MATHS - PURE

What is the cosine rule

$a^2=b^2+c^2-2bc\cos(A)$ - for finding a side

• rearrange for finding an angle

what are the special limits for differentiation by first principles that you are expected to know

• $$\lim_{h\rightarrow0}\frac{\sin\left(h\right)}{h}=1$$

• $$\lim_{h\rightarrow0}\frac{\cos(h)-1}{h}=0$$

How would you find out if a curve is convex, concave or has a point of inflection

• Concave ~ \frac{d^2y}{dx^2}<0

• Convex ~ \frac{d^2y}{dx^2}>0

• Point of inflection ~ $\frac{d^2y}{dx^2}=0$

How would you find out if a curve has two roots, one real root (repeated roots) or no real roots

• Two real roots ~ b^2-4ac>0

• One real root (repeated roots) ~ $b^2-4ac=0$

• No real roots ~ b^2-4ac<0

How would you find out if a curve is increasing, decreasing or stationary

How would you convert degrees into radians and radians into degrees

360° = 2$\pi$ radians

180° =$\pi$ radians

• Radians into degrees ~ divide by$\pi$ —> x180

• Degrees to radians ~ divide by 180 —> x$\pi$

What are all the pythagorean identities

• $$\sin^2x+\cos^2x=1$$

• $$\sec^2x=1+\tan^2x$$

• $$\operatorname{cosec}^2x=1+\cot^2x$$

How would you work out the nth term of an arithmetic sequence

• $$Un=a+\left(n-1\right)d$$

  • a is the first term

  • d is the common difference

What is the Newton-Raphson formula

• Used for finding the roots of a function

• Formula = $xn-\frac{f(x)}{f^{\prime}(x)}$

What is the formula for parametric integration $$\int_{t}^{t}y\cdot\frac{\differentialD x}{\differentialD t}dt

• \int_{t}^{t}y\frac{\differentialD x}{\differentialD t}dt

What is the formula for finding the nth term of a geometric sequence

• $$Un=ar^{n-1}$$

  • a is the first term

  • r is the common ratio

What is the general formula for exponential growth

• $$y=Ae^{kt}$$

What is the general formula for exponential decay

• $$y=Ae^{-kt}$$

What is the simplified version of the tan double angle formula

• $$\tan(2x)=\frac{2\tan x}{1-\tan^2x}$$

How would you find a unit vector of a given vector

• Divide by it’s magnitude

• e.g. if the vector is a, the unit vector would be $\frac{a}{\vert a\vert}$

What is the trapezium rule (in word form)

• Width of strip / 2 [1st height + 2(sum of middle heights) + last height]

What are the simplified versions of the cos(2A) double angle formula

• $$\cos^2A-\sin^2A$$

• $$1-2\sin^2A$$

How do you decide what to make as your “U” for integration by parts?

• Logarithms

• Algebra

• Trigonometry

• Exponentials

What is ln x integrated?

• xln(x) - x

how do you find the area of a sector in degrees and radians?

Why can’t you do the inverse of a many to one function?

• Because it would become a one to many function (which is not a function)

  • because each input must only have one output

What is $2^{x}$ differentiated and integrated?

• Differentiated = $2^{x}\ln2$

• Integrated =$\frac{2^{x}}{\ln2}$

What is domain and range?

• Domain = used as the “inputs” - range depends on the domain - x values

• Range = y values

What do the numbers in the number system mean?

How do you find the domain of the inverse of a function?

• The range of the original function