4.05b Transform equations: substitution for new roots

152 questions

Sort by: Default | Easiest first | Hardest first
CAIE Further Paper 1 2020 June Q2
8 marks Challenging +1.2
2 The cubic equation \(6 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } - 3 \mathrm { x } - 5 = 0\), where \(p\) is a constant, has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\).
  2. It is given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 ( \alpha + \beta + \gamma )\).
    1. Find the value of \(p\).
    2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
CAIE Further Paper 1 2020 June Q1
7 marks Standard +0.3
1 The cubic equation \(7 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { - 1 } , \beta ^ { - 1 } , \gamma ^ { - 1 }\).
  2. Find the value of \(\alpha ^ { - 2 } + \beta ^ { - 2 } + \gamma ^ { - 2 }\).
  3. Find the value of \(\alpha ^ { - 3 } + \beta ^ { - 3 } + \gamma ^ { - 3 }\).
CAIE Further Paper 1 2021 June Q3
9 marks Challenging +1.2
3 The equation \(x ^ { 4 } - 2 x ^ { 3 } - 1 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).
  1. Find a quartic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 } , \delta ^ { 3 }\) and state the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 }\). [4]
  2. Find the value of \(\frac { 1 } { \alpha ^ { 3 } } + \frac { 1 } { \beta ^ { 3 } } + \frac { 1 } { \gamma ^ { 3 } } + \frac { 1 } { \delta ^ { 3 } }\).
  3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
CAIE Further Paper 1 2021 June Q2
11 marks Challenging +1.2
2 The cubic equation \(2 x ^ { 3 } - 4 x ^ { 2 } + 3 = 0\) has roots \(\alpha , \beta , \gamma\). Let \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } } + \gamma ^ { \mathrm { n } }\).
  1. State the value of \(S _ { 1 }\) and find the value of \(S _ { 2 }\).
    1. Express \(\mathrm { S } _ { \mathrm { n } + 3 }\) in terms of \(\mathrm { S } _ { \mathrm { n } + 2 }\) and \(\mathrm { S } _ { \mathrm { n } }\).
    2. Hence, or otherwise, find the value of \(S _ { 4 }\).
  3. Use the substitution \(\mathrm { y } = \mathrm { S } _ { 1 } - \mathrm { x }\), where \(S _ { 1 }\) is the numerical value found in part (a), to find and simplify an equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
  4. Find the value of \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).
CAIE Further Paper 1 2022 June Q4
8 marks Challenging +1.2
4 The cubic equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 6 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\frac { 1 } { \alpha ^ { 3 } } , \frac { 1 } { \beta ^ { 3 } } , \frac { 1 } { \gamma ^ { 3 } }\).
  2. Find the value of \(\frac { 1 } { \alpha ^ { 6 } } + \frac { 1 } { \beta ^ { 6 } } + \frac { 1 } { \gamma ^ { 6 } }\).
  3. Find also the value of \(\frac { 1 } { \alpha ^ { 9 } } + \frac { 1 } { \beta ^ { 9 } } + \frac { 1 } { \gamma ^ { 9 } }\).
CAIE Further Paper 1 2023 June Q3
9 marks Challenging +1.2
3 The equation \(x ^ { 4 } - x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).
  1. Find a quartic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 } , \delta ^ { 2 }\) and state the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\).
  2. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
  3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
CAIE Further Paper 1 2024 June Q2
7 marks Challenging +1.2
2 The cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { 2 } + 1 , \beta ^ { 2 } + 1 , \gamma ^ { 2 } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-04_2714_34_143_2012}
  2. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 2 } + \left( \beta ^ { 2 } + 1 \right) ^ { 2 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 2 }\).
  3. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 3 } + \left( \beta ^ { 2 } + 1 \right) ^ { 3 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 3 }\).
CAIE Further Paper 1 2020 November Q3
11 marks Challenging +1.8
3 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { cx } + 1 = 0\), where \(c\) is a constant, has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\).
  2. Show that \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } = 3 - 2 c ^ { 3 }\).
  3. Find the real value of \(c\) for which the matrix \(\left( \begin{array} { c c c } 1 & \alpha ^ { 3 } & \beta ^ { 3 } \\ \alpha ^ { 3 } & 1 & \gamma ^ { 3 } \\ \beta ^ { 3 } & \gamma ^ { 3 } & 1 \end{array} \right)\) is singular.
