4.05b Transform equations: substitution for new roots

1. The cubic equation

$$z ^ { 3 } - 3 z ^ { 2 } + z + 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(2 \alpha + 1\) ), ( \(2 \beta + 1\) ) and ( \(2 \gamma + 1\) ), 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.

1. The cubic equation

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

1. The cubic equation

$$3 x ^ { 3 } + x ^ { 2 } - 4 x + 1 = 0$$ has roots \(\alpha , \beta\), and \(\gamma\).
Without solving the cubic equation,

1. determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\)
2. find a cubic equation that has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\), giving your answer in the form \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers to be determined.

1. The cubic equation

$$9 x ^ { 3 } - 5 x ^ { 2 } + 4 x + 7 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(3 \alpha - 2\) ), ( \(3 \beta - 2\) ) and ( \(3 \gamma - 2\) ), giving your answer in the form \(a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be determined.

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 \(n\).

3 A tree at the bottom of a garden needs to be reduced in height. The tree is known to increase in height by 15 centimetres each year. On the first day of every year, the height is measured and the tree is immediately trimmed by \(3 \%\) of this height. When the tree is measured, before trimming on the first day of year 1 , the height is 6 metres.
Let \(H _ { n }\) be the height of the tree immediately before trimming on the first day of year \(n\).

1. Explain, in the context of the problem, why the height of the tree may be modelled by the recurrence relation $$H _ { n + 1 } = 0.97 H _ { n } + 0.15 , \quad H _ { 1 } = 6 , \quad n \in \mathbb { Z } ^ { + }$$
2. Prove by induction that \(H _ { n } = 0.97 ^ { n - 1 } + 5 , \quad n \geqslant 1\)
3. Explain what will happen to the height of the tree immediately before trimming in the long term.
4. By what fixed percentage should the tree be trimmed each year if the height of the tree immediately before trimming is to be 4 metres in the long term?

1. On Jim's 11 th birthday his parents invest \(\pounds 1000\) for him in a savings account.

The account earns 2\% interest each year.
On each subsequent birthday, Jim's parents add another \(\pounds 500\) to this savings account.
Let \(U _ { n }\) be the amount of money that Jim has in his savings account \(n\) years after his 11th birthday, once the interest for the previous year has been paid and the \(\pounds 500\) has been added.

1. Explain, in the context of the problem, why the amount of money that Jim has in his savings account can be modelled by the recurrence relation of the form $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$
2. State an assumption that must be made for this model to be valid.
3. Solve the recurrence relation $$U _ { n } = 1.02 U _ { n - 1 } + 500 \quad U _ { 0 } = 1000 \quad n \in \mathbb { Z } ^ { + }$$ Jim hopes to be able to buy a car on his 18th birthday.
4. Use the answer to part (c) to find out whether Jim will have enough money in his savings account to buy a car that costs \(\pounds 4500\)

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.

1. The cubic equation

$$4 x ^ { 3 } + p x ^ { 2 } - 14 x + q = 0$$ where \(p\) and \(q\) are real positive constants, has roots \(\alpha , \beta\) and \(\gamma\) Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 16\)

1. show that \(p = 12\) Given that \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } = \frac { 14 } { 3 }\)
2. determine the value of \(q\) Without solving the cubic equation,
3. determine the value of \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)

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 \(x ^ { 3 } - 3 x ^ { 2 } + 19 x - 17 = 0\).

8 In this question you must show detailed reasoning. The equation \(\mathrm { x } ^ { 3 } + \mathrm { kt } ^ { 2 } + 15 \mathrm { x } - 25 = 0\) has roots \(\alpha , \beta\) and \(\frac { \alpha } { \beta }\). Given that \(\alpha > 0\), find, in any order,

• the roots of the equation,
• the value of \(k\).

1 The equation $$x ^ { 2 } - 5 x - 2 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\alpha ^ { 2 } \beta + \alpha \beta ^ { 2 }\).
3. Find a quadratic equation which has roots $$\alpha ^ { 2 } \beta \quad \text { and } \quad \alpha \beta ^ { 2 }$$

8

1. 1. It is given that \(\alpha\) and \(\beta\) are the roots of the equation $$x ^ { 2 } - 2 x + 4 = 0$$ Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ (6 marks)
   2. State, giving a reason, whether the roots of the equation $$x ^ { 2 } + 16 x + 64 = 0$$ are real and equal, real and distinct, or non-real.
2. Solve the equation $$x ^ { 2 } - 2 x + 4 = 0$$
3. Use your answers to parts (a) and (b) to show that $$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$

1 The quadratic equation $$3 x ^ { 2 } - 6 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 6\).
3. Find a quadratic equation, with integer coefficients, which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\).

1 The quadratic equation $$3 x ^ { 2 } - 6 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).

1. Write down the numerical values of \(\alpha + \beta\) and \(\alpha \beta\).
2. 1. Expand \(( \alpha + \beta ) ^ { 3 }\).
   2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } = 4\).
3. Find a quadratic equation with roots \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\), giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.

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 \(\gamma ^ { 2 }\).

12 The equation \(x ^ { 3 } - 2 x ^ { 2 } - x + 2 = 0\) has three roots. One of the roots is 2 12

1. Find the other two roots of the equation. 12
2. Hence, or otherwise, solve $$\cosh ^ { 3 } \theta - 2 \cosh ^ { 2 } \theta - \cosh \theta + 2 = 0$$ giving your answers in an exact form.

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

8 The three roots of the equation $$4 x ^ { 3 } - 12 x ^ { 2 } - 13 x + k = 0$$ where \(k\) is a constant, form an arithmetic sequence. Find the roots of the equation.

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

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 The roots of the equation \(4 x ^ { 4 } - 2 x ^ { 3 } - 3 x + 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\). By using a suitable substitution, find a quartic equation whose roots are \(\alpha + 2 , \beta + 2 , \gamma + 2\) and \(\delta + 2\) giving your answer in the form \(a t ^ { 4 } + b t ^ { 3 } + c t ^ { 2 } + d t + e = 0\), where \(a , b , c , d\), and \(e\) are integers.

2 The equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Use a substitution to find a cubic equation with integer coefficients whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).

2
The position vector of point \(A\) is \(\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }\).
The line \(l\) passes through \(A\) and is perpendicular to \(\mathbf { a }\).

1. Determine the shortest distance between the origin, \(O\), and \(l\). \(l\) is also perpendicular to the vector \(\mathbf { b }\) where \(\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\).
2. Find a vector which is perpendicular to both \(\mathbf { a }\) and \(\mathbf { b }\).
3. Write down an equation of \(l\) in vector form. \(P\) is a point on \(l\) such that \(P A = 2 O A\).
4. Find angle \(P O A\) giving your answer to 3 significant figures. \(C\) is a point whose position vector, \(\mathbf { c }\), is given by \(\mathbf { c } = p \mathbf { a }\) for some constant \(p\). The line \(m\) passes through \(C\) and has equation \(\mathbf { r } = \mathbf { c } + \mu \mathbf { b }\). The point with position vector \(9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }\) lies on \(m\).
5. Find the value of \(p\). \section*{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.