Edexcel Core 2 Maths - Key Facts

Question	Answer
$\log_a x+\log_a y = ?$	$\log_a x+\log_a y = \log_a(xy)$
$\log_a x-\log_a y = ?$	$\log_a x-\log_a y = \log_a\left (\frac{x}{y} \right )$ NOT $\frac{\log_a x}{\log_a y}$
$k \log_a x = ?$	$k \log_a x = \log_a\left (x^k \right )$
State the sine rule	$\frac{a}{\sin A}=\frac{b}{\sin B}$ or $\frac{\sin A}{a}=\frac{\sin B}{b}$
True or false? $\log_a\left (xy^k \right )=k \log_a\left ( xy \right )$	FALSE $\begin{align} \log_a\left (xy^k \right )&=\log_a x +\log_a \left (y^k \right )\\ &=\log_a x + k \log_a y \end{align}$
What is the trigonometric formula for the area of a triangle?	$Area=\frac{1}{2} ab \sin C$ Here, the sides $a$ and $b$ surround the angle $C$
What is the Pythagorean trigonometric identity? (Hint: it involves $\sin^2 x$ and $\cos^2 x$	$\sin^2 x + \cos^2 x = 1$
If $y=a^x$ , then $x=?$	If $y=a^x$ , then $x=\log_a y$
State the cosine rule [given in the formulae booklet]	$a^2=b^2+c^2-2bc \cos A$
$\log_a a =?$	$\log_a a =1$
$\log_a 1 =?$	$\log_a 1 =0$
State an identity relating $\sin x$ , $\cos x$ and $\tan x$	$\frac{\sin x}{\cos x}=\tan x$
How many degrees is $\pi$ radians?	$\pi$ radians is $180^{\circ}$
Formula for the area of a sector? Image: 246e5789-f0a7-4bdd-adf3-6c292cebdf77 (image/png)	$Area=\frac{1}{2}r^2 \theta$ Image: d9908c47-eea7-4a82-8040-ff473cf307cf (image/png)
Formula for the length of an arc? Image: 7b0c27d1-8771-4950-8671-09fd560d29b0.gif (image/gif)	$s=r \theta$ Image: 33c8b397-c996-4ddc-be92-21aa0cb46497.gif (image/gif)
How would you find the area of a segment of a circle? Image: 3f2f7ddf-b1c4-480f-86b7-8009148d4c6a.gif (image/gif)	$\begin{align} \mathrm{Segment}&= \mathrm{Sector}-\mathrm{Triangle}\\ &=\frac{1}{2}r^2\theta-\frac{1}{2}r^2 \sin\theta\\ &=\frac{1}{2}r^2\left ( \theta - \sin\theta \right ) \end{align}$
Formula for the $n$ th term of a geometric sequence... [given in the formulae booklet]	$u_n=ar^{n-1}$
Formula for the sum of the first $n$ terms of a geometric sequence... [given in the formulae booklet]	$S_n=\frac{a\left ( 1-r^n \right )}{1-r}$
Formula for the sum to infinity of a convergent geometric series (one where $\left \| r \right \|<1$ ) [given in the formulae booklet]	$S_\infty=\frac{a}{1-r}$
$\int ax^n \, dx=\, ?$	$\int ax^n \, dx= \frac{ax^{n+1}}{n+1}+c$
How would you find this shaded area? Image: 0a3690b2-1482-449a-aae6-dff7cd33d632 (image/png)	Work out $\int_{a}^{b} f(x) \, dx$
General equation of a circle, centre $\left ( a,b \right )$ and radius $r$	$\left ( x-a \right )^2+\left ( y-b \right )^2=r^2$
What is the angle between the tangent and radius at $P$ ? Image: 836b171c-39e7-4991-8c29-be862870372c (image/jpg)	$90^{\circ}$ This is always true at the point where a radius meets a tangent Image: 7b9e0001-91f1-483e-9305-f4ace70cf194 (image/png)
What does the graph of $y=a^x$ look like? Where does it cross the axes?	It goes through the $y$ -axis at $\left ( 0,1 \right )$ . It does not cross the $x$ -axis. The $x$ -axis is an asymptote. Image: 16493cc7-79d3-45f7-b601-63787610188f (image/jpg)
This is a triangle inside a semicircle, where one side of the triangle is the diameter of the circle. What is the size of angle $C$ ? Image: 7528b419-763e-4ed3-95f2-3fd66f8da729.gif (image/gif)	$90^{\circ}$
Draw the graph of $y=\sin x$ for $0\leq x \leq 2\pi$	Image: c5195b8d-e25a-460f-a4a3-3189515f3468.gif (image/gif)
Draw the graph of $y=\cos x$ for $0\leq x \leq 2\pi$	Image: eda70f8f-b1c3-4e65-9f56-d06139c3c3ad.gif (image/gif)
Draw the graph of $y=\tan x$ for $0\leq x \leq 2\pi$	The lines $x=\frac{\pi}{2}$ and $x=\frac{3\pi}{2}$ are asymptotes Image: 9e44c61a-ff49-4279-92e9-50bbaf346ee4 (image/png)
If $\left (x+a \right )$ is a factor of $f(x)$ , then...	$f(-a)=0$ This is known as the Factor Theorem
If the remainder, when $f(x)$ is divided by $(x+a)$ is R, then...	$f(-a)=R$ This is known as the Remainder Theorem
If we draw the perpendicular bisector of any chord on a circle, which point will it definitely go through?	The perpendicular bisector of a chord always passes through the centre of the circle Image: 4652850b-01de-4209-ac80-e44d81393c43 (image/png)
What does $n!$ mean?	$n!=n(n-1)(n-2)\times \ldots \times 3 \times 2 \times 1$ For example, $4!=4\times 3\times 2\times 1=24$
How would you use the second derivative, $\frac{d^2 y}{dx^2}$ to determine the nature of the stationary points on a graph?	Substitute the $x$ co-ordinates of the stationary points into $\frac{d^2 y}{dx^2}$ . If you get a positive answer, it's a MIN. If you get a negative answer, it's a MAX.
A function is said to be 'increasing' when its gradient is...	Positive
A function is said to be 'decreasing' when its gradient is...	Negative

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Edexcel Core 2 Maths - Key Facts

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