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How do I apply mathematical induction to answer questions

If you are familiar with induction but unsure how to answer the types of questions that require it, a structure can be useful. Even when confident, it will help an examiner to see where to award you marks if...
OM
3072 Views

z = -2 + (2root3)i. Find the modulus and argument of z.

The first step to answering this question is to pinpoint exactly what it means. We remind ourselves that when we are given a number in the form of a + bi, we are dealing with a complex number . A complex num...
MS
12569 Views

Find the Taylor Series expansion of tan(x) about π/4 up to the term in terms of (x-π/4)^3.

Firstly, set f(x) = tan(x). We can then differentiate this 3 times in order to find f'(x), f''(x) and f'''(x). This will require use of the chain rule and the product rule. We can find:f'(x) = sec 2 xf''(x) ...
JM
55931 Views

find all the roots to the equation: z^3 = 1 + i in polar form

1st write 1+ i in polar form, sketch a diagram to find the angle ( =pi/4) and find the modulus ( sqrt(2))z^3 = Sqrt(2) E^ipi/4This is true for all equivalent solutions (add 2kpi) z^3 = sqrt(2)e^(pi/4 +2kpi)i...
RM
11130 Views

Find the eigenvalues and eigenvectors of the matrix M , where M{2,2} = (1/2 2/3 ; 1/2 1/3) Hence express M in the form PDP^-1 where D is a diagonal matrix.

To start off, It's worth noting the definition of eigenvalues: for a Matrix A (n x n), it's ith eigenvalue (λ i ) is defined as the scalar constant the ith eigenvector (v i ) is multiplied for the matrix mul...
LT
3427 Views