Algebra

Find the values of $a$ and $b$.

When $a$ and $b$ have the values found in part (a), factorise $p(x)$ completely.

Hence solve the inequality $p(x) < 0$.

Sketch the graph of $y = |2x - 3|$.

Solve the inequality $3x - 1 < |2x - 3|$.

Expand $(6-x)(1-2x)^{-\frac{3}{2}}$ in ascending powers of $x$, up to and including the term in $x^2$, simplifying the coefficients.

State the set of values of $x$ for which the expansion is valid.

Sketch the graph of $y = |3x - 2a|$, where $a$ is a positive constant.

Hence or otherwise solve the inequality $|3x - 2a| < x + 5a$.

Express $f(x)$ in partial fractions.

Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$.

State the set of values of $x$ for which the expansion in part (b) is valid.

Express $\frac{12x^2 + 55x - 2}{(3x - 2)(x + 6)}$ in partial fractions.

Hence obtain the expansion of $\frac{12x^2 + 55x - 2}{(3x - 2)(x + 6)}$ in ascending powers of $x$, up to and including the term in $x^2$.

Express $f(x)$ in partial fractions.

Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$.

Sketch the graph of $y = |x + 3a|$, where $a$ is a positive constant.

Hence or otherwise solve the inequality $|x + 3a| > a - 2x$.

It is given that $f(x) = (x - a)^2 g(x)$, where $f(x)$ and $g(x)$ are polynomials.

Show that $(x - a)$ is a factor of $f'(x)$.

It is given that $(x - 3)^2$ is a factor of $2x^3 - 4x^2 + px + q$, where $p$ and $q$ are constants.

Find the values of $p$ and $q$.