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Flashcards by Daniel Cox, updated more than 1 year ago
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Created by Daniel Cox
almost 8 years ago
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Copied by Daniel Cox
almost 8 years ago
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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? | \[Area=\frac{1}{2}r^2 \theta \] |
| 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 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 \(r\) | \[\left ( x-a \right )^2+\left ( y-b \right )^2=r^2\] |
| 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 |
| 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 |
| 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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