📖 Concept Review
Topic 1 · Number & Algebra
▼
Arithmetic Sequence
\(u_n = u_1 + (n-1)d\)\(S_n = \dfrac{n}{2}(2u_1 + (n-1)d)\)
Geometric Sequence
\(u_n = u_1 \cdot r^{n-1}\)\(S_n = \dfrac{u_1(r^n-1)}{r-1},\; r\neq 1\)
Logarithm Laws
\(\log_a(mn) = \log_a m + \log_a n\)\(\log_a\!\left(\dfrac{m}{n}\right) = \log_a m - \log_a n\)
\(\log_a(m^k) = k\log_a m\)
Binomial Theorem
\((a+b)^n = \displaystyle\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\)
📝 Worked Example
AP: \(u_1=3,\;d=4,\;n=20\). Find \(S_{20}\).\(S_{20}=\dfrac{20}{2}(2\cdot3+19\cdot4)=10(6+76)=\mathbf{820}\)
Topic 2 · Functions
▼
Quadratic Formula
\(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)
Exponential / Log
\(a^x = b \Leftrightarrow x = \log_a b\)\(\ln(e^x) = x,\; e^{\ln x} = x\)
Inverse Function
\(f^{-1}(f(x)) = x\)Swap \(x\) and \(y\), then solve.
Composite Function
\((f \circ g)(x) = f(g(x))\)
📝 Worked Example
Solve \(3^{2x-1}=27\).\(3^{2x-1}=3^3 \Rightarrow 2x-1=3 \Rightarrow x=\mathbf{2}\)
Topic 3 · Geometry & Trigonometry
▼
Pythagorean Identity
\(\sin^2\theta + \cos^2\theta = 1\)
CAST / Unit Circle
\(\sin(30°)=\tfrac{1}{2},\; \sin(60°)=\tfrac{\sqrt{3}}{2}\)\(\cos(60°)=\tfrac{1}{2},\; \tan(45°)=1\)
Vector Magnitude
\(|\mathbf{v}|=\sqrt{v_1^2+v_2^2+v_3^2}\)
Dot Product
\(\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2\)\(\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\)
📝 Worked Example
Find \(|\mathbf{v}|\) for \(\mathbf{v}=(3,4)\).\(|\mathbf{v}|=\sqrt{9+16}=\sqrt{25}=\mathbf{5}\)
Topic 4 · Statistics & Probability
▼
Addition Rule
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Conditional Probability
\(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\)
Normal Distribution
\(X\sim N(\mu,\sigma^2)\)\(P(X<\mu)=0.5\)
Expected Value
\(E(X)=\mu,\; \text{Var}(X)=\sigma^2\)
📝 Worked Example
\(P(A)=0.4,\;P(B)=0.3,\;P(A\cap B)=0.1\). Find \(P(A\cup B)\).\(P(A\cup B)=0.4+0.3-0.1=\mathbf{0.6}\)
Topic 5 · Calculus
▼
Power Rule
\(\dfrac{d}{dx}(x^n) = nx^{n-1}\)
Chain Rule
\(\dfrac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\)
Definite Integral
\(\displaystyle\int_a^b f(x)\,dx = [F(x)]_a^b\)
Integration Power Rule
\(\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1}+C,\;n\neq -1\)
📝 Worked Example
\(f(x)=3x^4-2x^2+5\). Find \(f'(2)\).\(f'(x)=12x^3-4x \Rightarrow f'(2)=12(8)-4(2)=96-8=\mathbf{88}\)
✏️ Practice Questions
Final Score
0
/ 20
Answer Key