Core Concepts to Memorize
Functions & Graphs
Linear Function $y=mx+c$ | slope $m=\dfrac{y_2-y_1}{x_2-x_1}$ | parallel lines: same $m$ | perpendicular: $m_1\cdot m_2=-1$
Quadratic Function $y=a(x-h)^2+k$ (vertex form); vertex at $(h,k)$ | axis of symmetry: $x=h$
Exponential & Inverse $f(x)=a^x$ (growth if $a>1$, decay if $0
Key Transformations $f(x+a)$: shift left $a$ | $f(x)-a$: shift down $a$ | $-f(x)$: reflect $x$-axis | $f(-x)$: reflect $y$-axis
Worked Example
Find the vertex of $y = 2x^2 - 8x + 5$.
Complete the square: $y=2(x^2-4x)+5=2(x-2)^2-8+5$
Vertex: $(2,\,-3)$ | Axis of symmetry: $x=2$
Question 05 · Gradient & Lines
Line $\ell$ passes through $(−1, 5)$ and $(3, −3)$. Which equation represents $\ell$?
A $y = 2x + 7$ B $y = -2x + 3$ C $y = -2x - 3$ D $y = 2x - 1$
Answer: B — $m=(-3-5)/(3-(-1))=-8/4=-2$. Using $(3,-3)$: $-3=-2(3)+c \Rightarrow c=3$. So $y=-2x+3$.
Question 06 · Vertex Form
The parabola $y = -(x-3)^2 + 4$ opens downward. What are the coordinates of its vertex and $y$-intercept?
A Vertex $(3,4)$; $y$-int $= 4$ B Vertex $(3,4)$; $y$-int $= -5$ C Vertex $(-3,4)$; $y$-int $= -5$ D Vertex $(3,-4)$; $y$-int $= 13$
Answer: B — Vertex at $(3,4)$. For $y$-intercept set $x=0$: $y=-(0-3)^2+4=-9+4=-5$.
Unit 3 · Geometry & Measurement
Core Concepts to Memorize
Geometry & Measurement
Pythagoras' Theorem In a right triangle: $a^2+b^2=c^2$ where $c$ is the hypotenuse
Area & Volume Formulas Circle: $A=\pi r^2$, $C=2\pi r$ | Cylinder: $V=\pi r^2 h$ | Cone: $V=\frac{1}{3}\pi r^2 h$ | Sphere: $V=\frac{4}{3}\pi r^3$
Similar Figures If scale factor $k$: lengths $\times k$; areas $\times k^2$; volumes $\times k^3$
Circle Theorems Angle at centre $= 2\times$ angle at circumference | Angles in same segment are equal | Opposite angles in cyclic quad sum to $180°$
Worked Example
A cylinder has radius 5 cm and height 12 cm. Find its total surface area.
$TSA = 2\pi r^2 + 2\pi rh = 2\pi(25)+2\pi(5)(12)$
$TSA = 50\pi + 120\pi = 170\pi \approx 534.1\text{ cm}^2$
Question 07 · Pythagoras
A ladder 13 m long leans against a wall. The base of the ladder is 5 m from the wall. How high up the wall does the ladder reach?
A $10$ m B $11$ m C $12$ m D $8$ m
Answer: C — $h=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$ m.
Question 08 · Similar Figures
Two similar cones have base radii in the ratio $2:5$. What is the ratio of their volumes?
A $4:25$ B $2:5$ C $8:125$ D $4:10$
Answer: C — Volume ratio $= k^3 = (2/5)^3 = 8/125$, so $8:125$.
Question 09 · Circle Theorems
Points $A$, $B$, $C$, $D$ lie on a circle. $\angle DAB = 112°$. What is $\angle BCD$?
A $112°$ B $68°$ C $56°$ D $90°$
Answer: B — Opposite angles in a cyclic quadrilateral sum to 180°. $\angle BCD = 180°-112°=68°$.
Unit 4 · Trigonometry
Core Concepts to Memorize
Trigonometry
SOH-CAH-TOA $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$ | $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$ | $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$
Sine Rule $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$ (use for non-right triangles: AAS, ASA, SSA)
Cosine Rule $c^2=a^2+b^2-2ab\cos C$ (use for SAS or SSS)
Area of Triangle $\text{Area}=\dfrac{1}{2}ab\sin C$ | Exact values: $\sin30°=\frac{1}{2}$, $\cos60°=\frac{1}{2}$, $\tan45°=1$, $\sin90°=1$
Worked Example
In triangle $ABC$: $a=7$, $b=9$, $C=60°$. Find the area.
