4.02g Conjugate pairs: real coefficient polynomials

11 A point has Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) where \(r \geq 0\) and \(- \pi < \theta \leq \pi\) 11

1. Express \(r\) in terms of \(x\) and \(y\) 11
2. Express \(x\) in terms of \(r\) and \(\theta\) 11
3. The curve \(C _ { 1 }\) has the polar equation $$r ( 2 + \cos \theta ) = 1 \quad - \pi < \theta \leq \pi$$ 11 (c) (i) Show that the Cartesian equation of \(C _ { 1 }\) can be written as $$a y ^ { 2 } = ( 1 + b x ) ( 1 + x )$$ where \(a\) and \(b\) are integers to be determined.
   11 (c) (ii) The curve \(C _ { 2 }\) has the Cartesian equation $$a x ^ { 2 } = ( 1 + b y ) ( 1 + y )$$ where \(a\) and \(b\) take the same values as in part (c)(i). Describe fully a single transformation that maps the curve \(C _ { 1 }\) onto the curve \(C _ { 2 }\)

5 In this question you must show detailed reasoning.

1. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
2. Hence verify that \(2 + 3\) i is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
3. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.

3 In this question you must show detailed reasoning.
You are given that \(x = 2 + 5 \mathrm { i }\) is a root of the equation \(x ^ { 3 } - 2 x ^ { 2 } + 21 x + 58 = 0\).
Solve the equation.

9 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 \(\alpha = 1 + \mathrm { i } \sqrt { 2 }\).

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 \(\tan \theta = \sqrt { 2 }\).

2 In this question you must show detailed reasoning. The equation \(\mathrm { f } ( x ) = 0\), where \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 26 x + 169\), has a root \(x = 2 + 3 \mathrm { i }\).

1. Express \(\mathrm { f } ( x )\) as a product of two quadratic factors.
2. Hence write down all the roots of the equation \(\mathrm { f } ( x ) = 0\).

5

1. 1. Calculate \(( 2 + \mathrm { i } \sqrt { 5 } ) ( \sqrt { 5 } - \mathrm { i } )\).
   2. Hence verify that \(\sqrt { 5 } - \mathrm { i }\) is a root of the equation $$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$ where \(z ^ { * }\) is the conjugate of \(z\).
2. The quadratic equation $$x ^ { 2 } + p x + q = 0$$ in which the coefficients \(p\) and \(q\) are real, has a complex root \(\sqrt { 5 } - \mathrm { i }\).
   1. Write down the other root of the equation.
   2. Find the sum and product of the two roots of the equation.
   3. Hence state the values of \(p\) and \(q\).

2 The complex number \(2 + 3 \mathrm { i }\) is a root of the quadratic equation $$x ^ { 2 } + b x + c = 0$$ where \(b\) and \(c\) are real numbers.

1. Write down the other root of this equation.
2. Find the values of \(b\) and \(c\).

4

1. It is given that \(z = x + y \mathrm { i }\), where \(x\) and \(y\) are real numbers.
   1. Write down, in terms of \(x\) and \(y\), an expression for \(( z - 2 \mathrm { i } ) ^ { * }\).
   2. Solve the equation $$( z - 2 \mathrm { i } ) ^ { * } = 4 \mathrm { i } z + 3$$ giving your answer in the form \(a + b \mathrm { i }\).
2. It is given that \(p + q \mathrm { i }\), where \(p\) and \(q\) are real numbers, is a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\). Without finding the values of \(p\) and \(q\), state why \(p - q\) i is not a root of the equation \(z ^ { 2 } + 10 \mathrm { i } z - 29 = 0\).

