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Date	May 2019	Marks available	5	Reference code	19M.2.AHL.TZ1.H_11
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Show that	Question number	H_11	Adapted from	N/A

Question

Consider the equation ${x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0$ , where $m$ , $n$ , $p$ , $q \in \mathbb{R}$ .

The equation has three distinct real roots which can be written as ${\text{lo}}{{\text{g}}_2}\,a$ , ${\text{lo}}{{\text{g}}_2}\,b$ and ${\text{lo}}{{\text{g}}_2}\,c$ .

The equation also has two imaginary roots, one of which is $d{\text{i}}$ where $d \in \mathbb{R}$ .

The values $a$ , $b$ , and $c$ are consecutive terms in a geometric sequence.

Show that $abc = 8$ .

[5]

Show that one of the real roots is equal to 1.

[3]

Given that $q = 8{d^2}$ , find the other two real roots.

[9]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

recognition of the other root $= - d{\text{i}}$ (A1)

${\text{lo}}{{\text{g}}_2}\,a + {\text{lo}}{{\text{g}}_2}\,b + {\text{lo}}{{\text{g}}_2}\,c + d{\text{i}} - d{\text{i}} = 3$ M1A1

Note: Award M1 for sum of the roots, A1 for 3. Award A0M1A0 for just ${\text{lo}}{{\text{g}}_2}\,a + {\text{lo}}{{\text{g}}_2}\,b + {\text{lo}}{{\text{g}}_2}\,c = 3$ .

${\text{lo}}{{\text{g}}_2}\,abc = 3$ (M1)

$\Rightarrow abc = {2^3}$ A1

$abc = 8$ AG

[5 marks]

METHOD 1

let the geometric series be ${u_1}$ , ${u_1}r$ , ${u_1}{r^2}$

${\left( {{u_1}r} \right)^3} = 8$ M1

${u_1}r = 2$ A1

hence one of the roots is ${\text{lo}}{{\text{g}}_2}2 = 1$ R1

METHOD 2

$\frac{b}{a} = \frac{c}{b}$

${b^2} = ac \Rightarrow {b^3} = abc = 8$ M1

$b = 2$ A1

hence one of the roots is ${\text{lo}}{{\text{g}}_2}2 = 1$ R1

[3 marks]

METHOD 1

product of the roots is ${r_1} \times {r_2} \times 1 \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}$ (M1)(A1)

${r_1} \times {r_2} = - 8$ A1

sum of the roots is ${r_1} + {r_2} + 1 + d{\text{i}} + - d{\text{i}} = 3$ (M1)(A1)

${r_1} + {r_2} = 2$ A1

solving simultaneously (M1)

${r_1} = - 2$ , ${r_2} = 4$ A1A1

METHOD 2

product of the roots ${\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c \times d{\text{i}} \times - d{\text{i}} = - 8{d^2}$ M1A1

${\text{lo}}{{\text{g}}_2}\,a \times {\text{lo}}{{\text{g}}_2}\,b \times {\text{lo}}{{\text{g}}_2}\,c = - 8$ A1

EITHER

$a$ , $b$ , $c$ can be written as $\frac{2}{r}$ , $2$ , $2r$ M1

$\left( {{\text{lo}}{{\text{g}}_2}\frac{2}{r}} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2r} \right) = - 8$

attempt to solve M1

$\left( {1 - {\text{lo}}{{\text{g}}_2}\,r} \right)\left( {1 + {\text{lo}}{{\text{g}}_2}\,r} \right) = - 8$

${\text{lo}}{{\text{g}}_2}\,r = \pm 3$

$r = \frac{1}{8}{\text{,}}\,\,8$ A1A1

$a$ , $b$ , $c$ can be written as $a$ , $2$ , $\frac{4}{a}$ M1

$\left( {{\text{lo}}{{\text{g}}_2}\,a} \right)\left( {{\text{lo}}{{\text{g}}_2}\,2} \right)\left( {{\text{lo}}{{\text{g}}_2}\,\frac{4}{a}} \right) = - 8$