The roots of the cubic equation
\(x^{3}-5 x^{2}+13 x-4=0\)
are \(\alpha, \beta, \gamma\).
(i) Find the value of \(\alpha^{2}+\beta^{2}+\gamma^{2}\).
(ii) Find the value of \(\alpha^{3}+\beta^{3}+\gamma^{3}\).
By finding a cubic equation whose roots are \(\alpha, \beta\) and \(\gamma\), solve the set of simultaneous equations
\(\begin{aligned} \alpha+\beta+\gamma & =-1 \\ \alpha^{2}+\beta^{2}+\gamma^{2} & =29 \\ \frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} & =-1 \end{aligned}\)
The roots of the cubic equation \(x^{3}+2 x^{2}-3=0\) are \(\alpha, \beta\) and \(\gamma\).
(i) By using the substitution \(y=\frac{1}{x^{2}}\), find the cubic equation with roots \(\frac{1}{\alpha^{2}}, \frac{1}{\beta^{2}}\) and \(\frac{1}{\gamma^{2}}\).
(ii) Hence find the value of \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}\).
(iii) Find also the value of \(\frac{1}{\alpha^{2} \beta^{2}}+\frac{1}{\beta^{2} \gamma^{2}}+\frac{1}{\gamma^{2} \alpha^{2}}\).
The equation \(x^{3}+p x+q=0\), where \(p\) and \(q\) are constants, with \(q \neq 0\), has one root which is the reciprocal of another root. Prove that \(p+q^{2}=1\).
The cubic equation \(x^{3}+p x^{2}+q x+r=0\), where \(p, q\) and \(r\) are integers, has roots \(\alpha, \beta\) and \(\gamma\), such that
\(\begin{aligned} \alpha+\beta+\gamma & =15 \\ \alpha^{2}+\beta^{2}+\gamma^{2} & =83 . \end{aligned}\)
Write down the value of \(p\) and find the value of \(q\).
Given that \(\alpha, \beta\) and \(\gamma\) are all real and that \(\alpha \beta+\alpha \gamma=36\), find \(\alpha\) and hence find the value of \(r\).
The roots of the cubic equation \(x^{3}-7 x^{2}+2 x-3=0\) are \(\alpha, \beta\) and \(\gamma\). Find the values of
(i) \(\frac{1}{(\alpha \beta)(\beta \gamma)(\gamma \alpha)}\),
(ii) \(\frac{1}{\alpha \beta}+\frac{1}{\beta \gamma}+\frac{1}{\gamma \alpha}\),
(iii) \(\frac{1}{\alpha^{2} \beta \gamma}+\frac{1}{\alpha \beta^{2} \gamma}+\frac{1}{\alpha \beta \gamma^{2}}\).
Deduce a cubic equation, with integer coefficients, having roots \(\frac{1}{\alpha \beta}, \frac{1}{\beta \gamma}\) and \(\frac{1}{\gamma \alpha}\).
Find the cubic equation with roots \(\alpha, \beta\) and \(\gamma\) such that
\(\begin{aligned} \alpha+\beta+\gamma & =3 \\ \alpha^{2}+\beta^{2}+\gamma^{2} & =1 \\ \alpha^{3}+\beta^{3}+\gamma^{3} & =-30 \end{aligned}\)
giving your answer in the form \(x^{3}+p x^{2}+q x+r=0\), where \(p, q\) and \(r\) are integers to be found.
The cubic equation
\(z^{3}-z^{2}-z-5=0\)
has roots \(\alpha, \beta\) and \(\gamma\). Show that the value of \(\alpha^{3}+\beta^{3}+\gamma^{3}\) is 19 .
Find the value of \(\alpha^{4}+\beta^{4}+\gamma^{4}\).
Show that the cubic equation with roots \(\frac{\alpha-1}{\alpha}, \frac{\beta-1}{\beta}\) and \(\frac{\gamma-1}{\gamma}\) may be found using the substitution \(z=\frac{1}{1-x}\), and find this equation, giving your answer in the form \(p x^{3}+q x^{2}+r x+s=0\), where \(p, q, r\) and \(s\) are constants to be determined.
The roots of the cubic equation \(2 x^{3}+x^{2}-7=0\) are \(\alpha, \beta\) and \(\gamma\). Using the substitution \(y=1+\frac{1}{x}\), or otherwise, find the cubic equation whose roots are \(1+\frac{1}{\alpha}, 1+\frac{1}{\beta}\) and \(1+\frac{1}{\gamma}\), giving your answer in the form \(a y^{3}+b y^{2}+c y+d=0\), where \(a, b, c\) and \(d\) are constants to be found.
