# WJEC Core 2 Maths - Key Facts

Flashcards by Daniel Cox, updated more than 1 year ago Created by Daniel Cox over 4 years ago 92 2

### Description

Key facts and formulae which must be known for the WJEC Core 2 examination. ## Resource summary

 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 $a^2=b^2+c^2-2bc \cos A$ $\log_a a =?$ $\log_a a =1$ $\log_a 1 =?$ $\log_a 1 =0$ 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? $Area=\frac{1}{2}r^2 \theta$ Formula for the length of an arc? $s=r \theta$ How would you find the area of a segment of a circle? \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 an arithmetic sequence... [given in the formulae booklet] $u_n=a+(n-1)d$ Formula for the sum of the first $$n$$ terms of an arithmetic sequence... [given in the formulae booklet] $S_n=\frac{n}{2}\left ( 2a+(n-1)d \right )$ or $S_n=\frac{n}{2}\left ( a+l \right )$ where $$l$$ is the last term 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? Work out $$\int_{a}^{b} f(x) \, dx$$ General equation of a circle, centre $$\left ( a,b \right )$$ and radius $$r$$ $\left ( x-a \right )^2+\left ( y-b \right )^2=r^2$ If we are given $$\frac{dy}{dx}$$ or $$f'(x)$$ and told to find $$y$$ or $$f(x)$$, we need to... Integrate [remember to include $$+c$$] What is the angle between the tangent and radius at $$P$$? $90^{\circ}$ This is always true at the point where a radius meets a tangent 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. 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 $$C$$? $90^{\circ}$ Draw the graph of $$y=\sin x$$ for $$0\leq x \leq 2\pi$$ Draw the graph of $$y=\cos x$$ for $$0\leq x \leq 2\pi$$ 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 Differentiation is the reverse of ...? Integration Integration is the reverse of ...? Differentiation What is the condition for two circles to touch externally? The distance between their centres is the sum of their radii What is the condition for two circles to touch internally? The distance between their centres is the difference in their radii 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

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