ALGEBRA
1 / 20
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MAGIC WORDS
FOIL = First · Outer · Inner · Last |
DoS = Difference of Squares: \(a^2 - b^2 = (a+b)(a-b)\)
📘 EXAMPLE
Expand \((x+3)(x-5)\)
F: \(x \cdot x = x^2\)
O: \(x \cdot (-5) = -5x\)
I: \(3 \cdot x = 3x\)
L: \(3 \cdot (-5) = -15\)
Result: \(x^2 - 5x + 3x - 15 = \)
\(x^2 - 2x - 15\)
Middle term sign errors! Always write outer + inner separately before combining.
YOUR TURN — Try These!
✏️ QUESTION 1a
Factor completely: \(x^2 - 16\)
Hint: Can you see two perfect squares?
✏️ QUESTION 1b
Expand and simplify: \((2x-1)(3x+4)\)
ALGEBRA
2 / 20
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RULE: "MOVE IT, THEN DIVIDE"
Move all variables to one side → move all numbers to other side → divide both sides
📘 EXAMPLE
Solve: \(5x - 3 = 2x + 9\)
Subtract \(2x\): \(3x - 3 = 9\)
Add \(3\): \(3x = 12\)
Divide by \(3\): \(x = 4\) ✓
Check: \(5(4)-3 = 17\) and \(2(4)+9 = 17\) ✓
Forgetting to flip the sign when moving a term across the equals sign!
YOUR TURN
✏️ QUESTION 2a
Solve: \(7x + 2 = 4x - 10\)
✏️ QUESTION 2b TRICKY
Solve: \(3(x-2) = 2(x+5)\) Don't forget to distribute first!
QUADRATICS
3 / 20
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SING THIS: "NEGATIVE B..."
\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) — identify \(a, b, c\) from \(ax^2+bx+c=0\) FIRST
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
📘 EXAMPLE
Solve \(2x^2 - 3x - 5 = 0\)
\(a=2,\; b=-3,\; c=-5\)
Discriminant: \((-3)^2 - 4(2)(-5) = 9 + 40 = 49\)
\(x = \dfrac{3 \pm 7}{4}\)
\(x = \dfrac{10}{4} = 2.5\) or \(x = \dfrac{-4}{4} = -1\)
\(b = -3\), so \(-b = +3\) and \(b^2 = 9\) (NOT \(-9\)!). Squaring removes the negative!
YOUR TURN
✏️ QUESTION 3a
Solve using the quadratic formula: \(x^2 + 5x + 6 = 0\)
First write down: \(a =\)___ \(b =\)___ \(c =\)___
✏️ QUESTION 3b
Find the discriminant of \(x^2 - 4x + 5 = 0\). How many real solutions?
QUADRATICS
4 / 20
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"MULTIPLY & ADD" TRICK
For \(x^2 + bx + c\): find two numbers that multiply to \(c\) and add to \(b\)
📘 EXAMPLE
Factor \(x^2 + 7x + 12\)
Need: multiply to \(12\), add to \(7\)
Pairs: \(1\times12, 2\times6, 3\times4\) → \(3+4=7\) ✓
Answer: \((x+3)(x+4)\)
Both numbers negative when \(c>0, b<0\):
e.g. \(x^2-5x+6 = (x-2)(x-3)\)
One pos, one neg when \(c<0\):
e.g. \(x^2-x-6 = (x-3)(x+2)\)
YOUR TURN
✏️ QUESTION 4a
Factor: \(x^2 - 7x + 10\)
✏️ QUESTION 4b TRICKY
Factor: \(x^2 + x - 12\) Watch the signs!
FUNCTIONS
5 / 20
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BANNED LIST (domain killers)
① No division by zero ② No negative under square root ③ No log of zero or negative
📘 EXAMPLE
Find the domain of \(f(x) = \dfrac{1}{\sqrt{x-3}}\)
Need \(x - 3 > 0\) (no zero in denominator AND no negative under root)
So \(x > 3\)
Domain: \(x > 3\) or \((3, \infty)\)
Square root alone: \(x \geq 0\). But square root in denominator: \(x > 0\) (strict — can't be zero!)
YOUR TURN
✏️ QUESTION 5a
State the domain of \(g(x) = \sqrt{2x - 6}\)
✏️ QUESTION 5b
State the domain and range of \(h(x) = x^2 + 1\)
Draw a quick sketch in the space below to help!
FUNCTIONS
6 / 20
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ORDER MATTERS!
