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Prove by induction that (n^3)-n is divisible by 3 for all integers n>0 (typical fp1 problem)

Let P(n) be the statement “(n^3)-n is divisible by 3” First we’ll examine the base case: P(1) (n^3)-n=1^3-1=1-1=0 0=3*0, so is divisible by 3, and so P(1) is true Now assume for some k>0, an integer, that...
TN
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How do I sketch accurate graphs for rational functions in a short amount of time? (I.e. A step by step guide of sketching graphs)

Often in Further Pure modules for various exam boards a common question in the exam paper will be to sketch a graph for a rational function. The step by step aproach I used is as follows: Change the form (if...
DH
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solve the equation 4cos^2(x) -15sin(x) = 13

We first want to get every term in terms of the same variable, namely sin(x). to do so, we will use the identity sin^2(x) +cos^2(x) = 1 to get: 4(1-sin^2(x)) -15sin(x) -13 = 0. which we can then rewrite as: ...
MG
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The function f is defined for x > 0 by f (x) = x^1n x. Obtain an expression for f ′ (x).

By taking the natural log on both sides we can see that: ln(f(x)) = ln(x)^2 This is a more familiar expression that we know how to differentiate LHS: f '(x)/f(x), RHS: 2*ln(x)/x By rearranging this we can se...
SE
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Given z=cosx+isinx, show cosx=1/2(z+1/z)

We first consider 1/z=(cosx+isinx)^(-1). Application of De Moivre's theorem for integer n: (cosx+isinx)^(n)=cosnx+isinnx yields the result 1/z=cosx-isinx. Addition of the two forms z and z^(-1) steers us to ...
CS
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