Albegraic manipulation

a^3 - b^3 &= (a-b)(a^2+ab+b^2) \\ a^3 + b^3 &= (a+b)(a^2-ab+b^2) \\ (a +b+c)^2 &= a^2 + b^2 + c^3 + 2ab + 2bc + 2ac \end{align*}$$ ## powers #smtp $$\begin{align} a^n - b^n &= (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \dots + ab^{n-2} + b^{n-1}) \quad \text{for all positive intergers $n$}\\ a^n + b^n &= (a + b)(a^{n-1} - a^{n-2}b + a^{n-3}b^2 - \dots - ab^{n-2} + b^{n-1}) \quad \text{only for odd intergers $n$} \end{align}$$ # Surds and indices $$\begin{align} a^m \cdot a^n &= a^{m+n} \\ a^m \div a^n &= a^{m-n} \\ (a^m)^n &= a^{mn} \\ a^n \cdot b^n &= (ab)^n \\ a^n \div b^n &= (\frac{a}{b})^n \\ \end{align}$$

\left. \begin{aligned} a^{\frac{m}{n}} &= (\sqrt[n]{a })^m \ &= \sqrt[n]{ a^m } \end{aligned} \right} \quad \text{where } n \text{ and } m \text{ are positive integers}

$$$$

\text{For } a \in \mathbb{R} \setminus {-1, 0, 1}, \quad a^x = a^y \implies x = y.

$$rationalising multiple irrational parts$$

\begin{align*} \frac{1}{1 + \sqrt{2} + \sqrt{3}} &= \frac{1}{(1 + \sqrt{2}) + \sqrt{3}} \cdot \frac{(1 + \sqrt{2}) - \sqrt{3}}{(1 + \sqrt{2}) - \sqrt{3}} \ &= \frac{1 + \sqrt{2} - \sqrt{3}}{(1 + \sqrt{2})^2 - (\sqrt{3})^2} \ &= \frac{1 + \sqrt{2} - \sqrt{3}}{2\sqrt{2}} \ &= \frac{1 + \sqrt{2} - \sqrt{3}}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \ &= \frac{2 + \sqrt{2} - \sqrt{6}}{4} \end{align*}

# Linear graphs and simultaneous equations

\begin{align*} \text{distance} &= \sqrt{ (x_{2})-x_{1} )^2 + (y_{2} - y_{1})^2} \ \text{midpoint of $(x_1, y_1)$ and $(x_2, y_2)$} &= (\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}) \ \text{gradient} &= \tan(\theta) \end{align*}

# Quadratic equations and inequalities

\begin{align*} ax^2 + bx + c &= 0, \ a \ne 0 \ \text{discriminant} &= b^2 - 4ac \[10pt] \text{Nature of roots:} & \begin{cases} \text{discriminant} > 0 & \text{Two distinct real roots} \ & \text{(If } \text{discriminant} \text{ is a perfect square, roots are rational)} \ \text{discriminant} = 0 & \text{One repeated real root} \ \text{discriminant} < 0 & \text{Two complex conjugate roots} \ \text{discriminant is a perfect square} & \text{two distinct real rational roots} \end{cases} \end{align*}

$$$$

k(x - \alpha)(x - \beta) = 0 \quad \text{where } k \neq 0, \ \alpha \text{ and } \beta \text{ are roots}

$$$$

\begin{align*} x^2 + ax &= \left(x + \frac{a}{2}\right)^2 - \left(\frac{a}{2}\right)^2 \ a(x - h)^2 + k &= 0 \[15pt] \text{Where:} & \ a &\quad \text{Dictates the parabola's direction and stretch/compression.} \ (h, k) &\quad \text{The coordinate of the parabola's vertex (the turning point).} \end{align*}$$

$$\begin{align} ax^2 + bx + c &= 0 \\ \text{sum of roots} &= -\frac{b}{a} \\ \text{product of roots} &= \frac{c}{a} \\ x^2 - (\text{sum of roots})x + (\text{product of roots}) &= 0 \\ \text{line of symmetry} &\text{ is } x=-\frac{b}{2a} \end{align}$$ $$\begin{align*} a > c &\implies ab > cb \quad \text{if } b > 0 \\ a > c &\implies ab &lt; cb \quad \text{if } b &lt; 0 \\ \frac{a}{b} > 0&\implies ab>0, b \neq 0 \end{align*}$$

polynomials

$$\begin{align*} \frac{f(x)}{ (ax-b)} \text{ remainder } &= f(\frac{b}{a})\quad \text{by remainder theorem}\\ \text{if $(x-a)$ is a factor of $f(x)$} &\implies f(a) = 0 \quad \text{by Factor theorem} \\ &\text{To find a linear factor of } ax^3 + bx^2 + cx + d: \\[5pt] &\text{1. Test factors of } d \text{ works if $a=1$ } \\ &\text{2. Test } \frac{\text{factors of a}}{\text{factors of } a} \end{align*}$$