\(f(g(x))\): plug \(g(x)\) INTO \(f\) — work inside out, right to left
📘 EXAMPLE
If \(f(x) = x^2 + 1\) and \(g(x) = 2x - 3\), find \(f(g(x))\)
Replace every \(x\) in \(f\) with \(g(x) = 2x-3\)
\(f(g(x)) = (2x-3)^2 + 1\)
\(= 4x^2 - 12x + 9 + 1 = 4x^2 - 12x + 10\)
\(f(g(x)) \neq g(f(x))\) — they are usually different! Never swap the order.
YOUR TURN
✏️ QUESTION 6a
Using \(f(x) = 3x + 1\) and \(g(x) = x^2\), find \(g(f(x))\)
✏️ QUESTION 6b
Using the same functions, find \(f(g(2))\). Show all steps.
GEOMETRY
7 / 20
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SOH · CAH · TOA
Sin = Opp/Hyp · Cos = Adj/Hyp · Tan = Opp/Adj
LABEL: Hypotenuse = longest side (opposite right angle!)
📘 EXAMPLE
In a right triangle, hypotenuse = 10, one angle = 30°. Find the opposite side.
Use SOH: \(\sin(30°) = \dfrac{\text{opp}}{10}\)
\(\text{opp} = 10 \times \sin(30°) = 10 \times 0.5 = 5\)
Make sure your calculator is in DEGREE mode, not RADIAN mode!
YOUR TURN
✏️ QUESTION 7a
Find \(x\) in a right triangle where the adjacent side = 8 and angle = 40°. (Find hypotenuse)
✏️ QUESTION 7b
A ladder 13 m long leans against a wall. The base is 5 m from the wall. How high up the wall does the ladder reach?
GEOMETRY
8 / 20
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GRADIENT = "RISE OVER RUN"
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\) · Line: \(y = mx + c\) · Point-slope: \(y - y_1 = m(x - x_1)\)
📘 EXAMPLE
Find the equation of the line through \((1, 3)\) and \((4, 9)\)
Gradient: \(m = \dfrac{9-3}{4-1} = \dfrac{6}{3} = 2\)
Use \(y - 3 = 2(x - 1)\)
Simplify: \(y = 2x + 1\)
Always subtract coordinates in the SAME order: \((y_2 - y_1)\) on top, \((x_2 - x_1)\) on bottom!
YOUR TURN
✏️ QUESTION 8a
Find the gradient of the line through \((-2, 5)\) and \((3, -5)\)
✏️ QUESTION 8b TRICKY
Find the equation of the line perpendicular to \(y = 3x - 4\) passing through \((6, 1)\)
Hint: Perpendicular gradient: \(m_\perp = -\dfrac{1}{m}\)
STATISTICS
9 / 20
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M-M-M CHEAT
Mean = total ÷ count |
Median = middle (sort first!) |
Mode = most frequent |
Outliers → use Median!
📘 EXAMPLE
Data: \(3, 7, 7, 10, 13\)
Mean: \(\dfrac{3+7+7+10+13}{5} = \dfrac{40}{5} = 8\)
Median: Middle value = 7 (already sorted)
Mode: 7 (appears twice)
ALWAYS sort the data before finding the median! For even number of data points, average the middle two.
YOUR TURN
✏️ QUESTION 9a
Find the mean, median and mode of: \(12, 5, 8, 5, 20, 5, 9, 11\)
✏️ QUESTION 9b
The mean of 5 numbers is 8. Four of the numbers are 6, 7, 9, 10. Find the fifth number.
STATISTICS
10 / 20
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KEY WORDS
AND = multiply (×) | OR = add (+) − overlap |
\(P(A|B) = \dfrac{P(A \cap B)}{P(B)}\)
📘 EXAMPLE
A bag has 3 red, 2 blue, 5 green balls. One drawn at random.
P(red) = \(\dfrac{3}{10}\)
P(red or blue) = \(\dfrac{3+2}{10} = \dfrac{5}{10} = \dfrac{1}{2}\)
P(not green) = \(1 - \dfrac{5}{10} = \dfrac{1}{2}\)
"Not" events: Always use \(P(\text{not A}) = 1 - P(A)\). Much faster!
YOUR TURN
✏️ QUESTION 10a
A fair die is rolled. What is the probability of getting a number greater than 4?
✏️ QUESTION 10b HARD
Two dice are rolled. What is the probability that the sum equals 7?
Hint: Draw a 6×6 sample space table!
EXPONENTS & SURDS
11 / 20
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INDICES RULES (memorize all 5!)
