2026 RI P1 Q5
2026 RI P1 Q5
Junior College 2
8 marks
Do not use a calculator in answering this question.
It is given that \(1+2\mathrm i\) is a complex root of the quartic equation
\(z^4+pz^3+qz^2+rz+45=0,\)
where \(p\), \(q\) and \(r\) are real numbers.
- State, with justification, another complex root of the equation in the form \(a+\mathrm ib\), where \(a\) and \(b\) are real numbers and \(b\ne0\). Hence find a quadratic factor corresponding to these two roots.[2]
The quartic equation has a repeated negative real root.
- Find the value of the repeated real root and hence find the values of \(p\), \(q\) and \(r\).[4]
- By means of a suitable substitution, solve \(45w^4+rw^3+qw^2+pw+1=0\).[2]
Solution:
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Answer:(a) \(1-2\mathrm i\); \(z^2-2z+5\) (b) \(-3\); \(p=4,q=2,r=12\) (c) \(w=\frac{1\pm2\mathrm i}5,-\frac13\) (repeated real root)
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