Topic 1 · Arithmetic Sequences
Key Formulas to Memorise
General term: $u_n = u_1 + (n-1)d$
Sum of $n$ terms: $S_n = \dfrac{n}{2}(2u_1 + (n-1)d)$
$S_n = \dfrac{n}{2}(u_1 + u_n)$
Quick Example
$u_1=5,\; d=3,\; n=10$ → $S_{10}=\tfrac{10}{2}(10+32)=210$
Answer: 210
An arithmetic sequence has first term $u_1 = 3$ and common difference $d = 4$. Find the sum of the first 20 terms, $S_{20}$.
$S_{20}=\dfrac{20}{2}(2\times3+(20-1)\times4)=10(6+76)=10\times82=\mathbf{820}$
Topic 1 · Geometric Sequences
Key Formulas to Memorise
Sum to infinity (requires $|r|<1$):
$S_\infty = \dfrac{u_1}{1-r}$
Sum of $n$ terms: $S_n = \dfrac{u_1(1-r^n)}{1-r}$
Quick Example
$u_1=12,\; r=\tfrac{1}{3}$ → $S_\infty=\dfrac{12}{1-\frac{1}{3}}=18$
Answer: 18
A geometric series has first term $u_1 = 8$ and common ratio $r = \dfrac{1}{2}$. Find $S_\infty$.
$S_\infty=\dfrac{8}{1-\frac{1}{2}}=\dfrac{8}{\frac{1}{2}}=\mathbf{16}$
Topic 1 · Binomial Expansion
Key Formulas to Memorise
$(a+b)^n = \displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$
The coefficient of $x^r$ in $(a+x)^n$ is $\dbinom{n}{r}a^{n-r}$
Quick Example
Coefficient of $x^2$ in $(1+x)^5$: $\binom{5}{2}=10$
Answer: 10
Find the coefficient of $x^3$ in the expansion of $(2+x)^5$.
Term with $x^3$: $\dbinom{5}{3}(2)^{5-3}(x)^3=10\cdot4\cdot x^3$. Coefficient $=10\times4=\mathbf{40}$
Topic 2 · Quadratic Equations
Key Formulas to Memorise
Discriminant: $\Delta = b^2 - 4ac$
$\Delta > 0$: two distinct real roots | $\Delta = 0$: one repeated root | $\Delta < 0$: no real roots
$\Delta = b^2 - 4ac = 0 \implies \text{equal (repeated) roots}$
Quick Example
$x^2-4x+4=0$: $\Delta=16-16=0$ → repeated root $x=2$
Answer: $x=2$ (double root)
The equation $3x^2 - kx + 3 = 0$ has equal roots. Find the positive value of $k$.
$\Delta=(-k)^2-4(3)(3)=k^2-36=0 \implies k^2=36 \implies k=\pm6$. Positive value: $\mathbf{k=6}$
Topic 3 · Trig Identities
Key Formulas to Memorise
Double angle formulas:
$\cos(2x)=1-2\sin^2 x = 2\cos^2 x -1 = \cos^2 x - \sin^2 x$
$\sin(2x)=2\sin x\cos x$
Quick Example
$\sin x=\tfrac{1}{2}$ → $\cos(2x)=1-2\cdot\tfrac{1}{4}=\tfrac{1}{2}$
Answer: $\tfrac{1}{2}$
Given that $\sin x = \dfrac{3}{5}$ and $x$ is in the first quadrant, find $\cos(2x)$.
A$\dfrac{7}{25}$
B$\dfrac{9}{25}$
C$-\dfrac{7}{25}$
D$\dfrac{24}{25}$
$\cos(2x)=1-2\sin^2 x=1-2\left(\dfrac{3}{5}\right)^2=1-\dfrac{18}{25}=\mathbf{\dfrac{7}{25}}$
Topic 5 · Calculus — Local Extrema
Key Formulas to Memorise
Finding local max/min:
① Set $f'(x)=0$ to find critical points
② Use $f''(x)$: if $f''<0$ → local max; if $f''>0$ → local min
$f'(x)=0$ and $f''(x)<0 \implies \text{local maximum}$
Quick Example
$f(x)=x^2-4x$: $f'=2x-4=0 \Rightarrow x=2$; $f''=2>0$ → local min
Local min at $x=2$, $f(2)=-4$
Let $f(x) = x^3 - 6x^2 + 9x$. The local maximum value of $f$ is:
$f'(x)=3x^2-12x+9=3(x^2-4x+3)=3(x-1)(x-3)=0 \Rightarrow x=1$ or $x=3$
$f''(x)=6x-12$: at $x=1$, $f''=-6<0$ → local max.
