Equation with transformed roots

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. The quartic equation $$z ^ { 4 } + 5 z ^ { 2 } - 30 = 0$$ has roots \(p , q , r\) and \(s\).
   Without solving the equation, determine the quartic equation whose roots are $$( 3 p - 1 ) , ( 3 q - 1 ) , ( 3 r - 1 ) \text { and } ( 3 s - 1 )$$ Give your answer in the form \(w ^ { 4 } + a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be found.
2. The roots of the cubic equation $$4 x ^ { 3 } + n x + 81 = 0 \quad \text { where } n \text { is a real constant }$$ are \(\alpha , 2 \alpha\) and \(\alpha - \beta\)
   Determine
   (a) the values of the roots of the equation,
   (b) the value of \(n\).

1. The cubic equation

$$x ^ { 3 } - 7 x ^ { 2 } - 12 x + 6 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, determine a cubic equation whose roots are ( \(\alpha + 2\) ), \(( \beta + 2 )\) and \(( \gamma + 2 )\), giving your answer in the form \(w ^ { 3 } + p w ^ { 2 } + q w + r = 0\), where \(p , q\) and \(r\) are integers to be found.

4 In this question you must show detailed reasoning. The roots of the cubic equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { 2 } ( \alpha - 1 ) , \frac { 1 } { 2 } ( \beta - 1 )\) and \(\frac { 1 } { 2 } ( \gamma - 1 )\).
2. Hence or otherwise solve the equation \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).

3 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 3 } - 3 x ^ { 2 } - 2 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha \beta , \beta \gamma\) and \(\gamma \alpha\).

5 In this question you must show detailed reasoning. The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

3 In this question you must show detailed reasoning. The roots of the equation \(4 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 9 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

4 The cubic equation $$z ^ { 3 } + \mathrm { i } z ^ { 2 } + 3 z - ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of:
   1. \(\alpha + \beta + \gamma\);
   2. \(\alpha \beta + \beta \gamma + \gamma \alpha\);
   3. \(\alpha \beta \gamma\).
2. Find the value of:
   1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\);
   2. \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\);
   3. \(\alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
3. Hence write down a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).

7 The cubic equation \(27 z ^ { 3 } + k z ^ { 2 } + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. 1. In the case where \(\beta = \gamma\), find the roots of the equation.
   2. Find the value of \(k\) in this case.
3. 1. In the case where \(\alpha = 1 - \mathrm { i }\), find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\).
   2. Hence find the value of \(k\) in this case.
4. In the case where \(k = - 12\), find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1\) and \(\frac { 1 } { \gamma } + 1\).
   [0pt] [5 marks]

14 The graph of \(y = x ^ { 3 } - 3 x\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-22_718_771_370_632} The two stationary points have \(x\)-coordinates of - 1 and 1
The cubic equation $$x ^ { 3 } - 3 x + p = 0$$ where \(p\) is a real constant, has the roots \(\alpha , \beta\) and \(\gamma\).
The roots \(\alpha\) and \(\beta\) are not real.
14

1. Explain why \(\alpha + \beta = - \gamma\)
   14
2. Find the set of possible values for the real constant \(p\).
   14
3. \(\quad \mathrm { f } ( x ) = 0\) is a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\)
   14
4. 1. Show that the constant term of \(\mathrm { f } ( x )\) is \(p + 2\)
      14
5. (ii) Write down the \(x\)-coordinates of the stationary points of \(y = \mathrm { f } ( x )\)
   \includegraphics[max width=\textwidth, alt={}, center]{948391d8-10ad-44ce-b254-7f1aaac5c82c-24_2488_1719_219_150} Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin.

13 The cubic equation \(x ^ { 3 } - x - 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\) The cubic equation \(\mathrm { p } ( x ) = 0\) has roots \(\alpha - 1 , \beta - 1\) and \(\gamma - 1\)
The coefficient of \(x ^ { 3 }\) in \(\mathrm { p } ( x )\) is 1 13

1. Describe fully the transformation that maps the graph of \(y = x ^ { 3 } - x - 7\) onto the graph of \(y = \mathrm { p } ( x )\)
   13
2. Find \(\mathrm { p } ( x )\)
   Turn over for the next question

13 The equation \(z ^ { 3 } + k z ^ { 2 } + 9 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). 13

1. 1. Show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = k ^ { 2 }$$ 13
2. (ii) Show that $$\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } = - 18 k$$ 13
3. The equation \(9 z ^ { 3 } - 40 z ^ { 2 } + r z + s = 0\) has roots \(\alpha \beta + \gamma , \beta \gamma + \alpha\) and \(\gamma \alpha + \beta\). 13
4. 1. Show that $$k = - \frac { 40 } { 9 }$$ Question 13 continues on the next page 13
5. (ii) Without calculating the values of \(\alpha , \beta\) and \(\gamma\), find the value of \(s\). Show working to justify your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-23_2488_1716_219_153} A light spring is attached to the base of a long tube and has a mass \(m\) attached to the other end, as shown in the diagram. The tube is filled with oil. When the compression of the spring is \(\varepsilon\) metres, the thrust in the spring is \(9 m \varepsilon\) newtons.
   \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-24_506_250_721_895} The mass is held at rest in a position where the compression of the spring is \(\frac { 20 } { 9 }\) metres. The mass is then released from rest. During the subsequent motion the oil causes a resistive force of \(6 m v\) newtons to act on the mass, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the mass. At time \(t\) seconds after the mass is released, the displacement of the mass above its starting position is \(x\) metres.

5 The equation $$z ^ { 3 } + 2 z ^ { 2 } - 5 z - 3 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\)
Find a cubic equation with roots $$\frac { 1 } { 2 } \alpha - 1 , \frac { 1 } { 2 } \beta - 1 \text { and } \frac { 1 } { 2 } \gamma - 1$$

6 The cubic equation $$x ^ { 3 } + 5 x ^ { 2 } - 4 x + 2 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\)
Find a cubic equation, with integer coefficients, whose roots are \(3 \alpha , 3 \beta\) and \(3 \gamma\)

5 In this question you must show detailed reasoning. You are given that \(\alpha , \beta\) and \(\gamma\) are the roots of the equation \(5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0\).

1. Find the value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
2. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\) giving your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) where \(a , b , c\) and \(d\) are integers.

1 In this question you must show detailed reasoning.
The cubic equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 5 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). By making an appropriate substitution, or otherwise, find a cubic equation with integer coefficients whose roots are \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

3 In this question you must show detailed reasoning.
The cubic equation \(5 x ^ { 3 } + 3 x ^ { 2 } - 4 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

1 In this question you must show detailed reasoning.
The roots of the equation \(3 x ^ { 3 } - 2 x ^ { 2 } - 5 x - 4 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. Find a cubic equation with integer coefficients whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
2. Find the exact value of \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).