking along a graph. The derivative tells you how steep the path is under your feet. At the top of a hill or the bottom of a valley, the path is flat: the derivative is zero.

That single idea answers most calculus questions: tangents (use the gradient), turning points (gradient \(=0\)), increasing and decreasing (gradient positive or negative), and real-life maximum or minimum problems.

• ✗ Dropping “lim” halfway through first principles. ✓ Write \(\lim_{h\to0}\) on every line until \(h\) is replaced by 0. Examiners take a mark for it.
• ✗ Substituting the \(x\) of a turning point into \(f'\) to get the \(y\)-value. ✓ \(y\)-values always come from the original \(f\).
• ✗ Differentiating \(\frac{2}{x^2}\) as \(\frac{2}{2x}\). ✓ Rewrite as \(2x^{-2}\) first.
• ✗ Using \(f'(x)\) notation when asked for \(\frac{dy}{dx}\). ✓ Answer in the notation the question uses.

\(P(\text{event})\)​\({}=\dfrac{\text{ways it can happen}}{\text{all possible outcomes}}\). The hard part is never the division. It is counting correctly, so draw a Venn diagram, a tree, or a row of boxes.

• ✗ Thinking independent means mutually exclusive. ✓ Independent: multiply. Mutually exclusive: the overlap is zero. They are different tests.
• ✗ Filling the Venn diagram from the outside. ✓ Start with the middle (all three), then the pairs, then the singles.
• ✗ Saying “not independent” without numbers. ✓ Show \(P(A)\times P(B)\) and \(P(A\text{ and }B)\) and compare them.