AQA A-Level Maths: Advanced Algebra and Functions

The function $\text{f}$f is defined by $\text{f}(x) = \frac{4 + 20x - 2x^2}{(1 + x)(1 - 2x)(1 + 4x)}.$f(x)=(1+x)(1−2x)(1+4x)4+20x−2x2​.

(a) Express $\text{f}(x)$f(x) in partial fractions. (5 marks)

(b) Hence find the first three terms in the expansion of $\text{f}(x)$f(x) in ascending powers of $x$x, giving each coefficient as an integer. (4 marks)

(c) State the set of values of $x$x for which the expansion in part (b) is valid. (3 marks)

The functions $\text{f}$f and $\text{g}$g are defined by $\text{f}(x) = \frac{3x + 1}{x - 2}, \quad x \in \mathbb{R},\ x > 2, \qquad \text{g}(x) = x^2 - 3, \quad x \in \mathbb{R}.$f(x)=x−23x+1​,x∈R, x>2,g(x)=x2−3,x∈R.

(a) Find $\text{fg}(x)$fg(x), giving your answer as a single simplified fraction. (3 marks)

(b) Find $\text{f}^{-1}(x)$f−1(x). (4 marks)

(c) State the domain of $\text{f}^{-1}$f−1, justifying your answer. (3 marks)

(a) Solve the equation $|2x - 1| = |x + 4|.$∣2x−1∣=∣x+4∣. (4 marks)

(b) Hence, or otherwise, solve the inequality $|2x - 1| < |x + 4|.$∣2x−1∣<∣x+4∣. (4 marks)

The polynomial $\text{p}$p is defined by $\text{p}(x) = 2x^3 + ax^2 + bx - 6,$p(x)=2x3+ax2+bx−6, where $a$a and $b$b are constants. It is given that $(x - 2)$(x−2) is a factor of $\text{p}(x)$p(x), and that $\text{p}(x)$p(x) leaves a remainder of $-30$−30 when divided by $(x + 1)$(x+1).

(a) Find the values of $a$a and $b$b. (4 marks)

(b) Hence factorise $\text{p}(x)$p(x) completely. (2 marks)

The quadratic equation $x^2 + (k + 1)x + (k + 4) = 0,$x2+(k+1)x+(k+4)=0, where $k$k is a constant, has no real roots.

Find the set of possible values of $k$k. (5 marks)

The curve $y = \text{f}(x)$y=f(x) has a minimum turning point at $(3, -4)$(3,−4) and crosses the $x$x-axis at the points $(1, 0)$(1,0) and $(5, 0)$(5,0).

The curve is transformed to give the curve with equation $y = 2\text{f}(x + 1) - 3.$y=2f(x+1)−3.

State the coordinates of the image of the turning point and of each of the two crossing points on the transformed curve. (4 marks)