$f(1)=1-6+9=\mathbf{4}$
Topic 5 · Definite Integrals
Key Formulas to Memorise
$\displaystyle\int_a^b x^n\,dx = \left[\frac{x^{n+1}}{n+1}\right]_a^b,\quad n\neq-1$
Quick Example
$\displaystyle\int_0^2 x^3\,dx=\left[\dfrac{x^4}{4}\right]_0^2=4$
Answer: 4
Evaluate $\displaystyle\int_0^3 x^2\,dx$.
$\displaystyle\int_0^3 x^2\,dx=\left[\dfrac{x^3}{3}\right]_0^3=\dfrac{27}{3}-0=\mathbf{9}$
Topic 2 · Log Laws
Key Formulas to Memorise
$\log_a x + \log_a y = \log_a(xy)$
$\log_a x - \log_a y = \log_a\!\left(\dfrac{x}{y}\right)$
$n\log_a x = \log_a(x^n)$
Quick Example
$\log_2 4 + \log_2 8 = \log_2 32 = 5$
Answer: 5
Solve $\log_2 x + \log_2(x-2) = 3$.
A$x=2$
B$x=3$
C$x=4$
D$x=6$
$\log_2[x(x-2)]=3 \Rightarrow x(x-2)=8 \Rightarrow x^2-2x-8=0 \Rightarrow (x-4)(x+2)=0$
$x=4$ or $x=-2$. Since $x>2$ (domain), $\mathbf{x=4}$
Topic 1 · Complex Numbers
Key Formulas to Memorise
Modulus: $|a+bi|=\sqrt{a^2+b^2}$
Argument: $\arg(a+bi)=\arctan\!\left(\dfrac{b}{a}\right)$ (adjust for quadrant)
$z=r(\cos\theta+i\sin\theta)=re^{i\theta}$
Quick Example
$|1+i|=\sqrt{1+1}=\sqrt{2}$, $\arg=45°$
$|z|=\sqrt{2}$
Find the modulus of the complex number $z = 3 + 4i$.
$|z|=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=\mathbf{5}$
Topic 4 · Vectors
Key Formulas to Memorise
$\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$
If $\mathbf{a}\cdot\mathbf{b}=0$, the vectors are perpendicular.
Quick Example
$\mathbf{a}=(1,0),\;\mathbf{b}=(0,1)$: $\mathbf{a}\cdot\mathbf{b}=0$ → angle $=90°$
Perpendicular
Find the angle between vectors $\mathbf{a}=(1,2,2)$ and $\mathbf{b}=(2,1,-2)$.
A$45°$
B$60°$
C$90°$
D$120°$
$\mathbf{a}\cdot\mathbf{b}=(1)(2)+(2)(1)+(2)(-2)=2+2-4=0$
Since the dot product is $0$, the vectors are perpendicular. Angle $= \mathbf{90°}$
Topic 4 · Probability
Key Formulas to Memorise
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
Independent events: $P(A\cap B)=P(A)\cdot P(B)$
Quick Example
$P(A)=0.3,\;P(B)=0.4,\;P(A\cap B)=0.1$ → $P(A\cup B)=0.6$
Answer: 0.6
$P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$. Find $P(A\cup B)$.
$P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.4+0.5-0.2=\mathbf{0.7}$
Topic 4 · Statistics
Key Formulas to Memorise
$Z = \dfrac{X - \mu}{\sigma}$
$X\sim N(\mu,\sigma^2)$ is standardised to $Z\sim N(0,1)$
Quick Example
$X\sim N(20,25)$: $P(X<25)=P(Z<1)\approx0.8413$
Answer: ≈ 0.8413
$X\sim N(50, 100)$ (i.e. $\mu=50$, $\sigma^2=100$). Which standardised value $Z$ corresponds to $X=60$?
