OCR Further Pure Core AS (Further Pure Core AS) 2021 November

1 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 8
- 11
- 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
5
3 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6
11
8 \end{array} \right) + \mu \left( \begin{array} { r } - 3
1
- 1 \end{array} \right) \end{aligned}$$

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Write down the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).

2 The equation \(2 x ^ { 3 } + 3 x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
Use a substitution to find a cubic equation with integer coefficients whose roots are \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\).

3 In this question you must show detailed reasoning.
The equation \(x ^ { 4 } - 7 x ^ { 3 } - 2 x ^ { 2 } + 218 x - 1428 = 0\) has a root \(3 - 5 i\).
Find the other three roots of this equation.

4

1. A locus \(C _ { 1 }\) is defined by \(C _ { 1 } = \{ \mathrm { z } : | \mathrm { z } + \mathrm { i } | \leqslant \mid \mathrm { z } - 2 \}\).
   1. Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 1 }\).
   2. Find the cartesian equation of the boundary line of the region representing \(C _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\).
2. A locus \(C _ { 2 }\) is defined by \(C _ { 2 } = \{ \mathrm { z } : | \mathrm { z } + 1 | \leqslant 3 \} \cap \{ \mathrm { z } : | \mathrm { z } - 2 \mathrm { i } | \geqslant 2 \}\). Indicate by shading on the Argand diagram in the Printed Answer Booklet the region representing \(C _ { 2 }\).

5 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r l } - 1 & 0
0 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c } \frac { 5 } { 13 } & - \frac { 12 } { 13 }
\frac { 12 } { 13 } & \frac { 5 } { 13 } \end{array} \right)\).

1. Use \(\mathbf { A }\) and \(\mathbf { B }\) to disprove the proposition: "Matrix multiplication is commutative". Matrix \(\mathbf { B }\) represents the transformation \(\mathrm { T } _ { \mathrm { B } }\).
2. Describe the transformation \(\mathrm { T } _ { \mathrm { B } }\).
3. By considering the inverse transformation of \(\mathrm { T } _ { \mathrm { B } }\), determine \(\mathbf { B } ^ { - 1 }\). Matrix \(\mathbf { C }\) is given by \(\mathbf { C } = \left( \begin{array} { r r } 1 & 0
   0 & - 3 \end{array} \right)\) and represents the transformation \(\mathrm { T } _ { \mathrm { C } }\).
   The transformation \(\mathrm { T } _ { \mathrm { BC } }\) is transformation \(\mathrm { T } _ { \mathrm { C } }\) followed by transformation \(\mathrm { T } _ { \mathrm { B } }\).
   An object shape of area 5 is transformed by \(\mathrm { T } _ { \mathrm { BC } }\) to an image shape \(N\).
4. Determine the area of \(N\).

6 In this question you must show detailed reasoning.

1. Solve the equation \(2 z ^ { 2 } - 10 z + 25 = 0\) giving your answers in the form \(\mathrm { a } + \mathrm { bi }\).
2. Solve the equation \(3 \omega - 2 = \mathrm { i } ( 5 + 2 \omega )\) giving your answer in the form \(\mathrm { a } + \mathrm { bi }\).

7 Prove that \(2 ^ { 3 n } - 3 ^ { n }\) is divisible by 5 for all integers \(n \geqslant 1\).

8 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } t - 1 & t - 1 & t - 1
1 - t & 6 & t
2 - 2 t & 2 - 2 t & 1 \end{array} \right)\).

1. Find, in fully factorised form, an expression for \(\operatorname { det } \mathbf { A }\) in terms of \(t\).
2. State the values of \(t\) for which \(\mathbf { A }\) is singular. You are given the following system of equations in \(x , y\) and \(z\), where \(b\) is a real number. $$\begin{aligned} \left( b ^ { 2 } + 1 \right) x + \left( b ^ { 2 } + 1 \right) y + \left( b ^ { 2 } + 1 \right) z & = 5
   \left( - b ^ { 2 } - 1 \right) x + \quad 6 y + \left( b ^ { 2 } + 2 \right) z & = 10
   \left( - 2 b ^ { 2 } - 2 \right) x + \left( - 2 b ^ { 2 } - 2 \right) y + \quad z & = 15 \end{aligned}$$
3. Determine which one of the following statements about the solution of the equations is true.
   • There is a unique solution for all values of \(b\).
   • There is a unique solution for some, but not all, values of \(b\).
   • There is no unique solution for any value of \(b\).

9 The points \(P ( 3,5 , - 21 )\) and \(Q ( - 1,3 , - 16 )\) are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point \(M\) on the ceiling of the tunnel midway between \(P\) and \(Q\) to horizontal ground level (where the \(z\)-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) is the position vector of \(M\). You are given that \(\mathbf { b } = \left( \begin{array} { l } 1
\mathrm {~s}
\mathrm { t } \end{array} \right)\) where \(s\) and \(t\) are real numbers.

1. Show that \(\mathrm { S } = 2.5 \mathrm { t } - 2\).
2. Show that at the point where the ventilation shaft reaches the ground \(\lambda = \frac { \mathrm { C } } { \mathrm { t } }\), where \(c\) is a constant to be determined.
3. Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
4. Explain what the fact that \(\mathbf { b } \times \left( \begin{array} { l } 0
   0
   1 \end{array} \right) \neq \mathbf { O }\) means about the direction of the ventilation shaft.