Chapters Completed So Far: 1, 2, 3, 4
Chapter One - Further Inequalities
- Multiplying or dividing by a negative term necessitates the reversal of the inequality sign. A term such as (x - 2) may be positive or negative depending on the value of x. To avoid confusion, instead multiply by (x - 2)^2, since any real number squared is greater than or equal to zero.
- Try and keep the functions you are manipulating in factorised form wherever possible. For example, (x-1)(x-1)(x+3) - (x-1)(x+2), should NOT be expanded. Factor out the (x-1) term and then simplify what's left. This situation is very common if the squaring technique above is used.
- If your function does not factorise, replace the inequality by an equals sign and solve to get the critical values. Once the critical values have been identified, plot the graphs and identify the appropriate region based on the nature of the inequality symbol
- When dealing with modulus symbols, isolate the function in the modulus lines, plot the two graphs and identify the critical values
Chapter Two - Summation by the Method of Differences
- If you are asked to find the sum of f(r) to n terms, you will almost always be asked previously to express f(r) = g(r) - g(r + a), where a is an integer.
- a is usually less than or equal to 2
- Express your sum as pairs of terms (with the same values of r in each pair), with each successive pair under the last. Express the first four, then write .... underneath and continue expressing n-3 or n-2 up to n. Cancel according terms, then rewrite the remaining terms below and simplify. By doing this you make it clear to the examiner you understand the method of differences.
Chapter Three - Further Complex Numbers
- For a complex number z = x + iy,
- The modulus of z, |z| = r = sqrt(x^2 + y^2)
- The argument of z, arg(z) is more complicated. The method I use is as follows. Calculate arctan|x/y|, (note the modulus sign), which we will call a.
- First quadrant => arg(z) = a
- Second quadrant => arg(z) = pi - a
- Third quadrant => arg(z) = a - pi
- Fourth quadrant => arg(z) = - a
- In these notes, arg(z) = A
- Mod-Arg form: z = r(cosA + isinA)
- Exponential form: z = re^(iA)
- de Moivre's theorem: z^n = (r^n)(cosAn + isinAn)
- To solve an equation of the form z^n = r(cosA + isinA), replace A by (A + 2k(pi)), where k is an integer. Then, apply de Moivre's theorem in reverse (take the nth root of r, and divide the modified theta by n). Then, starting from k = 0, work outwards with integer values of k and reject the value of theta they give if it falls out of the accepted range (usually -pi < theta Using your values of theta, express the possible values of z in whatever notation the question requires
- To solve questions concerning the mapping of…