functions

#non-examinable

logirithms

• base is positive and not 1
• argument is positive

\log_{a}(b) &= \frac{\log_{c}(b)}{\log_{c}(a)} \\ \log_{b}(m\cdot n) &= \log_{b}(m) + \log_{b}(n) \\ \log_{b}\left( \frac{m}{n} \right) &= \log_{b}(m) - \log_{b}(n)\\ \log_{b}(1) &= 0 \\ \log_{b}(b) &= 1 \\ \log_{a}(b^k) &= k\cdot \log_{a}(b) \\ b^{\log_{b}(k)} &= k \\ lg(a) &= \log_{10}(a) \\ \ln(a) &= \log_{e}(a) \\ \frac{1}{\log_{b}a} &= \log_{a}b \end{align*}$$ # Graph transformations | Manipulation | Function | Effect on Graph | | :--------------------- | :------------------------- | :---------------------------------------------------------------------- | | Vertical Translation | $f(x) + c$ | Translates $c$ units upward, parallel to y-axis | | ^ | $f(x) - c$ | Translates $c$ units downward, parallel to y-axis | | Horizontal Translation | $f(x - c)$ | Translates $c$ units right, parallel to x-axis | | ^ | $f(x + c)$ | Translates $c$ units left, parallel to x-axis | | Horizontal Scale | $f(kx)$ | Scaled by factor of $\frac{1}{k}$ parallel to x-axis (y-axis invariant) | | Vertical Scale | $k f(x)$ | Scaled by factor of $k$ parallel to y-axis (x-axis invariant) | | Reflection | $-f(x)$ | Reflects across the x-axis | | ^ | $f(-x)$ | Reflects across the y-axis | | special reflection | $y=f(x) \implies x = f(y)$ | Reflects graph about the line $y=x$ | # Matrices - order of matrix with $m$ rows and $n$ columns is $m \times n$ - when writing what matrices represent, must write values - e.g. $PC$ represents the total production cost of two types of toys, 2800 automated and 2412098 clockwork respectively - properties:

\begin{align} A + B &= B + A \ A + (b + C) &= (A + B) + C \ A(B + C) &= AB + AC \ AB &\neq BA \ AI &= IA \ &= A \ AA^{-1} &= A^{-1}A \ &= I \end{align}

$$AB = \begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix} \begin{pmatrix} g & h \\ i & j \\ k & l \end{pmatrix} = \begin{pmatrix} ag + bi + ck & ah + bj + cl \\ dg + ei + fk & dh + ej + fl \end{pmatrix}$$ ## square matrices

\begin{align*} I &= \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \ &= \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}\ I &= I^2\ &= I^3 \end{align*}

$$$$

\begin{align*} A &= \begin{pmatrix} a & b \ c & d \end{pmatrix} \ A^{-1} &= \frac{1}{\text{determinant}} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \ \text{determinant} &= ad - bc \end{align*}

## System of equations - only works if $\det(\begin{pmatrix} a & b \\ d & e \end{pmatrix}) \neq 0$, else either no solutions or infinite solutions

\begin{aligned} ax + by &= c \ dx + ey &= f \ \begin{pmatrix} a & b \ d & e \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} &= \begin{pmatrix} c \ f \end{pmatrix}\ \begin{pmatrix} x \ y \end{pmatrix} &= \begin{pmatrix} a & b \ d & e \end{pmatrix}^{-1} \begin{pmatrix} c \ f \end{pmatrix} \end{aligned}$$

Straight lines and circles

Coordinate geom

gradient point form

$$\begin{align*} y - y_{1} &= m(x-x_{1}) \\ m_{\text{perpendicular line}} &= -\frac{1}{m} \end{align*}$$

Perpendicular lines

$m_{\text{perpendicular line}} = - \frac{1}{m}$

area of a triangle

• points need to be in anit-clockwise order

$$\begin{align} \text{area of polygon} &= \left| \begin{matrix} x_{1} & x_{2} & x_{3} & x_{1}\\ y_{1} & y_{2} & y_{3} & y_{1} \end{matrix} \right| \\ &= x_{1}y_{2} + x_{2}y_{3} + x_{3}y_{1} - x_{1}y_{3} - x_{3}y_{2} - x_{2}y_{1} \end{align}$$

Circles

$(x-a)^2 + (y-b)^2 = r^2$

• center of circle is (a, b)
• radius of circle is r units

$$\begin{align} x^2 + y^2 + 2gx + 2fy + c &= 0 \\ \text{radius} &= \sqrt{ f^2 + g^2 -c } \\ \end{align}$$

