Edexcel A-Level Maths: Algebra and Functions

The function $\text{f}$f is defined by

$\text{f}(x) = \frac{2 + 11x}{(1 - 2x)(1 + 3x)}.$f(x)=(1−2x)(1+3x)2+11x​.

(a) Express $\text{f}(x)$f(x) in partial fractions. (4 marks)

(b) Hence, or otherwise, 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. (5 marks)

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

The polynomial $\text{f}$f is given by

$\text{f}(x) = 2x^3 + 3x^2 - 11x - 6.$f(x)=2x3+3x2−11x−6.

(a) Use the factor theorem to show that $(x - 2)$(x−2) is a factor of $\text{f}(x)$f(x). (2 marks)

(b) Hence factorise $\text{f}(x)$f(x) completely. (4 marks)

(c) Hence solve the inequality $\text{f}(x) \geq 0$f(x)≥0. (4 marks)

The functions $\text{f}$f and $\text{g}$g are defined by

$\text{f}(x) = \frac{2x + 3}{x - 1}, \quad x \in \mathbb{R},\ x > 1, \qquad \text{g}(x) = x^2 + 2, \quad x \in \mathbb{R}.$f(x)=x−12x+3​,x∈R, x>1,g(x)=x2+2,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) and state its domain. (5 marks)

(a) Express $\dfrac{8}{3 + \sqrt{5}}$3+5​8​ in the form $a + b\sqrt{5}$a+b5​, where $a$a and $b$b are integers. (2 marks)

(b) Solve the equation

$x - 5\sqrt{x} + 6 = 0.$x−5x​+6=0.

(4 marks)

The function $\text{f}$f is defined by $\text{f}(x) = 2x^2 - 12x + 23$f(x)=2x2−12x+23 for $x \in \mathbb{R}$x∈R.

(a) Express $\text{f}(x)$f(x) in the form $a(x - p)^2 + q$a(x−p)2+q, where $a$a, $p$p and $q$q are constants to be found. (3 marks)

(b) Hence write down the coordinates of the turning point of the curve $y = \text{f}(x)$y=f(x), and state the range of $\text{f}$f. (2 marks)

Solve the inequality

$2x^2 - x - 6 > 0,$2x2−x−6>0,

giving your answer in set notation. (4 marks)