Edexcel Core 2 Maths - Key Facts

Daniel Cox
Flashcards by , created almost 3 years ago

Key facts and formulae which must be known for the Edexcel Core 2 examination.

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Daniel Cox
Created by Daniel Cox about 3 years ago
Daniel Cox
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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? 246e5789-f0a7-4bdd-adf3-6c292cebdf77.png (image/png) \[Area=\frac{1}{2}r^2 \theta \] d9908c47-eea7-4a82-8040-ff473cf307cf.png (image/png)
Formula for the length of an arc? 7b0c27d1-8771-4950-8671-09fd560d29b0.gif (image/gif) \[s=r \theta\] 33c8b397-c996-4ddc-be92-21aa0cb46497.gif (image/gif)
How would you find the area of a segment of a circle? 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? 0a3690b2-1482-449a-aae6-dff7cd33d632.png (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 \(P\)? 836b171c-39e7-4991-8c29-be862870372c.jpg (image/jpg) \[90^{\circ}\] This is always true at the point where a radius meets a tangent 7b9e0001-91f1-483e-9305-f4ace70cf194.png (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. 16493cc7-79d3-45f7-b601-63787610188f.jpg (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 \(C\)? 7528b419-763e-4ed3-95f2-3fd66f8da729.gif (image/gif) \[90^{\circ}\]
Draw the graph of \(y=\sin x\) for \(0\leq x \leq 2\pi\) c5195b8d-e25a-460f-a4a3-3189515f3468.gif (image/gif)
Draw the graph of \(y=\cos x\) for \(0\leq x \leq 2\pi\) 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 9e44c61a-ff49-4279-92e9-50bbaf346ee4.png (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 4652850b-01de-4209-ac80-e44d81393c43.png (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