4.05a Roots and coefficients: symmetric functions

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.

5 In this question you must show detailed reasoning. The roots of the equation \(5 x ^ { 2 } - 3 x + 12 = 0\) are \(\alpha\) and \(\beta\). By considering the symmetric functions of the roots, \(\alpha + \beta\) and \(\alpha \beta\), determine the exact value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }\).

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.

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 quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\). Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).

6 You are given that the cubic equation \(2 x ^ { 3 } + p x ^ { 2 } + q x - 3 = 0\), where \(p\) and \(q\) are real numbers, has a complex root \(\alpha = 1 + i \sqrt { 2 }\).

1. Write down a second complex root, \(\beta\).
2. Determine the third root, \(\gamma\).
3. Find the value of \(p\) and the value of \(q\).
4. Show that if \(n\) is an integer then \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 \times 3 ^ { \frac { 1 } { 2 } n } \times \cos n \theta + \frac { 1 } { 2 ^ { n } }\) where \(\tan \theta = \sqrt { 2 }\).

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 \(\frac { \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } } { \alpha \beta \gamma }\).

2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 2 } + ( \beta + r ) ^ { 2 } + ( \gamma + r ) ^ { 2 } \right) = n \left( n ^ { 2 } + a n + b \right)$$ where \(a\) and \(b\) are constants to be determined.

5 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 \(r\).

10 Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$ Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\). Deduce a quadratic equation, with integer coefficients, having roots \(\sec ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\sec ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
[0pt] [Question 11 is printed on the next page.]

2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants.

1. Show that $$2 p \alpha + q = 0$$
2. Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$

7

1. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
2. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.

7

1. Express \(x ^ { 3 } + y ^ { 3 }\) in terms of \(( x + y )\) and \(x y\).
2. The equation \(t ^ { 2 } - 3 t + \frac { 8 } { 9 } = 0\) has roots \(\alpha\) and \(\beta\).
   1. Determine the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\).
   2. Hence express 19 as the sum of the cubes of two positive rational numbers.

11

1. Determine \(p\) and \(q\) given that \(( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }\) and that \(p\) and \(q\) are real.
2. Let \(\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C\) for complex numbers \(A , B\) and \(C\).
   1. Given that the cubic equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }\) and \(\gamma = 4\), determine each of \(A , B\) and \(C\).
   2. Find the roots of the equation \(\mathrm { f } ^ { \prime } ( z ) = 0\).

9 The cubic equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. State, in terms of \(a , b\) and \(c\), the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. Find, in terms of \(a , b\) and \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
3. Show that \(( \alpha - 2 \beta \gamma ) ( \beta - 2 \gamma \alpha ) ( \gamma - 2 \alpha \beta ) = c ( 2 a + 1 ) ^ { 2 } - 2 ( b + 2 c ) ^ { 2 }\).
4. Deduce that one root of the equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) is twice the product of the other two roots if and only if \(c ( 2 a + 1 ) ^ { 2 } = 2 ( b + 2 c ) ^ { 2 }\).

2 The equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Evaluate \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and use your answer to comment on the nature of these roots.

2 The equation \(x ^ { 3 } - 14 x ^ { 2 } + 16 x + 21 = 0\) has roots \(\alpha , \beta , \gamma\). Determine the values of \(\alpha + \beta + \gamma\), \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

The equation \(x^2 + px + q = 0\), where \(p\) and \(q\) are constants, has roots \(-3\) and \(5\).

1. Find the values of \(p\) and \(q\). [2]
2. Using these values of \(p\) and \(q\), find the value of the constant \(r\) for which the equation \(x^2 + px + q + r = 0\) has equal roots. [3]

The quartic equation \(x^4 + 2x^3 - 1 = 0\) has roots \(\alpha, \beta, \gamma, \delta\).

1. Find a quartic equation whose roots are \(\alpha^4, \beta^4, \gamma^4, \delta^4\) and state the value of \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\). [5]
2. Find the value of \(\alpha^5 + \beta^5 + \gamma^5 + \delta^5\). [3]
3. Find the value of \(\alpha^8 + \beta^8 + \gamma^8 + \delta^8\). [2]

Find the sum of the squares of the roots of the equation $$x^3 + x + 12 = 0,$$ and deduce that only one of the roots is real. [4] The real root of the equation is denoted by \(\alpha\). Prove that \(-3 < \alpha < -2\), and hence prove that the modulus of each of the other roots lies between 2 and \(\sqrt{6}\). [5]

In the equation $$x^3 + ax^2 + bx + c = 0,$$ the coefficients \(a\), \(b\) and \(c\) are real. It is given that all the roots are real and greater than \(1\).

1. Prove that \(a < -3\). [1]
2. By considering the sum of the squares of the roots, prove that \(a^2 > 2b + 3\). [2]
3. By considering the sum of the cubes of the roots, prove that \(a^3 < -9b - 3c - 3\). [4]

The cubic equation \(x^3 + px^2 + qx + r = 0\), where \(p\), \(q\) and \(r\) are integers, has roots \(\alpha\), \(\beta\) and \(\gamma\), such that $$\alpha + \beta + \gamma = 15,$$ $$\alpha^2 + \beta^2 + \gamma^2 = 83.$$ Write down the value of \(p\) and find the value of \(q\). [3] Given that \(\alpha\), \(\beta\) and \(\gamma\) are all real and that \(\alpha\beta + \alpha\gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]

Using de Moivre's theorem, show that $$\tan 5\theta = \frac{5\tan\theta - 10\tan^3\theta + \tan^5\theta}{1 - 10\tan^2\theta + 5\tan^4\theta}.$$ [5] Hence show that the equation \(x^2 - 10x + 5 = 0\) has roots \(\tan^2\left(\frac{1}{5}\pi\right)\) and \(\tan^2\left(\frac{2}{5}\pi\right)\). [4] Deduce a quadratic equation, with integer coefficients, having roots \(\sec^2\left(\frac{1}{5}\pi\right)\) and \(\sec^2\left(\frac{2}{5}\pi\right)\). [3]

The roots of the equation $$x^3 + px^2 + qx + r = 0$$ are \(\alpha\), \(2\alpha\), \(4\alpha\), where \(p\), \(q\), \(r\) and \(\alpha\) are non-zero real constants.

1. Show that $$2p\alpha + q = 0.$$ [4]
2. Show that $$p^3 r - q^3 = 0.$$ [2]