The cubic equation \(2 x^{3}-3 x^{2}+4 x-10=0\) has roots \(\alpha, \beta\) and \(\gamma\).
(i) Find the value of \((\alpha+1)(\beta+1)(\gamma+1)\).
(ii) Find the value of \((\beta+\gamma)(\gamma+\alpha)(\alpha+\beta)\).
Answer only one of the following two alternatives.
EITHER
The roots of the quartic equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\) are \(\alpha, \beta, \gamma\) and \(\delta\). Find the values of
(i) \(\alpha+\beta+\gamma+\delta\),
(ii) \(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\),
(iii) \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}\),
(iv) \(\frac{\alpha}{\beta \gamma \delta}+\frac{\beta}{\alpha \gamma \delta}+\frac{\gamma}{\alpha \beta \delta}+\frac{\delta}{\alpha \beta \gamma}\).
Using the substitution \(y=x+1\), find a quartic equation in \(y\). Solve this quartic equation and hence find the roots of the equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\).
OR
The square matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that if \(\mathbf{A}\) is non-singular then
(i) \(\lambda \neq 0\),
(ii) the matrix \(\mathbf{A}^{-1}\) has \(\lambda^{-1}\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector.
The \(3 \times 3\) matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by
\(\mathbf{A}=\left(\begin{array}{rrr} -2 & 2 & -4 \\ 0 & -1 & 5 \\ 0 & 0 & 3 \end{array}\right) \quad \text { and } \quad \mathbf{B}=(\mathbf{A}+3 \mathbf{I})^{-1}\)
where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf{P}\), and a diagonal matrix \(\mathbf{D}\), such that \(\mathbf{B}=\mathbf{P D P}^{-1}\).
Answer only one of the following two alternatives.
EITHER
The roots of the quartic equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\) are \(\alpha, \beta, \gamma\) and \(\delta\). Find the values of
(i) \(\alpha+\beta+\gamma+\delta\),
(ii) \(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\),
(iii) \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}\),
(iv) \(\frac{\alpha}{\beta \gamma \delta}+\frac{\beta}{\alpha \gamma \delta}+\frac{\gamma}{\alpha \beta \delta}+\frac{\delta}{\alpha \beta \gamma}\).
Using the substitution \(y=x+1\), find a quartic equation in \(y\). Solve this quartic equation and hence find the roots of the equation \(x^{4}+4 x^{3}+2 x^{2}-4 x+1=0\).
OR
The square matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that if \(\mathbf{A}\) is non-singular then
(i) \(\lambda \neq 0\),
(ii) the matrix \(\mathbf{A}^{-1}\) has \(\lambda^{-1}\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector.
The \(3 \times 3\) matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by
\(\mathbf{A}=\left(\begin{array}{rrr} -2 & 2 & -4 \\ 0 & -1 & 5 \\ 0 & 0 & 3 \end{array}\right) \quad \text { and } \quad \mathbf{B}=(\mathbf{A}+3 \mathbf{I})^{-1}\)
where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf{P}\), and a diagonal matrix \(\mathbf{D}\), such that \(\mathbf{B}=\mathbf{P D P}^{-1}\).
The cubic equation \(x^{3}-2 x^{2}-3 x+4=0\) has roots \(\alpha, \beta, \gamma\). Given that \(c=\alpha+\beta+\gamma\), state the value of \(c\).
Use the substitution \(y=c-x\) to find a cubic equation whose roots are \(\alpha+\beta, \beta+\gamma, \gamma+\alpha\).
Find a cubic equation whose roots are \(\frac{1}{\alpha+\beta}, \frac{1}{\beta+\gamma}, \frac{1}{\gamma+\alpha}\).
Hence evaluate \(\frac{1}{(\alpha+\beta)^{2}}+\frac{1}{(\beta+\gamma)^{2}}+\frac{1}{(\gamma+\alpha)^{2}}\).
