SSCE HSC Mathematics Extension 2 Modulus and argument: Consider the function $f(x) = \f

Question

Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$.

(i) Show that $f(x)$ is increasing for all $x$.

(ii) Show that $f(x)$ is an odd function.

(iii) Describe the behaviour of $f(x)$ for large positive values of $x$.

(iv) Hence sketch the graph of $f(x) = \frac{e^x - 1}{e^x + 1}$.

(v) Hence, or otherwise, sketch the graph of $y = \frac{1}{f(x)}$.

Solve the quadratic equation $z^2 + (2 + 3i)z + (1 + 3i) = 0$, giving your answers in the form $a + bi$, where $a$ and $b$ are real numbers.

(c) Find $\int x 	an^{-1} x \, dx$.

(d) Let $P(x)$ be a polynomial.

(i) Given that $(x - a^2)$ is a factor of $P(x)$, show that $P(a) = P(a^2) = 0$.

(ii) Given that the polynomial $P(x) = x^4 - 3x^3 + x^2 + 4$ has a factor $(x - a^2)$, find the value of $\alpha$.

Consider the function $f(x) = \frac{e^x - 1}{e^x + 1}$ - HSC - SSCE Mathematics Extension 2 - Question 12 - 2017 - Paper 1