Edexcel CP AS (Core Pure AS) 2024 June

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. \(( 5 - \alpha ) ( 5 - \beta ) ( 5 - \gamma )\)

1. \(\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about }
   \text { the } z \text {-axis is represented by the matrix }
   \qquad \left( \begin{array} { c c c } \cos \theta & - \sin \theta & 0
   \sin \theta & \cos \theta & 0
   0 & 0 & 1 \end{array} \right) \end{array} \right]\)

Given that the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { c c c } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } & 0
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } & 0
0 & 0 & 1 \end{array} \right)$$ represents a rotation through \(\alpha ^ { \circ }\) anticlockwise about the \(z\)-axis with respect to the right-hand rule,

1. determine the value of \(\alpha\).
2. Hence determine the smallest possible positive integer value of \(k\) for which \(\mathbf { M } ^ { k } = \mathbf { I }\) The \(3 \times 3\) matrix \(\mathbf { N }\) represents a reflection in the plane with equation \(y = 0\)
3. Write down the matrix \(\mathbf { N }\). The point \(A\) has coordinates (-2, 4, 3)
   The point \(B\) is the image of the point \(A\) under the transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\).
4. Show that the coordinates of \(B\) are \(( 2 + \sqrt { 3 } , 2 \sqrt { 3 } - 1,3 )\) Given that \(O\) is the origin,
5. show that, to 3 significant figures, the size of angle \(A O B\) is \(66.9 ^ { \circ }\)
6. Hence determine the area of triangle \(A O B\), giving your answer to 3 significant figures.

1. (a) Use the standard results for summations to show that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.
(b) Hence show that, for all positive integers \(k\), $$\sum _ { r = k + 1 } ^ { 3 k } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 3 } k ( 3 k + 1 ) \left( A k ^ { 2 } + B k + C \right)$$ where \(A , B\) and \(C\) are integers to be determined.
(c) Hence, using algebra and making your method clear, determine the value of \(k\) for which $$25 \sum _ { r = k + 1 } ^ { 3 k } r ^ { 2 } ( r + 1 ) = 192 k ^ { 3 } ( 3 k + 1 )$$

4. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & - 2 & - 7
3 & k & 2
1 & 1 & 4 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { c c c } 4 k - 2 & 1 & 7 k - 4
- 10 & 3 & - 19
3 - k & - 1 & 6 - k \end{array} \right)$$ where \(k\) is a constant.

1. Determine the value of the constant \(c\) for which $$\mathbf { A B } = ( 3 k + c ) \mathbf { I }$$
2. Hence determine the value of \(k\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Given that \(\mathbf { A } ^ { - 1 }\) does exist,
3. write down \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
4. Use the answer to part (c) to solve the simultaneous equations $$\begin{aligned} - x - 2 y - 7 z & = 10
   3 x + k y + 2 z & = 3
   x + y + 4 z & = 1 \end{aligned}$$ giving the values of \(x , y\) and \(z\) in simplest form in terms of \(k\).

1. Given that on an Argand diagram the locus of points defined by \(| z + 5 - 12 i | = 10\) is a circle,
   1. write down,
      1. the coordinates of the centre of this circle,
      2. the radius of this circle.
   2. Show, by shading on an Argand diagram, the set of points defined by
   $$| z + 5 - 12 i | \leqslant 10$$
2. For the set of points defined in part (b), determine the maximum value of \(| z |\) The set of points \(A\) is defined by $$A = \{ z : 0 \leqslant \arg ( z + 5 - 20 i ) \leqslant \pi \} \cap \{ z : | z + 5 - 12 i | \leqslant 10 \}$$
3. Determine the area of the region defined by \(A\), giving your answer to 3 significant figures.

1. The drainage system for a sports field consists of underground pipes.

This situation is modelled with respect to a fixed origin \(O\).
According to the model,

• the surface of the sports field is a plane with equation \(z = 0\)
• the pipes are straight lines
• one of the pipes, \(P _ { 1 }\), passes through the points \(A ( 3,4 , - 2 )\) and \(B ( - 2 , - 8 , - 3 )\)
• a different pipe, \(P _ { 2 }\), has equation \(\frac { x - 1 } { 2 } = \frac { y - 3 } { 4 } = \frac { z + 1 } { - 2 }\)
• the units are metres
  1. Determine a vector equation of the line representing the pipe \(P _ { 1 }\)
  2. Determine the coordinates of the point at which the pipe \(P _ { 1 }\) meets the surface of the playing field, according to the model.

Determine, according to the model,

the acute angle between pipes \(P _ { 1 }\) and \(P _ { 2 }\), giving your answer in degrees to 3 significant figures,

the shortest distance between pipes \(P _ { 1 }\) and \(P _ { 2 }\)

1. (i) Prove by induction that, for all positive integers \(n\),

$$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }$$ (ii) Prove by induction that, for all positive integers \(n\), $$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5

8. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows the central vertical cross-section, \(O A B C D E O\), of the design for a solid glass ornament. Figure 2 shows the finite region, \(R\), which is bounded by the \(y\)-axis, the horizontal line \(C B\), the vertical line \(B A\), and the curve \(A O\). The ornament is formed by rotating the region \(R\) through \(360 ^ { \circ }\) about the \(y\)-axis.
The curve \(A O\) is modelled by the equation $$x = k y ^ { 2 } + \sqrt { y } \quad 0 \leqslant y \leqslant 4$$ where \(k\) is a constant.
The point \(A\) has coordinates ( \(0.4,4\) ) and the point \(B\) has coordinates ( \(0.4,4.5\) )
The units are centimetres.

1. Determine the value of \(k\) according to this model.
2. Use algebraic integration to determine the exact volume of glass that would be required to make the ornament, according to the model.
3. State a limitation of the model. When the ornament was manufactured, \(9 \mathrm {~cm} ^ { 3 }\) of glass was required.
4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.