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Further Mathematics

A Level

Express the complex number (1+i)/(1-i) in the form x+iy

First of all calculate the complex conjugate of the denominator. The complex conjugate of (1-i) is 1+i.Now multiply the given complex number by (1+i)/(1+i), note that we are not modifying the starting num...

Answered by Claudio M. • Further Mathematics tutor

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Given a curve with parametric equations, x=acos^3(t) and y=asin^3(t), find the length of the curve between points A and B, where t=0 and t=2pi respectively.

The length of an arc between two points on a curve can be calculated in two ways; as the integral of ((dy/dx)^2 + 1)^1/2 between the values of the points, or as the integral of ((dy/dt)^2+(dx/dt)^2)^1/2 b...

Answered by Jack W. • Further Mathematics tutor

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A parabola with equation y^2=4ax for constant a is translated by the vector (2,3) to give the curve C. The curve C passes through the point (4,7), what is the value of a?

Invert the translation of (2,3) to get the parabola passing through the point (4,7)-(2,3)=(2,4). This is the same as saying that y=4 when x=2, substitute this into your equation y^2=4ax to get a=2.This wi...

Answered by Gabriel V. • Further Mathematics tutor

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Express f(x) = ln(x+1) as an infinite series in ascending powers of x up to the 3rd power of x

Recall that the Maclaurin series for f(x) is f(x) = f(0) + f'(0)x + f''(0)x²/2! + ... + f^(r)(0)x<...

Answered by Caspar P. • Further Mathematics tutor

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prove by induction that, f(n) = 2^(3n+1) + 3(5^(2n+1)) is divisible by 17 for all n>0.

With induction we start with the base case n = 1. So setting n=1 we find that f(1) = 391 which is equal to 17x23. So indeed the base case holds.We assume that for positive integers k, f(k) is divisible by...

Answered by Matt B. • Further Mathematics tutor

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