SPS SPS ASFM (SPS ASFM) 2020 May

1. You are given that \(z = 3 - 4 \mathrm { i }\).

1. Find
   • \(| z |\),
   • \(\arg ( z )\),
   • \(z ^ { * }\).
   On an Argand diagram the complex number \(w\) is represented by the point \(A\) and \(w ^ { * }\) is represented by the point \(B\).
2. Describe the geometrical relationship between the points \(A\) and \(B\).

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 \(p\).

3. \section*{In this question you must show detailed reasoning.} You are given that \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).

1. Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
2. Solve \(\mathrm { f } ( z ) = 0\). Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
3. Find the exact area of \(R\).
4. \(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\). Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).

4. \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.

5. A transformation T is represented by the matrix \(\mathbf { T }\) where \(\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4
3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)\).
A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q ^ { \prime }\). Find the smallest possible value of the area of \(Q ^ { \prime }\).

6. In this question you must show detailed reasoning.
\(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6
0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right)
0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\).

7. A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).
   \includegraphics[max width=\textwidth, alt={}, center]{4718e448-71e5-452a-a905-608331f743a5-6_675_412_296_221} As shown in the diagram, \(A B\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m _ { 1 } \mathrm {~kg}\). One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m _ { 2 } \mathrm {~kg}\), which is free to move on \(A B\). Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). The magnitude of the tension in string \(A P\) is denoted by \(T _ { 1 } \mathrm {~N}\) while that in string \(P R\) is denoted by \(T _ { 2 } \mathrm {~N}\).
3. By considering forces on \(R\), express \(T _ { 2 }\) in terms of \(m _ { 2 }\).
4. Show that
   1. \(T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)\),
   2. \(\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }\).
5. Deduce that, in the case where \(m _ { 1 }\) is much bigger than \(m _ { 2 } , \omega \approx 3.5\). In a different case, where \(m _ { 1 } = 2.5\) and \(m _ { 2 } = 2.8 , P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
6. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm { rads } ^ { - 1 }\) to zero. Three particles, \(P , Q\) and \(R\), are at rest on a smooth horizontal plane. The particles lie along a straight line with \(Q\) between \(P\) and \(R\). The particles \(Q\) and \(R\) have masses \(m\) and \(k m\) respectively, where \(k\) is a constant. Particle \(Q\) is projected towards \(R\) with speed \(u\) and the particles collide directly.
   The coefficient of restitution between each pair of particles is \(e\).
7. Find, in terms of \(e\), the range of values of \(k\) for which there is a second collision. Given that the mass of \(P\) is \(k m\) and that there is a second collision,
8. write down, in terms of \(u , k\) and \(e\), the speed of \(Q\) after this second collision.

10. On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution \(\operatorname { Po } ( 120 )\).

1. Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
2. Find the probability that in a randomly chosen 10 -minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
3. State a necessary assumption for the validity of your calculation in part (b).

11. The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other.
2. Find the probability that no two men are next to one another.

12. Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to \(n\) in an order of Jane's own choice and Alex tells Jane what order he believes was chosen. They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round. They use a value of \(n\) such that, if Alex chooses the order of the integers at random, the probability that Alex's claim will be accepted is less than \(1 \%\). Determine the smallest possible value of \(n\).