21M.1.SL.TZ2.4

pestleMathematicsAASLPaper 121M· sl-1-9-binomial-theorem-where-n-is-an-integersource ↗

In the expansion of $(x + k)^{7}$ , where $k \in ℝ$ , the coefficient of the term in $x^{5}$ is $63$ .

Find the possible values of $k$ .

Markscheme / solution

EITHER

attempt to use the binomial expansion of ${(x + k)}^{7}$ (M1)

${}^{7}C_{0} x^{7} k^{0} + {}^{7}C_{1} x^{6} k^{1} + {}^{7}C_{2} x^{5} k^{2} + \dots$ (or ${}^{7}C_{0} k^{7} x^{0} + {}^{7}C_{1} k^{5} x^{1} + {}^{7}C_{2} k^{5} x^{2} + \dots$ )

identifying the correct term ${}^{7}C_{2} x^{5} k^{2}$ (or ${}^{7}C_{5} k^{2} x^{5}$ ) (A1)

attempt to use the general term ${}^{7}C_{r} x^{r} k^{7 - r}$ (or ${}^{7}C_{r} k^{r} x^{7 - r}$ ) (M1)

$r = 2$ (or $r = 5$ ) (A1)

THEN

${}^{7}C_{2} = 21$ (or ${}^{7}C_{5} = 21$ (seen anywhere) (A1)

$21 x^{5} k^{2} = 63 x^{5} (21 k^{2} = 63, k^{2} = 3)$ A1

$k = \pm \sqrt{3}$ A1

Note: If working shown, award M1A1A1A1A0 for $k = \sqrt{3}$ .

[5 marks]

Examiners’ report

[N/A]