The roots of the equation \(x^{4}-4 x^{2}+3 x-2=0\) are \(\alpha, \beta, \gamma\) and \(\delta\); the sum \(\alpha^{n}+\beta^{n}+\gamma^{n}+\delta^{n}\) is denoted by \(S_{n}\). By using the relation \(y=x^{2}\), or otherwise, show that \(\alpha^{2}, \beta^{2}, \gamma^{2}\) and \(\delta^{2}\) are the roots of the equation
\(y^{4}-8 y^{3}+12 y^{2}+7 y+4=0\)
State the value of \(S_{2}\) and hence show that
\(S_{8}=8 S_{6}-12 S_{4}-72 .\)
The cubic equation \(x^{3}-p x-q=0\), where \(p\) and \(q\) are constants, has roots \(\alpha, \beta, \gamma\). Show that
(i) \(\alpha^{2}+\beta^{2}+\gamma^{2}=2 p\),
(ii) \(\alpha^{3}+\beta^{3}+\gamma^{3}=3 q\),
(iii) \(6\left(\alpha^{5}+\beta^{5}+\gamma^{5}\right)=5\left(\alpha^{3}+\beta^{3}+\gamma^{3}\right)\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\).
Answer only one of the following two alternatives.
EITHER
State the fifth roots of unity in the form \(\cos \theta+\mathrm{i} \sin \theta\), where \(-\pi\lt \theta \leqslant \pi\).
Simplify
\(\left(x-\left[\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right]\right)\left(x-\left[\cos \frac{2}{5} \pi-i \sin \frac{2}{5} \pi\right]\right) .\)
Hence find the real factors of
\(x^{5}-1\)
Express the six roots of the equation
\(x^{6}-x^{3}+1=0\)
as three conjugate pairs, in the form \(\cos \theta \pm \mathrm{i} \sin \theta\).
Hence find the real factors of
\(x^{6}-x^{3}+1\)
OR
Given that
\(y^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}-6 y^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}+2 y\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2}+3 y^{3}=25 \mathrm{e}^{-2 x}\)
and that \(v=y^{3}\), show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}-6 \frac{\mathrm{~d} v}{\mathrm{~d} x}+9 v=75 \mathrm{e}^{-2 x}\)
Find the particular solution for \(y\) in terms of \(x\), given that when \(x=0, y=2\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=1\).
The cubic equation \(x^{3}-p x-q=0\), where \(p\) and \(q\) are constants, has roots \(\alpha, \beta, \gamma\). Show that
(i) \(\alpha^{2}+\beta^{2}+\gamma^{2}=2 p\),
(ii) \(\alpha^{3}+\beta^{3}+\gamma^{3}=3 q\),
(iii) \(6\left(\alpha^{5}+\beta^{5}+\gamma^{5}\right)=5\left(\alpha^{3}+\beta^{3}+\gamma^{3}\right)\left(\alpha^{2}+\beta^{2}+\gamma^{2}\right)\).
Answer only one of the following two alternatives.
EITHER
State the fifth roots of unity in the form \(\cos \theta+\mathrm{i} \sin \theta\), where \(-\pi\lt \theta \leqslant \pi\).
Simplify
\(\left(x-\left[\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right]\right)\left(x-\left[\cos \frac{2}{5} \pi-i \sin \frac{2}{5} \pi\right]\right) .\)
Hence find the real factors of
\(x^{5}-1\)
Express the six roots of the equation
\(x^{6}-x^{3}+1=0\)
as three conjugate pairs, in the form \(\cos \theta \pm \mathrm{i} \sin \theta\).
Hence find the real factors of
\(x^{6}-x^{3}+1\)
OR
Given that
\(y^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}-6 y^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}+2 y\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^{2}+3 y^{3}=25 \mathrm{e}^{-2 x}\)
and that \(v=y^{3}\), show that
\(\frac{\mathrm{d}^{2} v}{\mathrm{~d} x^{2}}-6 \frac{\mathrm{~d} v}{\mathrm{~d} x}+9 v=75 \mathrm{e}^{-2 x}\)
Find the particular solution for \(y\) in terms of \(x\), given that when \(x=0, y=2\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=1\).
The equation
\(8 x^{3}+36 x^{2}+k x-21=0\)
where \(k\) is a constant, has roots \(a-d, a, a+d\). Find the numerical values of the roots and determine the value of \(k\).
The equation
\(x^{3}+x-1=0\)
has roots \(\alpha, \beta, \gamma\). Show that the equation with roots \(\alpha^{3}, \beta^{3}, \gamma^{3}\) is
\(y^{3}-3 y^{2}+4 y-1=0 .\)
Hence find the value of \(\alpha^{6}+\beta^{6}+\gamma^{6}\).