① \(a^m \cdot a^n = a^{m+n}\) ② \(\dfrac{a^m}{a^n} = a^{m-n}\) ③ \((a^m)^n = a^{mn}\)
④ \(a^0 = 1\) ⑤ \(a^{-n} = \dfrac{1}{a^n}\)
📘 EXAMPLE
Simplify: \(\dfrac{x^5 \cdot x^3}{x^4}\)
Numerator: \(x^5 \cdot x^3 = x^{5+3} = x^8\)
Divide: \(x^8 \div x^4 = x^{8-4} = x^4\)
\((ab)^n = a^n b^n\) but \((a+b)^n \neq a^n + b^n\) — huge exam trap!
YOUR TURN
✏️ QUESTION 11a
Simplify: \(\dfrac{2^6 \times 2^{-3}}{2^2}\)
✏️ QUESTION 11b
Simplify: \((3x^2y)^3\)
EXPONENTS & SURDS
12 / 20
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SURDS TRICK
Simplify: find largest perfect square factor |
Rationalise: multiply top & bottom by the surd |
\(\dfrac{a}{\sqrt{b}} = \dfrac{a\sqrt{b}}{b}\)
📘 EXAMPLE
Simplify \(\sqrt{72}\) and rationalise \(\dfrac{5}{\sqrt{3}}\)
\(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\)
\(\dfrac{5}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{5\sqrt{3}}{3}\)
Always multiply BOTH numerator AND denominator when rationalising. Multiply by 1 in a clever form!
YOUR TURN
✏️ QUESTION 12a
Simplify: \(\sqrt{50} + \sqrt{8}\)
✏️ QUESTION 12b
Rationalise the denominator: \(\dfrac{6}{2 + \sqrt{3}}\)
Hint: Multiply by the conjugate \((2 - \sqrt{3})\)
SIMULTANEOUS EQ.
13 / 20
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TWO METHODS
SUBSTITUTION: isolate one variable, plug into other equation
ELIMINATION: make coefficients equal, then add or subtract — kill one variable!
📘 EXAMPLE (Elimination)
Solve: \(2x + 3y = 12\) and \(x - y = 1\)
Multiply 2nd equation ×2: \(2x - 2y = 2\)
Subtract: \((2x+3y)-(2x-2y) = 12-2\) → \(5y = 10\) → \(y = 2\)
Sub back: \(x - 2 = 1\) → \(x = 3\)
Answer: \((x, y) = (3, 2)\) — Always check in BOTH equations!
Sign errors when subtracting equations! Write out each step. Don't skip to mental math.
YOUR TURN
✏️ QUESTION 13
Solve simultaneously: \(3x + y = 11\) and \(x - 2y = 0\)
SEQUENCES
14 / 20
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AP vs GP
Arithmetic: add constant → \(u_n = u_1 + (n-1)d\) Geometric: multiply constant → \(u_n = u_1 \cdot r^{n-1}\)
AP sum: \(S_n = \dfrac{n}{2}(u_1 + u_n)\) GP sum: \(S_n = \dfrac{u_1(r^n - 1)}{r-1}\)
📘 EXAMPLE — Arithmetic
Find the 10th term of: \(3, 7, 11, 15, \ldots\)
\(u_1 = 3,\; d = 4\)
\(u_{10} = 3 + (10-1) \times 4 = 3 + 36 = 39\)
Formula uses \((n-1)\) NOT \(n\)! For the 1st term: \(u_1 + (1-1)d = u_1\) ✓
YOUR TURN
✏️ QUESTION 14a
An arithmetic sequence has first term 5 and common difference 3. Find the 15th term.
✏️ QUESTION 14b
A geometric sequence: \(2, 6, 18, 54, \ldots\) — find the 6th term.
SEQUENCES
15 / 20
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SIGMA = SUM
\(\displaystyle\sum_{k=1}^{n} u_k\) = add up terms from \(k=1\) to \(k=n\) |
Always find the first term and last term by substituting \(k\) values
📘 EXAMPLE
Evaluate \(\displaystyle\sum_{k=1}^{4} (2k + 1)\)
\(k=1\): \(2(1)+1=3\)
\(k=2\): \(2(2)+1=5\)
\(k=3\): \(7\), \(k=4\): \(9\)
Sum: \(3+5+7+9 = 24\)
The number below sigma is where you START, not how many terms there are. Always substitute first!
YOUR TURN
✏️ QUESTION 15
Evaluate \(\displaystyle\sum_{k=2}^{5} k^2\) (Start from \(k=2\), not \(k=1\)!)
