Symmetric functions of roots

1 The cubic equation \(2 x ^ { 3 } + x ^ { 2 } - p x - 5 = 0\), where \(p\) is a positive constant, has roots \(\alpha , \beta , \gamma\).

1. State, in terms of \(p\), the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
2. Find the value of \(\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }\).
3. Deduce a cubic equation whose roots are \(\alpha \beta , \beta \gamma , \alpha \gamma\).
4. Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = \frac { 1 } { 3 }\), find the value of \(p\).

3 It is given that $$\begin{aligned} & \alpha + \beta + \gamma + \delta = 2 \\ & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 3 \\ & \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = 4 \end{aligned}$$

1. Find the value of \(\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta\).
2. Find the value of \(\alpha ^ { 2 } \beta + \alpha ^ { 2 } \gamma + \alpha ^ { 2 } \delta + \beta ^ { 2 } \alpha + \beta ^ { 2 } \gamma + \beta ^ { 2 } \delta + \gamma ^ { 2 } \alpha + \gamma ^ { 2 } \beta + \gamma ^ { 2 } \delta + \delta ^ { 2 } \alpha + \delta ^ { 2 } \beta + \delta ^ { 2 } \gamma\).
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3. It is given that \(\alpha , \beta , \gamma , \delta\) are the roots of the equation $$6 x ^ { 4 } - 12 x ^ { 3 } + 3 x ^ { 2 } + 2 x + 6 = 0 .$$
   1. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
   2. Find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } + \delta ^ { 5 }\).

10 The roots of the equation $$x ^ { 3 } - 9 x ^ { 2 } + 27 x - 29 = 0$$ are denoted by \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.

1. Write down the value of \(\alpha + \beta + \gamma\).
2. It is given that \(\beta = p + \mathrm { i } q\), where \(q > 0\). Find the value of \(p\), in terms of \(\alpha\).
3. Write down the value of \(\alpha \beta \gamma\).
4. Find the value of \(q\), in terms of \(\alpha\) only.

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

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 22\).

9 The roots of the equation \(x ^ { 3 } - k x ^ { 2 } - 2 = 0\) are \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.

1. Show that \(k = \alpha - \frac { 2 } { \alpha ^ { 2 } }\).
2. Given that \(\beta = u + \mathrm { i } v\), where \(u\) and \(v\) are real, find \(u\) in terms of \(\alpha\).
3. Find \(v ^ { 2 }\) in terms of \(\alpha\).

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

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Find the value of \(\left( \alpha + \frac { 1 } { \alpha } \right) \left( \beta + \frac { 1 } { \beta } \right)\) in terms of \(k\).

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

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

4 The roots of the equation $$x ^ { 3 } - 8 x ^ { 2 } + 5 = 0$$ are \(\alpha , \beta , \gamma\). Show that $$\alpha ^ { 2 } = \frac { 5 } { \beta + \gamma } .$$ It is given that the roots are all real. Without reference to a graph, show that one of the roots is negative and the other two roots are positive.

14 Substitute appropriate values of \(t _ { 1 }\) and \(t _ { 2 }\) to verify that the expression \(t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 } + t _ { 1 } t _ { 2 } + \frac { 1 } { 2 }\) gives the correct value for the \(y\)-coordinate of the point of intersection of the normals at the points A and B in Fig. C2.

4 The roots of the equation $$5 x ^ { 3 } + 2 x ^ { 2 } - 3 x + p = 0$$ are \(\alpha , \beta\) and \(\gamma\) Given that \(p\) is a constant, state the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\) Circle your answer.
\(- \frac { 3 } { 5 }\)
\(- \frac { 2 } { 5 }\)
\(\frac { 2 } { 5 }\)
\(\frac { 3 } { 5 }\)

2 In this question you must show detailed reasoning.
The equation \(\mathrm { x } ^ { 2 } - \mathrm { kx } + 2 \mathrm { k } = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\).
Find \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha }\) in terms of \(k\), simplifying your answer.

5

1. Find the volume scale factor of the transformation with associated matrix \(\left( \begin{array} { r r r } 1 & 2 & 0 \\ 0 & 3 & - 1 \\ - 1 & 0 & 2 \end{array} \right)\).
2. The transformations S and T of the plane have associated \(2 \times 2\) matrices \(\mathbf { P }\) and \(\mathbf { Q }\) respectively.
   1. Write down an expression for the associated matrix of the combined transformation S followed by T. The determinant of \(\mathbf { P }\) is 3 and \(\mathbf { Q } = \left( \begin{array} { r r } k & 3 \\ - 1 & 2 \end{array} \right)\), where \(k\) is a constant.
   2. Given that this combined transformation preserves both orientation and area, determine the value of \(k\).

1. The roots of the quartic equation

$$3 x ^ { 4 } + 5 x ^ { 3 } - 7 x + 6 = 0$$ are \(\alpha , \beta , \gamma\) and \(\delta\)
Making your method clear and without solving the equation, determine the exact value of

1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\)
2. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma } + \frac { 2 } { \delta }\)
3. \(( 3 - \alpha ) ( 3 - \beta ) ( 3 - \gamma ) ( 3 - \delta )\)

1. The cubic equation

$$2 x ^ { 3 } - 3 x ^ { 2 } + 5 x + 7 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, determine the exact value of

1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)
2. \(\frac { 3 } { \alpha } + \frac { 3 } { \beta } + \frac { 3 } { \gamma }\)
3. \(( 5 - \alpha ) ( 5 - \beta ) ( 5 - \gamma )\)