• center of circle is (-g, -f)
• given two circles with radius $r_{1}$ and $r_{2}$, and $r_{2} \geq r_{1}$
  • r_{1} + r_{2} &lt; d
    • No points of intersection
    • Two circles are completely outside each other
  • $r_{1} + r_{2} = d$
    • 1 point of intersection
    • circles touch on the outside
  • r_{2}-r_{1} &lt; d &lt; r_{1}+r_{2}
    • two points of intersection
  • $r_{2}-r_{1} = d$
    • 1 point of intersection
    • one circle is inside the other
  • d &lt; r_{2}-r_{1}
    • no points of intersection
    • one circle is completely inside another
  • $d=0$
    • $r_{1} \neq r_{2}$
      • one circle is inside another
    • $r_{1}=r_{2}$
      • infinite points of intersection
      • circles touch

Linear law

Linear conversion

$Y=mX + C$

Experimental data

• Graph is Y against X
• plots are done using X
• Values taken from the graph (e.g. predicting values, evaluating gradient) done using $\cdot$
• label intervals at least every 10-20 squares
• When presenting solutions to constants, use $\approx$ when appropriate; e.g.
  • $lg(k) \approx 3.48\quad(\text{"from the graph" or "from gradient point form" if y-intercept canot be found})$
  • \begin{align}k &\approx 10^{3.48} \\ &= 3020 \quad (\text{to 3s.f.})\end{align}

Trig

Circular measure

Radian /area of circle

1^\circ &= 2\pi \\ \pi &= 180^\circ \\ \Theta &= \frac{s\text{ (arc length)}}{r\text{ (radius)}} \\ \text{area of sector} &= \frac{1}{2}r^2\Theta \quad \text{$\Theta$ is in radians, never in degrees}\\ &= \frac{1}{2}rs \end{align}$$ ## Trig ratios ### Quadrants - if $\Theta$ is in the $1^\text{st}$ quadrant - $\sin(\Theta)$, $\cos(\Theta)$, $\tan(\Theta)$ are positive - if $\Theta$ is in the $2^\text{nd}$ quadrant - $\sin(\Theta)$ is positive - if $\Theta$ is in the $3^\text{rd}$ quadrant - $\tan(\Theta)$ is positive - if $\Theta$ is in the $4^\text{th}$ quadrant - $\cos(\Theta)$ is positive ### Trig ratios for acute angles #### Special angles | $\theta$ (Degrees) | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ | | :----------------- | :-------: | :------------------: | :------------------: | :------------------: | :-------------: | | $\theta$ (Radians) | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | | $\sin\theta$ | $0$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | $1$ | | $\cos\theta$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $0$ | | $\tan\theta$ | $0$ | $\frac{1}{\sqrt{3}}$ | $1$ | $\sqrt{3}$ | $-$ | #### complemntary angles $$\begin{align} \sin(90^\circ - \Theta) &= \cos(\Theta) \\ \cos(90^\circ - \Theta) &= \tan(\Theta) \\ \tan(90^\circ - \Theta) &= \frac{1}{\tan(\Theta)} \end{align}$$ ### trig ratios of any angle $$\begin{align} \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \end{align}$$ #### Negative angles $$\begin{align} \sin(-\Theta) &= -\sin(\Theta) \\ \sin(-\Theta) &= \cos(\Theta) \\ \tan(-\Theta) &= -\tan(\Theta) \end{align}$$ #### Suplementary angles $$\begin{align} \sin(180^\circ -\Theta) &= \sin(\Theta) \\ \cos(180^\circ - \Theta) &= -\cos(\Theta) \\ \tan(180^\circ - \Theta) &= -\tan(\Theta) \end{align}$$ ## sin and cosine rule ### Area of triangle

\text{area of triangle} = \frac{1}{2}ab\sin(C)$$

Sin rule

$\frac{\sin(A)}{a} = \frac{\sin(B)}{b}$

Cosine rule

$c^2 = a^2 + b^2 - 2ab\cos(C)$

Heron's formula #smtp

s &= \frac{a+b+c}{2} \\ \text{area of triangle} &= \sqrt{ s(s-a)(s-b)(s-c) } \end{align}$$ # Bearings and 3d problems ## Bearings - ambiguous triangles occur when there are more than 1 possible triangle that satisfies the requirements - SAS and ASA triangles are always non-ambiguous - SSA triangles may be ambiguous, given sides c and b, and angle B - assume points are counter-clockwise ![[IMG-20260923221217925.png300]] - side a is the baseline - side c is the fixed adjecent side - side b is the one that "swings" - altitude droped from A to side c is $h=c\sin(B)$ - $b<h$ - no triangle - $b=h$ - 1 right-angled triangle - $h<b<c$ - 2 ambiguous triangles, 1 obtuse C and 1 acute C