QUADRATICS — GRAPH
16 / 20
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VERTEX = TURNING POINT
Vertex form: \(y = a(x - h)^2 + k\) → vertex at \((h, k)\)
CTS step: \(x^2 + bx = \left(x + \dfrac{b}{2}\right)^2 - \left(\dfrac{b}{2}\right)^2\)
📘 EXAMPLE
Write \(y = x^2 - 6x + 11\) in vertex form
Take \(x^2 - 6x\): half of \(-6\) is \(-3\), square is \(9\)
\(y = (x^2 - 6x + 9) - 9 + 11\)
\(y = (x-3)^2 + 2\) → Vertex: \((3, 2)\)
After adding \(\left(\dfrac{b}{2}\right)^2\), you MUST subtract it again to keep the equation balanced!
YOUR TURN
✏️ QUESTION 16a
Write \(y = x^2 + 4x + 1\) in vertex form. State the vertex.
✏️ QUESTION 16b
State whether the vertex is a maximum or minimum. Give a reason.
INEQUALITIES
17 / 20
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FLIP THE SIGN WHEN...
You multiply or divide by a NEGATIVE number — the inequality direction FLIPS!
\(> \) becomes \(<\) | \(\leq\) becomes \(\geq\)
📘 EXAMPLE
Solve: \(-3x + 5 > 14\)
Subtract 5: \(-3x > 9\)
Divide by \(-3\) → FLIP! \(x < -3\)
Graph: open circle at \(-3\), arrow pointing left ←
Most students forget to flip the inequality when dividing by negative. Circle it as a reminder!
YOUR TURN
✏️ QUESTION 17a
Solve and show on a number line: \(-2x + 3 \leq 9\)
✏️ QUESTION 17b
Solve: \(1 < 3x - 2 \leq 10\) Double inequality — work all three parts together!
RATIOS & PERCENTAGES
18 / 20
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% CHANGE & REVERSE
% change = \(\dfrac{\text{change}}{\text{original}} \times 100\) |
Reverse: Divide by the multiplier (e.g. after 20% increase, original = new ÷ 1.2)
📘 EXAMPLE
A shirt now costs $60 after a 25% discount. What was the original price?
25% off → multiplier = \(1 - 0.25 = 0.75\)
Original = \(60 \div 0.75 = \$80\)
NEVER find 25% of $60 and add it back — that gives the wrong answer! Always divide by the multiplier.
YOUR TURN
✏️ QUESTION 18a
A population increased from 4000 to 4600. Find the percentage increase.
✏️ QUESTION 18b
After a 15% price increase, a laptop costs \$920. Find the original price.
MENSURATION
19 / 20
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KEY FORMULAS
Sphere: \(V = \dfrac{4}{3}\pi r^3\), \(SA = 4\pi r^2\) |
Cone: \(V = \dfrac{1}{3}\pi r^2 h\), \(SA = \pi r l + \pi r^2\)
Cylinder: \(V = \pi r^2 h\), \(SA = 2\pi r h + 2\pi r^2\)
📘 EXAMPLE
Find the volume of a cone with radius 4 cm and height 9 cm.
\(V = \dfrac{1}{3} \pi r^2 h = \dfrac{1}{3} \times \pi \times 16 \times 9\)
\(= \dfrac{144\pi}{3} = 48\pi \approx 150.8 \text{ cm}^3\)
Slant height \(l \neq h\)! Use Pythagoras to find slant height: \(l = \sqrt{r^2 + h^2}\)
YOUR TURN
✏️ QUESTION 19a
Find the total surface area of a cylinder with radius 5 cm and height 12 cm. Leave answer in terms of \(\pi\).
✏️ QUESTION 19b
A sphere has a surface area of \(100\pi\) cm². Find its radius.
MIXED CHALLENGE
20 / 20
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EXAM STRATEGY
Read twice → Write what you know → Pick the right formula → Show ALL working → Check your answer!
✏️ QUESTION 20a — Algebra + Geometry
A rectangle has length \((3x + 2)\) cm and width \((x - 1)\) cm. Its area is 30 cm².
Form a quadratic equation and solve for \(x\). State the dimensions of the rectangle.
✏️ QUESTION 20b — Functions + Sequences
The function \(f(n) = 3n - 1\) generates a sequence.
(i) Write the first 4 terms. (ii) Is it arithmetic or geometric? (iii) Find the sum of the first 20 terms.
🎉 Congratulations — You finished all 20 questions! 🎉
Review your mistakes, and you'll ace the exam!