4 \sqrt { 3 } & - 4 \end{array} \right)$$

1. Determine
   1. the value of \(k\),
   2. the smallest value of \(\theta\) A square \(S\) has vertices at the points with coordinates ( 0,0 ), ( \(a , - a\) ), ( \(2 a , 0\) ) and ( \(a , a\) ) where \(a\) is a constant. The square \(S\) is transformed to the square \(S ^ { \prime }\) by the transformation represented by \(\mathbf { M }\).
2. Determine, in terms of \(a\), the area of \(S ^ { \prime }\)
   1. (a) Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos ^ { 2 } \left( \frac { x } { 3 } \right)\) in ascending powers of \(x\), up to and including the term in \(x ^ { 4 }\)
   Give each term in simplest form.
3. Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$
4. Use the integration function on your calculator to evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \left( \frac { 1 } { x } \cos ^ { 2 } \left( \frac { x } { 3 } \right) \right) \mathrm { d } x$$ Give your answer to 5 decimal places.
5. Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b).
   1. The cubic equation
   $$a x ^ { 3 } + b x ^ { 2 } - 19 x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha , \beta\) and \(\gamma\)\\ The cubic equation $$w ^ { 3 } - 9 w ^ { 2 } - 97 w + c = 0$$ where \(c\) is a constant, has roots \(( 4 \alpha - 1 ) , ( 4 \beta - 1 )\) and \(( 4 \gamma - 1 )\)\\ Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\).
   1. 1. \(\mathbf { A }\) is a 2 by 2 matrix and \(\mathbf { B }\) is a 2 by 3 matrix.
   Giving a reason for your answer, explain whether it is possible to evaluate
6. \(\mathbf { A B }\)
7. \(\mathbf { A } + \mathbf { B }\)\\ (ii) Given that $$\left( \begin{array} { r r r } - 5 & 3 & 1
   a & 0 & 0
   b & a & b \end{array} \right) \left( \begin{array} { r r r } 0 & 5 & 0
   2 & 12 & - 1
   - 1 & - 11 & 3 \end{array} \right) = \lambda \mathbf { I }$$ where \(a\), \(b\) and \(\lambda\) are constants,
8. determine
   • the value of \(\lambda\)
   • the value of \(a\)
   • the value of \(b\)
   • Hence deduce the inverse of the matrix \(\left( \begin{array} { r r r } - 5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{array} \right)\)\\ (iii) Given that
   $$\mathbf { M } = \left( \begin{array} { c c c } 1 & 1 & 1
   0 & \sin \theta & \cos \theta
   0 & \cos 2 \theta & \sin 2 \theta \end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf { M }\) is singular.

1. The roots of the equation

$$2 x ^ { 3 } - 3 x ^ { 2 } + 12 x + 7 = 0$$ are \(\alpha , \beta\) and \(\gamma\)
Without solving the equation,

1. write down the value of each of $$\alpha + \beta + \gamma \quad \alpha \beta + \alpha \gamma + \beta \gamma \quad \alpha \beta \gamma$$
2. Use the answers to part (a) to determine the value of
   1. \(\frac { 2 } { \alpha } + \frac { 2 } { \beta } + \frac { 2 } { \gamma }\)
   2. \(( \alpha - 1 ) ( \beta - 1 ) ( \gamma - 1 )\)
   3. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)

1. The roots of the equation

$$x ^ { 3 } - 2 x ^ { 2 } + 4 x - 5 = 0$$ are \(p , q\) and \(r\).
Without solving the equation, find the value of

1. \(\frac { 2 } { p } + \frac { 2 } { q } + \frac { 2 } { r }\)
2. \(( p - 4 ) ( q - 4 ) ( r - 4 )\)
3. \(p ^ { 3 } + q ^ { 3 } + r ^ { 3 }\)

1. The roots of the equation

$$x ^ { 3 } - 8 x ^ { 2 } + 28 x - 32 = 0$$ are \(\alpha , \beta\) and \(\gamma\)
Without solving the equation, find the value of

1. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\)
2. \(( \alpha + 2 ) ( \beta + 2 ) ( \gamma + 2 )\)
3. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)

5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the position vector of \(P\).
2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
   \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
   The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
   The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\).

2 In this question you must show detailed reasoning.
The quadratic equation \(3 x ^ { 2 } - 7 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).

1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Hence find the values of the following expressions.
   (a) \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
   (b) \(\alpha ^ { 2 } + \beta ^ { 2 }\)
   \(3 \quad l _ { 1 }\) and \(l _ { 2 }\) are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c }

2 The quadratic equation \(x ^ { 2 } + p x + q = 0\) has roots \(\alpha\) and \(\beta\)
Which of the following is equal to \(\alpha \beta\) ?
Circle your answer.
[0pt] [1 mark]
\(p - p - q - q\)

3 The quadratic equation \(a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )\) has real roots \(\alpha\) and \(\beta\). One of the four statements below is incorrect. Which statement is incorrect? Tick ( \(\checkmark\) ) one box.
\(c = 0 \Rightarrow \alpha = 0\) or \(\beta = 0\) □
\(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) □
\(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) □
\(b = 0 \Rightarrow \alpha = - \beta\) □

1 The roots of the equation \(20 x ^ { 3 } - 16 x ^ { 2 } - 4 x + 7 = 0\) are \(\alpha , \beta\) and \(\gamma\)
Find the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\)
Circle your answer.
\(- \frac { 4 } { 5 }\)
\(- \frac { 1 } { 5 }\)
\(\frac { 1 } { 5 }\)
\(\frac { 4 } { 5 }\)