CAIE FP1 2002 November — Question 2 5 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2002
SessionNovember
Marks5
PaperDownload PDF ↗
TopicRoots of polynomials
TypeEquation with nonlinearly transformed roots
DifficultyStandard +0.8 This is a standard Further Maths question on transformed roots requiring the reciprocal substitution technique (replacing x with 1/y) and using relationships between sums of powers of roots. While it involves multiple steps and symmetric function manipulation, these are well-practiced techniques in FP1. The condition linking sum of squares to sum of reciprocal squares adds modest complexity but follows a predictable approach using Vieta's formulas.
Spec4.05b Transform equations: substitution for new roots
2 The equation $$x ^ { 4 } + x ^ { 3 } + A x ^ { 2 } + 4 x - 2 = 0$$ where \(A\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\). Find a polynomial equation whose roots are $$\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$$ Given that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$$ find the value of \(A\). 
2 The equation

$$x ^ { 4 } + x ^ { 3 } + A x ^ { 2 } + 4 x - 2 = 0$$

where $A$ is a constant, has roots $\alpha , \beta , \gamma , \delta$. Find a polynomial equation whose roots are

$$\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$$

Given that

$$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$$

find the value of $A$.

\hfill \mbox{\textit{CAIE FP1 2002 Q2 [5]}}
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