CAIE Further Paper 1 2020 November Q1
8 marks Standard +0.8
1 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\), where \(b , c\) and \(d\) are constants, has roots \(\alpha , \beta , \gamma\). It is given that \(\alpha \beta \gamma = - 1\).
  1. State the value of \(d\).
  2. Find a cubic equation, with coefficients in terms of \(b\) and \(c\), whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\).
  3. Given also that \(\gamma + 1 = - \alpha - 1\), deduce that \(( \mathrm { c } - 2 \mathrm {~b} + 3 ) ( \mathrm { b } - 3 ) = \mathrm { b } - \mathrm { c }\).
CAIE Further Paper 1 2022 November Q1
8 marks Challenging +1.2
1 The cubic equation \(x ^ { 3 } + b x ^ { 2 } + d = 0\) has roots \(\alpha , \beta , \gamma\), where \(\alpha = \beta\) and \(d \neq 0\).
  1. Show that \(4 b ^ { 3 } + 27 d = 0\).
  2. Given that \(2 \alpha ^ { 2 } + \gamma ^ { 2 } = 3 b\), find the values of \(b\) and \(d\).
CAIE Further Paper 1 2022 November Q2
9 marks Challenging +1.2
2 The equation \(x ^ { 4 } + 3 x ^ { 2 } + 2 x + 6 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).
  1. Find a quartic equation whose roots are \(\frac { 1 } { \alpha ^ { 2 } } , \frac { 1 } { \beta ^ { 2 } } , \frac { 1 } { \gamma ^ { 2 } } , \frac { 1 } { \delta ^ { 2 } }\) and state the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
  2. Find the value of \(\beta ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
  3. Find the value of \(\frac { 1 } { \alpha ^ { 4 } } + \frac { 1 } { \beta ^ { 4 } } + \frac { 1 } { \gamma ^ { 4 } } + \frac { 1 } { \delta ^ { 4 } }\).
CAIE Further Paper 1 2023 November Q4
10 marks Challenging +1.2
4 The cubic equation \(27 x ^ { 3 } + 18 x ^ { 2 } + 6 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Show that a cubic equation with roots \(3 \alpha + 1,3 \beta + 1,3 \gamma + 1\) is $$y ^ { 3 } - y ^ { 2 } + y - 2 = 0$$ The sum \(( 3 \alpha + 1 ) ^ { n } + ( 3 \beta + 1 ) ^ { n } + ( 3 \gamma + 1 ) ^ { n }\) is denoted by \(\mathrm { S } _ { \mathrm { n } }\).
  2. Find the values of \(S _ { 2 }\) and \(S _ { 3 }\).
  3. Find the values of \(S _ { - 1 }\) and \(S _ { - 2 }\).
CAIE Further Paper 1 2020 Specimen Q4
9 marks Standard +0.8
4 The cubic equation $$z ^ { 3 } - z ^ { 2 } - z - 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
  1. Show that the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\) is 19 .
  2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 }\).
  3. Find a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), 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.
Edexcel F1 2014 January Q2
8 marks Moderate -0.3
2. The quadratic equation $$5 x ^ { 2 } - 4 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots $$\frac { 1 } { \alpha ^ { 2 } } \text { and } \frac { 1 } { \beta ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{4da2bb2c-a51b-493c-a9f2-f4ff008a3aac-07_70_51_2663_1896}
Edexcel F1 2015 January Q5
8 marks Standard +0.3
5. The quadratic equation $$4 x ^ { 2 } + 3 x + 1 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\).
  2. Find the value of \(\left( \alpha ^ { 2 } + \beta ^ { 2 } \right)\).
  3. Find a quadratic equation which has roots $$( 4 \alpha - \beta ) \text { and } ( 4 \beta - \alpha )$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.
Edexcel F1 2018 January Q4
8 marks Standard +0.8
  1. The quadratic equation
$$3 x ^ { 2 } + 2 x + 5 = 0$$ has roots \(\alpha\) and \(\beta\). Without solving the equation,
  1. find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
  2. show that \(\alpha ^ { 3 } + \beta ^ { 3 } = \frac { 82 } { 27 }\)
  3. find a quadratic equation which has roots $$\left( \alpha + \frac { \alpha } { \beta ^ { 2 } } \right) \text { and } \left( \beta + \frac { \beta } { \alpha ^ { 2 } } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.
