4.02i Quadratic equations: with complex roots

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where \(a\) and \(b\) are constants.It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) ,and that the remainder is 24 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) .

1. Find the values of \(a\) and \(b\) . \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
3. Solve the equation \(\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and \(y = \frac { 27 } { 2 x + 5 }\) for \(x \geqslant 0\).
   The curves meet at the point \(( 2,3 )\).
   Region \(A\) is bounded by the curve \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and the straight lines \(x = 0 , x = 2\) and \(y = 0\).
   Region \(B\) is bounded by the two curves and the straight line \(x = 0\).

9 The complex number \(1 + i \sqrt { } 3\) is denoted by \(u\).

1. Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Hence, or otherwise, find the modulus and argument of \(u ^ { 2 }\) and \(u ^ { 3 }\).
2. Show that \(u\) is a root of the equation \(z ^ { 2 } - 2 z + 4 = 0\), and state the other root of this equation.
3. Sketch an Argand diagram showing the points representing the complex numbers \(i\) and \(u\). Shade the region whose points represent every complex number \(z\) satisfying both the inequalities $$| z - \mathrm { i } | \leqslant 1 \quad \text { and } \quad \arg z \geqslant \arg u .$$

8

1. Find the roots of the equation \(z ^ { 2 } - z + 1 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Obtain the modulus and argument of each root.
3. Show that each root also satisfies the equation \(z ^ { 3 } = - 1\).

3

1. Solve the equation \(z ^ { 2 } - 2 \mathrm { i } z - 5 = 0\), giving your answers in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.
2. Find the modulus and argument of each root.
3. Sketch an Argand diagram showing the points representing the roots.

7

1. Solve the equation \(z ^ { 2 } + ( 2 \sqrt { } 3 ) \mathrm { i } z - 4 = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. Sketch an Argand diagram showing the points representing the roots.
3. Find the modulus and argument of each root.
4. Show that the origin and the points representing the roots are the vertices of an equilateral triangle.

7

1. Find the roots of the equation $$z ^ { 2 } + ( 2 \sqrt { } 3 ) z + 4 = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. State the modulus and argument of each root.
3. Showing all your working, verify that each root also satisfies the equation $$z ^ { 6 } = - 64$$

10

1. The complex numbers \(u\) and \(w\) satisfy the equations $$u - w = 4 \mathrm { i } \quad \text { and } \quad u w = 5$$ Solve the equations for \(u\) and \(w\), giving all answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. 1. On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 2 + 2 \mathrm { i } | \leqslant 2 , \arg z \leqslant - \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \geqslant 1\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
   2. Calculate the greatest possible value of \(\operatorname { Re } z\) for points lying in the shaded region.

8 The complex number \(w\) is defined by \(w = \frac { 22 + 4 \mathrm { i } } { ( 2 - \mathrm { i } ) ^ { 2 } }\).

1. Without using a calculator, show that \(w = 2 + 4 \mathrm { i }\).
2. It is given that \(p\) is a real number such that \(\frac { 1 } { 4 } \pi \leqslant \arg ( w + p ) \leqslant \frac { 3 } { 4 } \pi\). Find the set of possible values of \(p\).
3. The complex conjugate of \(w\) is denoted by \(w ^ { * }\). The complex numbers \(w\) and \(w ^ { * }\) are represented in an Argand diagram by the points \(S\) and \(T\) respectively. Find, in the form \(| z - a | = k\), the equation of the circle passing through \(S , T\) and the origin.

10

1. Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. 1. On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
   2. Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.

7

1. Showing all working and without using a calculator, solve the equation $$( 1 + \mathrm { i } ) z ^ { 2 } - ( 4 + 3 \mathrm { i } ) z + 5 + \mathrm { i } = 0$$ Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. The complex number \(u\) is given by $$u = - 1 - \mathrm { i }$$ On a sketch of an Argand diagram show the point representing \(u\). Shade the region whose points represent complex numbers satisfying the inequalities \(| z | < | z - 2 \mathrm { i } |\) and \(\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi\).

7 The equation \(2 x ^ { 3 } + x ^ { 2 } + 25 = 0\) has one real root and two complex roots.

1. Verify that \(1 + 2 \mathrm { i }\) is one of the complex roots.
2. Write down the other complex root of the equation.
3. Sketch an Argand diagram showing the point representing the complex number \(1 + 2 \mathrm { i }\). Show on the same diagram the set of points representing the complex numbers \(z\) which satisfy $$| z | = | z - 1 - 2 \mathrm { i } |$$

7 The complex number \(- 2 + \mathrm { i }\) is denoted by \(u\).

1. Given that \(u\) is a root of the equation \(x ^ { 3 } - 11 x - k = 0\), where \(k\) is real, find the value of \(k\).
2. Write down the other complex root of this equation.
3. Find the modulus and argument of \(u\).
4. Sketch an Argand diagram showing the point representing \(u\). Shade the region whose points represent the complex numbers \(z\) satisfying both the inequalities $$| z | < | z - 2 | \quad \text { and } \quad 0 < \arg ( z - u ) < \frac { 1 } { 4 } \pi$$

10 The polynomial \(\mathrm { p } ( z )\) is defined by $$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$ where \(m\) is a constant. It is given that \(( z + 2 )\) is a factor of \(\mathrm { p } ( z )\).

1. Find the value of \(m\).
2. Hence, showing all your working, find
   1. the three roots of the equation \(\mathrm { p } ( z ) = 0\),
   2. the six roots of the equation \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).