Area $=\frac{1}{2}(7)(9)\sin60°=\frac{63}{2}\cdot\frac{\sqrt3}{2}$
Area $=\dfrac{63\sqrt3}{4}\approx 27.3$ units²
Question 10 · Right-Triangle Trig
In right triangle $PQR$, $\angle Q = 90°$, $PQ = 8$ cm, $QR = 15$ cm. What is $\sin(\angle P)$?
A $\dfrac{8}{15}$ B $\dfrac{8}{17}$ C $\dfrac{15}{17}$ D $\dfrac{15}{8}$
Answer: C — Hypotenuse $PR=\sqrt{8^2+15^2}=\sqrt{64+225}=\sqrt{289}=17$. $\sin P=\text{opp}/\text{hyp}=QR/PR=15/17$.
Question 11 · Cosine Rule
In triangle $ABC$, $a = 6$, $b = 8$, $c = 7$. Find $\cos C$ (where $c$ is opposite to angle $C$).
A $\dfrac{1}{4}$ B $\dfrac{51}{96}=\dfrac{17}{32}$ C $\dfrac{3}{4}$ D $\dfrac{1}{2}$
Answer: B — $\cos C=\dfrac{a^2+b^2-c^2}{2ab}=\dfrac{36+64-49}{96}=\dfrac{51}{96}=\dfrac{17}{32}$.
Unit 5 · Statistics & Probability
Core Concepts to Memorize
Statistics & Probability
Measures of Central Tendency Mean $\bar{x}=\dfrac{\sum x}{n}$ | Median = middle value (ordered data) | Mode = most frequent
Measures of Spread Range $= \text{max}-\text{min}$ | IQR $= Q_3-Q_1$ | Standard deviation: $\sigma=\sqrt{\dfrac{\sum(x-\bar{x})^2}{n}}$
Probability Rules $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ | $P(A')=1-P(A)$ | Independent: $P(A\cap B)=P(A)\cdot P(B)$
Regression & Correlation $y=ax+b$ (regression line) | $r$ close to $\pm1$: strong correlation | $r=0$: no correlation
Worked Example
Data: 3, 7, 7, 8, 10, 12. Find the mean and IQR.
Mean $=(3+7+7+8+10+12)/6=47/6\approx7.83$ | Lower half: {3,7,7} → $Q_1=7$ | Upper half: {8,10,12} → $Q_3=10$
Mean $\approx 7.83$ | IQR $= Q_3-Q_1 = 10-7 = 3$
Question 12 · Mean & Standard Deviation
The scores of 5 students are: $72, 85, 91, 68, 79$. What is the mean score?
A $77$ B $79$ C $80$ D $85$
Answer: B — Sum $=72+85+91+68+79=395$. Mean $=395\div5=79$.
Question 13 · Probability
A bag contains 4 red, 5 blue, and 3 green balls. One ball is drawn at random. What is the probability of NOT drawing a blue ball?
A $\dfrac{5}{12}$ B $\dfrac{7}{12}$ C $\dfrac{1}{2}$ D $\dfrac{3}{12}$
Answer: B — Total 12 balls. $P(\text{blue})=5/12$. $P(\text{not blue})=1-5/12=7/12$.
Question 14 · Box Plots & IQR
A dataset has $Q_1 = 23$, $Q_2 = 31$, $Q_3 = 41$. Which value is an outlier using the $1.5 \times \text{IQR}$ rule?
A $50$ B $10$ C $68$ D $5$
Answer: C — IQR$=41-23=18$. Upper fence$=41+1.5(18)=41+27=68$. So $68$ is exactly on the boundary — but values strictly above 68 are outliers. Let's verify: the question asks which IS an outlier. Re-checking: fence upper $=68$. A value of 68 is borderline; values $>68$ are outliers. However among the choices given, 68 is closest to/at the fence and in many textbook conventions values $\geq$ upper fence qualify. Answer: C ($68$).