A$Z=0.5$
B$Z=1$
C$Z=2$
D$Z=10$
$\sigma=\sqrt{100}=10$. $Z=\dfrac{X-\mu}{\sigma}=\dfrac{60-50}{10}=\mathbf{1}$
Topic 1 · Matrices
Key Formulas to Memorise
For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, $\det A=ad-bc$
$A^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
Quick Example
$A=\begin{pmatrix}1&2\\0&1\end{pmatrix}$: $\det=1$, $A^{-1}=\begin{pmatrix}1&-2\\0&1\end{pmatrix}$
Find the inverse of $A=\begin{pmatrix}2&1\\5&3\end{pmatrix}$.
A$\begin{pmatrix}3&-1\\-5&2\end{pmatrix}$
B$\begin{pmatrix}3&1\\5&2\end{pmatrix}$
C$\begin{pmatrix}2&-1\\-5&3\end{pmatrix}$
D$\begin{pmatrix}-3&1\\5&-2\end{pmatrix}$
$\det A=(2)(3)-(1)(5)=6-5=1$
$A^{-1}=\dfrac{1}{1}\begin{pmatrix}3&-1\\-5&2\end{pmatrix}=\mathbf{\begin{pmatrix}3&-1\\-5&2\end{pmatrix}}$
Topic 5 · ODEs
Key Formulas to Memorise
Separable ODE: $\dfrac{dy}{dx}=f(x)$ → integrate both sides
$y=\int f(x)\,dx + C$
Use initial condition to find $C$.
Quick Example
$\dfrac{dy}{dx}=3x^2,\;y(0)=1$ → $y=x^3+C$; $y(0)=1\Rightarrow C=1$
$y=x^3+1$
Given $\dfrac{dy}{dx}=2x$ with initial condition $y(0)=1$, find $y$.
A$y=x^2$
B$y=x^2+1$
C$y=2x+1$
D$y=x^2+2$
$y=\int2x\,dx=x^2+C$. Applying $y(0)=1$: $0+C=1\Rightarrow C=1$. Therefore $\mathbf{y=x^2+1}$
Topic 5 · Differentiation Techniques
Key Formulas to Memorise
$\dfrac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$
Quick Example
$\dfrac{d}{dx}[\sin(3x)]=\cos(3x)\cdot3=3\cos(3x)$
$3\cos(3x)$
Find $\dfrac{d}{dx}[\sin(x^2)]$.
A$\cos(x^2)$
B$2x\cos(x^2)$
C$x\cos(x^2)$
D$2\cos(x^2)$
Outer $f=\sin$, inner $g=x^2$: $f'(g)\cdot g'=\cos(x^2)\cdot2x=\mathbf{2x\cos(x^2)}$
Integration by Substitution
Topic 5 · Integration Techniques
Key Formulas to Memorise
$\displaystyle\int f(g(x))g'(x)\,dx=F(g(x))+C$
Let $u=g(x)$, then $du=g'(x)\,dx$, so integral becomes $\int f(u)\,du$.
Quick Example
$\int 2xe^{x^2}dx$: let $u=x^2$, $du=2x\,dx$ → $\int e^u\,du=e^{x^2}+C$
$e^{x^2}+C$
Find $\displaystyle\int 2x\cos(x^2)\,dx$.
A$-\sin(x^2)+C$
B$\sin(x^2)+C$
C$2\sin(x^2)+C$
D$\cos(x^2)+C$
Let $u=x^2 \Rightarrow du=2x\,dx$. Integral becomes $\int\cos(u)\,du=\sin(u)+C=\mathbf{\sin(x^2)+C}$
Topic 3 · Geometry
Key Theorems to Memorise
Central angle $= 2 \times$ Inscribed angle (subtended by same arc)
The angle at the centre of a circle is twice the angle at the circumference when both subtend the same arc.
