Grade 12 Advance Mathematics Formula Sheet : PNG

The Papua New Guinea Grade 12 Advance Mathematics Formula Sheet is an essential reference for students preparing for examinations, assignments, and revision. This formula sheet contains the key mathematical formulas prescribed for Grade 12 Advanced Mathematics in Papua New Guinea, covering mensuration, trigonometry, calculus, algebra, statistics, probability, vectors, logarithms, and analytical geometry.

📐 ADVANCED MATHEMATICS

Upper Secondary School Certificate Examinations — Formulae Sheet

📏 MENSURATION

• Arc Length   \[ L = \frac{\theta}{360} \times 2\pi r \]
• Area of Sector   \[ A = \frac{\theta}{360} \pi r^2 \]
• Volume of right prism   \[ V = A \times h \]
• Volume of Sphere   \[ V = \frac{4}{3} \pi r^3 \]
• Interior Angle Sum of Polygon   \[ S_n = (n-2) \times 180 \]

💰 INTEREST

• Compound Interest   \[ A = P \left( 1 + \frac{r}{100} \right)^n \]

📐 TRIGONOMETRY

• Sin Rule   \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
• Cosine Rule   \[ c^2 = a^2 + b^2 - 2ab \cos C \]
• Area of Triangle   \[ A = \frac{1}{2} ab \sin C \]
• Conversion   \[ \pi^c = 180^\circ \]
• Arc Length   \[ L = r \theta^c \]
• Area of Sector   \[ A = \frac{1}{2} r^2 \theta^c \]
• Area of Minor Segment   \[ A = \frac{1}{2} r^2 \left( \theta^c - \sin \theta^c \right) \]

🔢 PERMUTATION & COMBINATION

• Permutation   \[ ^nP_r = \frac{n!}{(n-r)!} \]
• Combination   \[ ^nC_r = \frac{n!}{r!(n-r)!} \]

📊 SERIES

• Arithmetic Progression   \[ T_n = a + (n-1)d \]
• Geometric Progression   \[ S_n = \frac{a}{r} \left( \frac{1-r^n}{1-r} \right) \]
• \[ S_n = \frac{a}{r-1} \left( \frac{1-r^n}{1-r} \right) ,\quad r \neq 1 \]
• \[ S_n = \frac{a}{1-r} ,\quad -1 < r < 1 \]

⚡ ALGEBRA & CALCULUS

• Quadratic Formula   \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
• First Derivative   \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x} \]
• Product Rule   \[ (f(x) \times g(x))' = f(x) \times g'(x) + g(x) \times f'(x) \]
• Quotient Rule   \[ \left( \frac{f(x)}{g(x)} \right)' = \frac{g(x) \times f'(x) - f(x) \times g'(x)}{[g(x)]^2} ,\quad g(x) \neq 0 \]
• Chain Rule   \[ \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \]

📌 ANALYTIC GEOMETRY

• Absolute Value   \[ |x| = \begin{cases} -x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases} \]
• Mid-Point   \[ \left( x = \frac{x_1 + x_2}{2},\ y = \frac{y_1 + y_2}{2} \right) \]

📦 BINOMIAL EXPANSION

\[ (x + y)^n = x^n + \binom{n}{1}x^{n-1}y + \binom{n}{2}x^{n-2}y^2 + \cdots + y^n, \text{ where } \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

📈 AREA UNDER GRAPH

\[ A = \int_a^b f(x) \, dx \]

📊 AREA BETWEEN TWO FUNCTIONS

\[ A = \int_a^b [f(x) - g(x)] \, dx, \quad f(x) \geq g(x) \]

∫ STANDARD INTEGRALS

1. \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1 \]
2. \[ \int \cos x \, dx = \sin x + C \]
3. \[ \int \sin x \, dx = -\cos x + C \]
4. \[ \int \sec^2 x \, dx = \tan x + C \]
5. \[ \int \frac{1}{x} \, dx = \ln|x| + C, \quad x \neq 0 \]

↗ ANGLE BETWEEN TWO VECTORS

\[ a \cdot b = \|a\| \|b\| \cos \theta \]

📊 LOGARITHM

1. \[ \log_b a^n = n \log_b a \]
2. \[ \log_b (ac) = \log_b a + \log_b c \]
3. \[ \log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c, \quad c \neq 0 \]
4. \[ \log_b b = 1 \]
5. \[ \log_b 1 = 0 \]

d/dx STANDARD DERIVATIVES

1. \[ \frac{d}{dx} (x^n) = nx^{n-1}, \quad n \neq 0 \]
2. \[ \frac{d}{dx} (\sin x) = \cos x \]
3. \[ \frac{d}{dx} (\cos x) = -\sin x \]
4. \[ \frac{d}{dx} (\tan x) = \sec^2 x \]
5. \[ \frac{d}{dx} (e^x) = e^x \]
6. \[ \frac{d}{dx} (\ln x) = \frac{1}{x} \]

⭕ ANGLES IN A CIRCLE

1. Angles in the same segment of a circle are equal.
2. The angle which an arc of a circle subtends at the centre is twice the angle which the arc subtends at the circumference.
3. The opposite angles of any quadrilateral inscribed in a circle are supplementary.

🎲 PROBABILITY

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

📊 STATISTICS

1. Mean   \[ \bar{x} = \frac{\sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i} \]
2. Standard Deviation   \[ s = \sqrt{\frac{\sum_{i=1}^n (f_i - \bar{x})^2}{\sum_{i=1}^n f_i}} \]

Formulae Sheet for Advanced Mathematics — Upper Secondary School Certificate Examinations 2022