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Prove algebraically that n^3+3n^2+2n+1 is odd for all integers n

To show this we should consider when n is odd and when n is even,If n is odd then we can find an m such that n=2m+1. Substituting n=2m+1 and expanding gives 8m^3+12m^2+6m+1+12m^2+12m+3+4m+2+1=8m^3+24m^2+22m+...
AP
Answered by Alistair P. • Maths tutor
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Calculate the volume obtained when rotating the curve y=x^2 360 degrees around the x axis for 0<x<2

I haven't yet figured out how to write in proper mathematical notation on here, and my drawing pad is yet to arrive, so please forgive the formatFirst, the formula for the volume of revolution is V= pi * the...
Answered by • Maths tutor
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Work out the equation of the tangent at x = 3, knowing that f(x) =x^2

Currently, we know that:f(x) = x 2 To find the equation of the tangent you need the gradient and the y-value of the graph at x = 3So, you can differentiate f(x):f'(x) = 2xlet x = 3: f(x) = 9 f'(x) = 6the equ...
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Answered by Arnav A. • Maths tutor
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For which values of k does the quadratic equation 2x^2+kx+3=0 only have one unique solution?

A quadratic equation ax 2 +bx+c=0 will have only one unique solution if and only if b 2 -4ac=0. If we apply this to the equation in the question, we get that k 2 -24=0. If we solve this equation we get two s...
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Answered by Ewan R. • Maths tutor
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FInd the equation of the line tangent to the graph g(x)=integral form 1 to x of cos(x*pi/3)/t at the point x=1

This problem requires us to find out the equation of the tangent line, which has form y=mx+q.M is the angular coefficient or slope of the line and can be evaluated through the derivative of the function at t...
Answered by • Maths tutor
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