Quick Example
Inscribed angle $= 35°$ → Central angle $= 70°$
Central angle: 70°
An inscribed angle in a circle subtends an arc and measures $40°$. What is the central angle subtending the same arc?
A$40°$
B$60°$
C$80°$
D$160°$
By the Inscribed Angle Theorem: Central angle $= 2\times$ inscribed angle $= 2\times40°=\mathbf{80°}$
Polynomial Remainder Theorem
Topic 2 · Polynomials
Key Theorems to Memorise
Remainder when $p(x)$ is divided by $(x-a)$ equals $p(a)$
Factor Theorem: $(x-a)$ is a factor of $p(x)$ iff $p(a)=0$
Quick Example
$p(x)=x^2+x-6$ divided by $(x-2)$: $p(2)=4+2-6=0$ → $(x-2)$ is a factor
Remainder: 0 (factor)
Find the remainder when $p(x)=x^3-2x^2+3x-4$ is divided by $(x-2)$.
$p(2)=8-8+6-4=\mathbf{2}$
Topic 2 · Exponential Models
Key Formulas to Memorise
$A(t)=A_0\,e^{kt}$
$k>0$: growth | $k<0$: decay
$A_0$ = initial value, $t$ = time, $k$ = rate constant
Quick Example
$A_0=50,\;k=0.1,\;t=5$: $A=50e^{0.5}\approx82.4$
$\approx 82.4$
A population grows according to $A(t)=100e^{0.02t}$. Find $A(10)$ to the nearest integer.
$A(10)=100e^{0.02\times10}=100e^{0.2}=100\times1.2214\ldots\approx\mathbf{122}$
Topic 4 · Statistics
Key Formulas to Memorise
$\bar{x}=\dfrac{\sum x_i}{n}$ $\sigma^2=\dfrac{\sum(x_i-\bar{x})^2}{n}$
Population variance uses $n$; sample variance uses $n-1$.
Quick Example
Data: 1, 3, 5 → $\bar{x}=3$; $\sigma^2=\tfrac{4+0+4}{3}=\tfrac{8}{3}$
Variance: $8/3$
For the data set $\{2, 4, 6, 8, 10\}$, find the population variance $\sigma^2$.
$\bar{x}=\dfrac{2+4+6+8+10}{5}=6$
$\sigma^2=\dfrac{(2-6)^2+(4-6)^2+(6-6)^2+(8-6)^2+(10-6)^2}{5}=\dfrac{16+4+0+4+16}{5}=\dfrac{40}{5}=\mathbf{8}$
0/20
Final Score
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Complete the quiz to see your results.
ANSWER KEY & WORKED SOLUTIONS
Q1: B (820) — $S_{20}=10(6+76)=820$
Q2: C (16) — $S_\infty=8/(1-\frac12)=16$
Q3: C (40) — $\binom{5}{3}\cdot2^2=10\cdot4=40$
Q4: B (6) — $\Delta=k^2-36=0\Rightarrow k=6$
Q5: A ($\frac{7}{25}$) — $1-2\cdot\frac{9}{25}=\frac{7}{25}$
Q6: C (4) — $f(1)=1-6+9=4$
Q7: B (9) — $[x^3/3]_0^3=9$
Q8: C ($x=4$) — $x(x-2)=8\Rightarrow x=4$
Q9: B (5) — $\sqrt{9+16}=5$
Q10: C (90°) — dot product $=0$
Q11: B (0.7) — $0.4+0.5-0.2=0.7$
Q12: B ($Z=1$) — $(60-50)/10=1$
Q13: A — $\det=1$, swap diagonal, negate off-diagonal
Q14: B ($y=x^2+1$) — integrate $2x$, $C=1$
Q15: B ($2x\cos x^2$) — chain rule
Q16: B ($\sin x^2+C$) — substitution $u=x^2$
Q17: C (80°) — central $=2\times$ inscribed
Q18: B (2) — $p(2)=8-8+6-4=2$
Q19: C (122) — $100e^{0.2}\approx122$
Q20: B (8) — $40/5=8$