Specialist2025 NHT Exam 2 Section B · Q1CAS active10 marks

Rational function family: domain, partial fractions, graph and inflection

Consider the family of functions $f$ with rule $f(x) = \frac{x^3 + ax^2 + ax + 3}{x^2 - 1}$ , where is a real constant.

Write down the implied domain of $f$ given that $a \neq -2$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

b.i.

The rule for $f$ can be written as $f(x) = x + a + g(x)$ .

Using partial fractions, show that $g(x) = \frac{a+2}{x-1} - \frac{1}{x+1}$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

b.ii.

Show that when $a = -2$ the graph of $y = f(x)$ has no stationary points.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Sketch the graph of $y = f(x)$ when $a = -2$ on the axes below.

Label all asymptotes with their equations, and label any intercepts with the coordinate axes with their values.

Blank Cartesian axes with a grid, x from about −5 to 5 and y from about −5 to 5, with ticks labelled −4, −2, 2, 4 on each axis, for sketching y = f(x).

3 marks

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

Using $f(x) = x + a + \frac{a+2}{x-1} - \frac{1}{x+1}$ , find $f''(x)$ where $a$ is an arbitrary real constant.

Give your answer in the form $f''(x) = \frac{A(a+2)}{(x-1)^p} + \frac{B}{(x+1)^p}$ where $A$ , $B$ and $p \in R$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Specialist2025 NHT Exam 2 Section B · Q1CAS active10 marks

Rational function family: domain, partial fractions, graph and inflection

Consider the family of functions $f$ with rule $f(x) = \frac{x^3 + ax^2 + ax + 3}{x^2 - 1}$ , where is a real constant.

Write down the implied domain of $f$ given that $a \neq -2$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

b.i.

The rule for $f$ can be written as $f(x) = x + a + g(x)$ .

Using partial fractions, show that $g(x) = \frac{a+2}{x-1} - \frac{1}{x+1}$ .

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

b.ii.

Show that when $a = -2$ the graph of $y = f(x)$ has no stationary points.

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Sketch the graph of $y = f(x)$ when $a = -2$ on the axes below.

Label all asymptotes with their equations, and label any intercepts with the coordinate axes with their values.

Blank Cartesian axes with a grid, x from about −5 to 5 and y from about −5 to 5, with ticks labelled −4, −2, 2, 4 on each axis, for sketching y = f(x).

3 marks

Hints, answer checking and worked solutions come with a Season Pass.

d.i.

Using $f(x) = x + a + \frac{a+2}{x-1} - \frac{1}{x+1}$ , find $f''(x)$ where $a$ is an arbitrary real constant.

Give your answer in the form $f''(x) = \frac{A(a+2)}{(x-1)^p} + \frac{B}{(x+1)^p}$ where $A$ , $B$ and $p \in R$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.