3

1. Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) can be expressed in the form \(2 + b \mathrm { i }\), where \(b\) is an integer.
2. It is given that \(2 + \mathrm { i }\) is a root of the equation $$z ^ { 3 } + p z + q = 0$$ where \(p\) and \(q\) are real numbers.
   1. Show that \(p = - 11\) and find the value of \(q\).
   2. Given that \(2 - \mathrm { i }\) is also a root of \(z ^ { 3 } + p z + q = 0\), find a quadratic factor of \(z ^ { 3 } + p z + q\) with real coefficients.
   3. Find the real root of the equation \(z ^ { 3 } + p z + q = 0\).
      \includegraphics[max width=\textwidth, alt={}]{e45b07a3-e303-4caf-8f3a-5341bad7560a-06_1568_1707_1139_155}

3 The cubic equation $$2 z ^ { 3 } + p z ^ { 2 } + q z + 16 = 0$$ where \(p\) and \(q\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha = 2 + 2 \sqrt { 3 } \mathrm { i }\).

1. 1. Write down another root, \(\beta\), of the equation.
   2. Find the third root, \(\gamma\).
   3. Find the values of \(p\) and \(q\).
2. 1. Express \(\alpha\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. Show that $$( 2 + 2 \sqrt { 3 } \mathrm { i } ) ^ { n } = 4 ^ { n } \left( \cos \frac { n \pi } { 3 } + \mathrm { i } \sin \frac { n \pi } { 3 } \right)$$
   3. Show that $$\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 ^ { 2 n + 1 } \cos \frac { n \pi } { 3 } + \left( - \frac { 1 } { 2 } \right) ^ { n }$$ where \(n\) is an integer.

3

1. Show that \(( 1 + \mathrm { i } ) ^ { 3 } = 2 \mathrm { i } - 2\).
2. The cubic equation $$z ^ { 3 } - ( 5 + \mathrm { i } ) z ^ { 2 } + ( 9 + 4 \mathrm { i } ) z + k ( 1 + \mathrm { i } ) = 0$$ where \(k\) is a real constant, has roots \(\alpha , \beta\) and \(\gamma\).
   It is given that \(\alpha = 1 + \mathrm { i }\).
   1. Find the value of \(k\).
   2. Show that \(\beta + \gamma = 4\).
   3. Find the values of \(\beta\) and \(\gamma\).

7 The numbers \(\alpha , \beta\) and \(\gamma\) satisfy the equations $$\begin{aligned} & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 10 - 12 \mathrm { i } \\ & \alpha \beta + \beta \gamma + \gamma \alpha = 5 + 6 \mathrm { i } \end{aligned}$$

1. Show that \(\alpha + \beta + \gamma = 0\).
2. The numbers \(\alpha , \beta\) and \(\gamma\) are also the roots of the equation $$z ^ { 3 } + p z ^ { 2 } + q z + r = 0$$ Write down the value of \(p\) and the value of \(q\).
3. It is also given that \(\alpha = 3 \mathrm { i }\).
   1. Find the value of \(r\).
   2. Show that \(\beta\) and \(\gamma\) are the roots of the equation $$z ^ { 2 } + 3 \mathrm { i } z - 4 + 6 \mathrm { i } = 0$$
   3. Given that \(\beta\) is real, find the values of \(\beta\) and \(\gamma\).

4 The cubic equation $$z ^ { 3 } + p z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\).
   2. Express \(\alpha \beta \gamma\) in terms of \(q\).
2. Show that $$\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 \alpha \beta \gamma$$
3. Given that \(\alpha = 4 + 7 \mathrm { i }\) and that \(p\) and \(q\) are real, find the values of:
   1. \(\beta\) and \(\gamma\);
   2. \(p\) and \(q\).
4. Find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

3 In this question you must show detailed reasoning.
The function \(\mathrm { f } ( \mathrm { z } )\) is given by \(\mathrm { f } ( \mathrm { z } ) = 2 \mathrm { z } ^ { 3 } - 7 \mathrm { z } ^ { 2 } + 16 \mathrm { z } - 15\).
By first evaluating \(\mathrm { f } \left( \frac { 3 } { 2 } \right)\), find the roots of \(\mathrm { f } ( \mathrm { z } ) = 0\).

1 The quadratic equation \(\mathrm { x } ^ { 2 } + \mathrm { ax } + \mathrm { b } = 0\), where \(a\) and \(b\) are real constants, has a root 2-3.