9 The complex number \(1 + ( \sqrt { } 2 ) \mathrm { i }\) is denoted by \(u\). The polynomial \(x ^ { 4 } + x ^ { 2 } + 2 x + 6\) is denoted by \(\mathrm { p } ( x )\).

1. Showing your working, verify that \(u\) is a root of the equation \(\mathrm { p } ( x ) = 0\), and write down a second complex root of the equation.
2. Find the other two roots of the equation \(\mathrm { p } ( x ) = 0\).

10

1. Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
2. 1. Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where $$| z - 4 - 4 i | \leqslant 2$$
   2. For the complex numbers represented by points in the region \(R\), it is given that $$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$ Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures.

9

1. Without using a calculator, use the formula for the solution of a quadratic equation to solve $$( 2 - \mathrm { i } ) z ^ { 2 } + 2 z + 2 + \mathrm { i } = 0$$ Give your answers in the form \(a + b \mathrm { i }\).
2. The complex number \(w\) is defined by \(w = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\). In an Argand diagram, the points \(A , B\) and \(C\) represent the complex numbers \(w , w ^ { 3 }\) and \(w ^ { * }\) respectively (where \(w ^ { * }\) denotes the complex conjugate of \(w\) ). Draw the Argand diagram showing the points \(A , B\) and \(C\), and calculate the area of triangle \(A B C\).

9

1. The complex numbers \(u\) and \(w\) are such that $$u - w = 2 \mathrm { i } \quad \text { and } \quad u w = 6$$ Find \(u\) and \(w\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities $$| z - 2 - 2 \mathbf { i } | \leqslant 2 , \quad 0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi \quad \text { and } \quad \operatorname { Re } z \leqslant 3$$
   \includegraphics[max width=\textwidth, alt={}]{c1bd46f8-a33a-4927-af59-718b1c9dd4e1-16_462_709_260_719}
   A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is \(A\) and the radius is \(r\), as shown in the diagram. The depth of water at time \(t\) is \(h\). At time \(t = 0\) the tank is full and the depth of the water is \(r\). At this instant a tap at \(A\) is opened and water begins to flow out at a rate proportional to \(\sqrt { h }\). The tank becomes empty at time \(t = 14\). The volume of water in the tank is \(V\) when the depth is \(h\). It is given that \(V = \frac { 1 } { 3 } \pi \left( 3 r h ^ { 2 } - h ^ { 3 } \right)\).
   1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { B } { 2 r h ^ { \frac { 1 } { 2 } } - h ^ { \frac { 3 } { 2 } } } ,$$ where \(B\) is a positive constant.
   2. Solve the differential equation and obtain an expression for \(t\) in terms of \(h\) and \(r\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5

1. Solve the equation \(z ^ { 2 } - 2 p \mathrm { i } z - q = 0\), where \(p\) and \(q\) are real constants.
   In an Argand diagram with origin \(O\), the roots of this equation are represented by the distinct points \(A\) and \(B\).
2. Given that \(A\) and \(B\) lie on the imaginary axis, find a relation between \(p\) and \(q\).
3. Given instead that triangle \(O A B\) is equilateral, express \(q\) in terms of \(p\).

10

1. Verify that \(- 1 + \sqrt { 2 } \mathrm { i }\) is a root of the equation \(z ^ { 4 } + 3 z ^ { 2 } + 2 z + 12 = 0\).
2. Find the other roots of this equation.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 The complex number \(- 1 + \sqrt { 7 } \mathrm { i }\) is denoted by \(u\). It is given that \(u\) is a root of the equation $$2 x ^ { 3 } + 3 x ^ { 2 } + 14 x + k = 0$$ where \(k\) is a real constant.

1. Find the value of \(k\).
2. Find the other two roots of the equation.
3. On an Argand diagram, sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - u | = 2\).
4. Determine the greatest value of \(\arg z\) for points on this locus, giving your answer in radians.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 The polynomial \(x ^ { 3 } + 5 x ^ { 2 } + 31 x + 75\) is denoted by \(\mathrm { p } ( x )\).

1. Show that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).
2. Show that \(z = - 1 + 2 \sqrt { 6 } \mathrm { i }\) is a root of \(\mathrm { p } ( z ) = 0\).
3. Hence find the complex numbers \(z\) which are roots of \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 Find the complex numbers \(w\) which satisfy the equation \(w ^ { 2 } + 2 \mathrm { i } w ^ { * } = 1\) and are such that \(\operatorname { Re } w \leqslant 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.

7

1. Verify that \(- 1 + \sqrt { 5 } \mathrm { i }\) is a root of the equation \(2 x ^ { 3 } + x ^ { 2 } + 6 x - 18 = 0\).
2. Find the other roots of this equation.

3

1. Given the complex numbers \(u = a + \mathrm { i } b\) and \(w = c + \mathrm { i } d\), where \(a , b , c\) and \(d\) are real, prove that \(( u + w ) ^ { * } = u ^ { * } + w ^ { * }\).
2. Solve the equation \(( z + 2 + \mathrm { i } ) ^ { * } + ( 2 + \mathrm { i } ) z = 0\), giving your answer in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.

5

1. Solve the equation \(z ^ { 2 } - 6 \mathrm { i } z - 12 = 0\), giving the answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
2. On a sketch of an Argand diagram with origin \(O\), show points \(A\) and \(B\) representing the roots of the equation in part (a).
3. Find the exact modulus and argument of each root.
4. Hence show that the triangle \(O A B\) is equilateral.