| Date | None Specimen | Marks available | 6 | Reference code | SPNone.1.hl.TZ0.2 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Solve | Question number | 2 | Adapted from | N/A |
Question
Consider the equation \(9{x^3} - 45{x^2} + 74x - 40 = 0\) .
Write down the numerical value of the sum and of the product of the roots of this equation.
[1]
a.
The roots of this equation are three consecutive terms of an arithmetic sequence.
Taking the roots to be \(\alpha {\text{ , }}\alpha \pm \beta \) , solve the equation.
[6]
b.
Markscheme
\({\text{sum}} = \frac{{45}}{9},{\text{ product}} = \frac{{40}}{9}\) A1
[1 mark]
a.
it follows that \(3\alpha = \frac{{45}}{9}\) and \(\alpha ({\alpha ^2} - {\beta ^2}) = \frac{{40}}{9}\) A1A1
solving, \(\alpha = \frac{5}{3}\) A1
\(\frac{5}{3}\left( {\frac{{25}}{9} - {\beta ^2}} \right) = \frac{{40}}{9}\) M1
\(\beta = ( \pm )\frac{1}{3}\) A1
the other two roots are 2, \(\frac{4}{3}\) A1
[6 marks]
b.
Examiners report
[N/A]
a.
[N/A]
b.
Syllabus sections
Topic 1 - Core: Algebra » 1.1 » Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.
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