21N.1.AHL.TZ0.9

pestleMathematicsAAHLPaper 121N· ahl-1-10-perms-and-combs-binomial-with-negative-and-fractional-indicessource ↗

Consider the expression $\frac{1}{\sqrt{1 + a x}} - \sqrt{1 - x}$ where $a \in ℚ, a \neq 0$ .

The binomial expansion of this expression, in ascending powers of $x$ , as far as the term in $x^{2}$ is $4 b x + b x^{2}$ , where $b \in ℚ$ .

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

[6]

State the restriction which must be placed on $x$ for this expansion to be valid.

[1]

Markscheme / solution

attempt to expand binomial with negative fractional power (M1)

$\frac{1}{\sqrt{1 + a x}} = {(1 + a x)}^{- \frac{1}{2}} = 1 - \frac{a x}{2} + \frac{3 a^{2} x^{2}}{8} + \dots$ A1

$\sqrt{1 - x} = {(1 - x)}^{\frac{1}{2}} = 1 - \frac{x}{2} - \frac{x^{2}}{8} + \dots$ A1

$\frac{1}{\sqrt{1 + a x}} - \sqrt{1 - x} = \frac{(1 - a)}{2} x + (\frac{3 a^{2} + 1}{8}) x^{2} + \dots$

attempt to equate coefficients of $x$ or $x^{2}$ (M1)

$x : \frac{1 - a}{2} = 4 b; x^{2} : \frac{3 a^{2} + 1}{8} = b$

attempt to solve simultaneously (M1)

$a = - \frac{1}{3}, b = \frac{1}{6}$ A1

[6 marks]

$|x| < 1$ A1

[1 mark]

Examiners’ report

[N/A]