CAIE Further Paper 1 2022 November — Question 1

Exam BoardCAIE
ModuleFurther Paper 1 (Further Paper 1)
Year2022
SessionNovember
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TopicRoots of polynomials
1 The cubic equation \(x ^ { 3 } + b x ^ { 2 } + d = 0\) has roots \(\alpha , \beta , \gamma\), where \(\alpha = \beta\) and \(d \neq 0\).
  1. Show that \(4 b ^ { 3 } + 27 d = 0\).
  2. Given that \(2 \alpha ^ { 2 } + \gamma ^ { 2 } = 3 b\), find the values of \(b\) and \(d\).
1 The cubic equation $x ^ { 3 } + b x ^ { 2 } + d = 0$ has roots $\alpha , \beta , \gamma$, where $\alpha = \beta$ and $d \neq 0$.\\
(a) Show that $4 b ^ { 3 } + 27 d = 0$.\\

(b) Given that $2 \alpha ^ { 2 } + \gamma ^ { 2 } = 3 b$, find the values of $b$ and $d$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q1}}
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Q1 Q2 6 Q3 Q4 Q5 Q6 Q7