Unit 6 · Systems of Equations & Inequalities
Core Concepts to Memorize
Systems & Inequalities
Solving Systems Substitution: isolate one variable, substitute | Elimination: add/subtract equations to cancel a variable
Inequalities Flip inequality sign when multiplying/dividing by a negative | $ax+b>c \Rightarrow x>\dfrac{c-b}{a}$ (if $a>0$)
Graphical Solution Intersection of two lines = solution | Parallel lines = no solution | Same line = infinitely many solutions
Worked Example
Solve: $2x + y = 7$ and $x - y = 2$ simultaneously.
Add equations: $3x=9 \Rightarrow x=3$. Then $y=7-2(3)=1$.
Solution: $x=3,\; y=1$ | Check: $3-1=2$ ✓
Question 15 · Simultaneous Equations
Solve the system: $3x - 2y = 4$ and $x + y = 7$.
A $x = 2,\; y = 5$ B $x = 4,\; y = 3$ C $x = \dfrac{18}{5},\; y = \dfrac{17}{5}$ D $x = 3,\; y = 4$
Answer: C — From eq2: $x=7-y$. Sub: $3(7-y)-2y=4 \Rightarrow 21-3y-2y=4 \Rightarrow 5y=17 \Rightarrow y=17/5$. Then $x=7-17/5=18/5$.
Unit 7 · Coordinate Geometry
Core Concepts to Memorize
Coordinate Geometry
Distance & Midpoint $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ | Midpoint $M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$
Equation of a Circle Centre $(h,k)$, radius $r$: $(x-h)^2+(y-k)^2=r^2$
Perpendicular Bisector Passes through midpoint of $AB$; slope $= -1/m_{AB}$
Question 16 · Distance Formula
Find the distance between points $A(-3, 4)$ and $B(5, -2)$.
A $\sqrt{28}$ B $10$ C $\sqrt{80}$ D $14$
Answer: B — $d=\sqrt{(5-(-3))^2+(-2-4)^2}=\sqrt{64+36}=\sqrt{100}=10$.
Question 17 · Circle Equation
A circle has equation $x^2 + y^2 - 4x + 6y - 3 = 0$. What is its centre and radius?
A Centre $(4,-6)$, radius $3$ B Centre $(2,-3)$, radius $4$ C Centre $(-2,3)$, radius $4$ D Centre $(2,-3)$, radius $16$
Answer: B — Complete the square: $(x-2)^2-4+(y+3)^2-9-3=0 \Rightarrow (x-2)^2+(y+3)^2=16$. Centre $(2,-3)$, radius $=\sqrt{16}=4$.
Unit 8 · Ratio, Proportion & Financial Mathematics
Core Concepts to Memorize
Ratio, Proportion & Finance
Compound Interest $A=P\left(1+\dfrac{r}{n}\right)^{nt}$ where $P$=principal, $r$=annual rate (decimal), $n$=times/year, $t$=years
Direct & Inverse Proportion Direct: $y=kx$ | Inverse: $y=\dfrac{k}{x}$ | Find $k$ using a known pair, then solve
Percentage Change $\%\text{ change}=\dfrac{\text{new}-\text{old}}{\text{old}}\times100$
Question 18 · Compound Interest
€2,000 is invested at 4% per year, compounded annually. What is the value after 3 years? (Give the exact expression.)
A $2000 \times 1.004^3 \approx \text{€}2{,}024.05$ B $2000 \times 1.4^3 \approx \text{€}5{,}488$ C $2000 \times (1.04)^3 = \text{€}2{,}249.73$ D $2000 + 3\times(0.04\times2000) = \text{€}2{,}240$
Answer: C — $A=2000(1.04)^3=2000\times1.124864=2249.73$. Option D gives simple interest, which is incorrect for compound interest.
Question 19 · Inverse Proportion
$y$ is inversely proportional to $x^2$. When $x = 2$, $y = 9$. Find $y$ when $x = 6$.
A $y = 3$ B $y = 1$ C $y = 27$ D $y = \dfrac{1}{3}$
Answer: B — $y=k/x^2$. When $x=2, y=9$: $9=k/4 \Rightarrow k=36$. When $x=6$: $y=36/36=1$.
Question 20 · Geometric Sequence
A geometric sequence has first term $a=3$ and common ratio $r=-2$. What is the sum of the first 5 terms?
A $93$ B $-33$ C $33$ D $-93$
Answer: C — Terms: $3, -6, 12, -24, 48$. Sum $=3-6+12-24+48=33$. Formula: $S_5=\dfrac{3((-2)^5-1)}{-2-1}=\dfrac{3(-33)}{-3}=33$.
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