The complex number z is such that |z| = 2 and arg(z) = \frac{\pi}{4} - HSC - SSCE Mathematics Extension 2 - Question 12 - 2015 - Paper 1

To find u, we calculate:

$u = z^2 = (\sqrt{2} + i \sqrt{2})^2 = 2 + 2i\sqrt{2} - 2 = 2i\sqrt{2}$

Now in Cartesian form, we plot the point (0, 2\sqrt{2}) on the Argand diagram, which lies on the positive imaginary axis.

First, we need to compute the conjugate of z:

$\overline{z} = \sqrt{2} - i\sqrt{2}$

Now, substituting for v:

$v = z^2 - \overline{z} = 2i\sqrt{2} - (\sqrt{2} - i\sqrt{2}) = (0 + 2i\sqrt{2}) - (\sqrt{2} - i\sqrt{2}) = (\sqrt{2} + i\sqrt{2})$

Plot the resultant vector on the Argand diagram.

To find the roots of polynomial P(x), we need to evaluate the polynomial at various x-values. By applying the Rational Root Theorem and synthetic division, we can determine that:

The roots are x = 1, 2, 3 + i, 3 - i. Here, a = 3 and b = 1.

The complex roots (3 + i) and (3 - i) suggest a quadratic factor:

$Q(x) = (x - (3 + i))(x - (3 - i)) = (x - 3 - i)(x - 3 + i) = (x - 3)^2 + 1 = x^2 - 6x + 10$

This quadratic polynomial has real coefficients and is a factor of P(x).

To express (\frac{(x - 2)(x - 5)}{x - 1}) in the desired form: Perform polynomial long division:

1. Divide (x - 2)(x - 5) by (x - 1):

Using synthetic division, we arrive at: $m = 2, b = -7, a = -3, \implies \frac{(x - 2)(x - 5)}{x - 1} = 2x - 7 + \frac{-3}{x - 1}$

The expression simplifies, showing vertical asymptotes at x = 1 and x-intercepts at x = 2 and x = 5. Additionally, the oblique asymptote derived from previous calculations is y = 2x - 7. Therefore, sketching the graph involves indicating intercepts and asymptotes in the Cartesian plane.