1. Write down the other root.
2. Hence or otherwise determine the values of \(a\) and \(b\).

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

1. Find \(\alpha\) and \(\beta\) in modulus-argument form.
2. Hence or otherwise show that \(\alpha\) and \(\beta\) are both roots of \(x ^ { 3 } + \lambda = 0\), where \(\lambda\) is a real constant to be determined.

7 In the quartic equation \(2 x ^ { 4 } - 20 x ^ { 3 } + a x ^ { 2 } + b x + 250 = 0\), the coefficients \(a\) and \(b\) are real. One root of the equation is \(2 + \mathrm { i }\). Find the other roots.

3. Given that \(5 - \mathrm { i }\) is a root of the equation \(x ^ { 4 } - 10 x ^ { 3 } + 10 x ^ { 2 } + 160 x - 416 = 0\),

1. write down another root of the equation,
2. find the remaining roots.

2. Given that \(x ^ { 2 } + 4 x + 5\) is a factor of \(x ^ { 3 } + x ^ { 2 } - 7 x - 15\), solve the equation \(x ^ { 3 } + x ^ { 2 } - 7 x - 15 = 0\).

7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35 Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).

5. \begin{figure}[h] \end{figure} The complex numbers \(z _ { 1 } = - 2 , z _ { 2 } = - 1 + 2 \mathrm { i }\) and \(z _ { 3 } = 1 + \mathrm { i }\) are plotted in Figure 1, on an Argand diagram for the complex plane with \(z = x + \mathrm { i } y\)

1. Explain why \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) cannot all be roots of a quartic polynomial equation with real coefficients.
2. Show that \(\arg \left( \frac { z _ { 2 } - z _ { 1 } } { z _ { 3 } - z _ { 1 } } \right) = \frac { \pi } { 4 }\)
3. Hence show that \(\arctan ( 2 ) - \arctan \left( \frac { 1 } { 3 } \right) = \frac { \pi } { 4 }\) A copy of Figure 1, labelled Diagram 1, is given on page 12.
4. Shade, on Diagram 1, the set of points of the complex plane that satisfy the inequality $$| z + 2 | \leqslant | z - 1 - \mathrm { i } |$$
   \includegraphics[max width=\textwidth, alt={}]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-12_479_524_296_776}
   \section*{Diagram 1}

7. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a\), \(b\), \(c\) and \(d\) are real constants.
The equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has complex roots \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) and \(\mathrm { z } _ { 4 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 } , z _ { 3 }\) and \(z _ { 4 }\) form the vertices of a square, with one vertex in each quadrant.
Given that \(z _ { 1 } = 2 + 3 i\), determine the values of \(a , b , c\) and \(d\).

7. $$f ( z ) = z ^ { 4 } - 6 z ^ { 3 } + p z ^ { 2 } + q z + r$$ where \(p , q\) and \(r\) are real constants.
The roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\) where \(\alpha = 3\) and \(\beta = 2 + \mathrm { i }\) Given that \(\gamma\) is a complex root of \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. 1. write down the root \(\gamma\),
   2. explain why \(\delta\) must be real.
2. Determine the value of \(\delta\).
3. Hence determine the values of \(p , q\) and \(r\).
4. Write down the roots of the equation \(\mathrm { f } ( - 2 \mathrm { z } ) = 0\)

1. \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } ^ { 2 } + b \mathrm { z } + 175 \quad\) where \(a\) and \(b\) are real constants

Given that \(- 3 + 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. determine the value of \(a\) and the value of \(b\).
2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
3. Write down the roots of the equation \(\mathrm { f } ( \mathrm { z } + 2 ) = 0\)

1. $$f ( z ) = z ^ { 4 } + a z ^ { 3 } + b z ^ { 2 } + c z + d$$ where \(a , b , c\) and \(d\) are real constants.
Given that \(- 1 + 2 \mathrm { i }\) and \(3 - \mathrm { i }\) are two roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)

1. show all the roots of \(f ( z ) = 0\) on a single Argand diagram,
2. find the values of \(a , b , c\) and \(d\).