Edexcel F1 2021 January Q4
8 marks Standard +0.8
  1. The equation \(2 x ^ { 2 } + 5 x + 7 = 0\) has roots \(\alpha\) and \(\beta\)
Without solving the equation
  1. determine the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. form a quadratic equation, with integer coefficients, which has roots $$\frac { \alpha ^ { 2 } } { \beta } \text { and } \frac { \beta ^ { 2 } } { \alpha }$$ \includegraphics[max width=\textwidth, alt={}, center]{f8660b02-384e-460f-a0e4-282ef5fef475-09_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel F1 2023 January Q5
9 marks Standard +0.8
  1. The quadratic equation
$$4 x ^ { 2 } + 3 x + k = 0$$ where \(k\) is an integer, has roots \(\alpha\) and \(\beta\)
  1. Write down, in terms of \(k\) where appropriate, the value of \(\alpha + \beta\) and the value of \(\alpha \beta\)
  2. Determine, in simplest form in terms of \(k\), the value of \(\frac { \alpha } { \beta ^ { 2 } } + \frac { \beta } { \alpha ^ { 2 } }\)
  3. Determine a quadratic equation which has roots $$\frac { \alpha } { \beta ^ { 2 } } \text { and } \frac { \beta } { \alpha ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integer values in terms of \(k\)
Edexcel F1 2024 January Q5
9 marks Standard +0.8
  1. The quadratic equation
$$2 x ^ { 2 } - 3 x + 7 = 0$$ has roots \(\alpha\) and \(\beta\) Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. determine the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
  3. find a quadratic equation which has roots $$\left( \alpha - \frac { 1 } { \beta ^ { 2 } } \right) \text { and } \left( \beta - \frac { 1 } { \alpha ^ { 2 } } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.
Edexcel F1 2014 June Q6
8 marks Standard +0.8
6. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(3 x ^ { 2 } + 5 x - 1 = 0\)
  1. Find the exact value of \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. Find a quadratic equation which has roots \(\frac { \alpha ^ { 2 } } { \beta }\) and \(\frac { \beta ^ { 2 } } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers.
Edexcel F1 2015 June Q3
6 marks Standard +0.3
3. It is given that \(\alpha\) and \(\beta\) are roots of the equation $$2 x ^ { 2 } - 7 x + 4 = 0$$
  1. Find the exact value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
  2. Find a quadratic equation which has roots \(\frac { \alpha } { \beta }\) and \(\frac { \beta } { \alpha }\), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel F1 2016 June Q9
9 marks Standard +0.8
9. The quadratic equation $$2 x ^ { 2 } + 4 x - 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the quadratic equation,
  1. find the exact value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  2. Find a quadratic equation which has roots ( \(\alpha ^ { 2 } + \beta\) ) and ( \(\beta ^ { 2 } + \alpha\) ), giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a , b\) and \(c\) are integers.
    \includegraphics[max width=\textwidth, alt={}, center]{0b7ef4a1-51bf-4f0c-908a-7caf26a144dc-27_99_332_2622_1466}
Edexcel F1 2018 June Q7
9 marks Standard +0.3
7. It is given that \(\alpha\) and \(\beta\) are roots of the equation \(5 x ^ { 2 } - 4 x + 3 = 0\) Without solving the quadratic equation,
  1. find the exact value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\)
  2. find a quadratic equation which has roots \(\frac { 3 } { \alpha ^ { 2 } }\) and \(\frac { 3 } { \beta ^ { 2 } }\) giving your answer in the form \(a x ^ { 2 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel F1 2020 June Q2
9 marks Standard +0.3
2
2. The quadratic equation $$5 x ^ { 2 } - 2 x + 3 = 0$$ has roots \(\alpha\) and \(\beta\).
Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. determine, giving each answer as a simplified fraction, the value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  3. determine a quadratic equation that has roots $$\left( \alpha + \beta ^ { 2 } \right) \text { and } \left( \beta + \alpha ^ { 2 } \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.
Edexcel F1 2022 June Q5
10 marks Standard +0.8
  1. The quadratic equation
$$2 x ^ { 2 } - 3 x + 5 = 0$$ has roots \(\alpha\) and \(\beta\) Without solving the equation,
  1. write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
  2. determine the value of
    1. \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    2. \(\alpha ^ { 3 } + \beta ^ { 3 }\)
  3. find a quadratic equation which has roots $$\left( \alpha ^ { 3 } - \beta \right) \text { and } \left( \beta ^ { 3 } - \alpha \right)$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.