Edexcel PURE 2024 October

1. The line \(l _ { 1 }\) passes through the point \(A ( - 5,20 )\) and the point \(B ( 3 , - 4 )\).
   1. Find an equation for \(l _ { 1 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\)
2. Find an equation for \(l _ { 2 }\) giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

1. Simplify fully $$\frac { 3 y ^ { 3 } \left( 2 x ^ { 4 } \right) ^ { 3 } } { 4 x ^ { 2 } y ^ { 4 } }$$
2. Find the exact value of \(a\) such that $$\frac { 16 } { \sqrt { 3 } + 1 } = a \sqrt { 27 } + 4$$ Write your answer in the form \(p \sqrt { 3 } + q\) where \(p\) and \(q\) are fully simplified rational constants.

1. In this question you must show all stages of your working.

$$f ( x ) = \frac { ( x + 5 ) ^ { 2 } } { \sqrt { x } } \quad x > 0$$

1. Find \(\int f ( x ) d x\)
2. 1. Show that when \(\mathrm { f } ^ { \prime } ( x ) = 0\) $$3 x ^ { 2 } + 10 x - 25 = 0$$
   2. Hence state the value of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) = 0$$

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\)

• has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic function
• cuts the \(x\)-axis at the origin and at \(x = 4\)
• has a minimum turning point at ( \(2 , - 4.8\) )
  1. find \(\mathrm { f } ( x )\)

Given that \(C _ { 2 }\)

The curves \(C _ { 1 }\) and \(C _ { 2 }\) meet in the first quadrant at the point \(P\), shown in Figure 1.

Use algebra to find the coordinates of \(P\).

1. A plot of land \(O A B\) is in the shape of a sector of a circle with centre \(O\).

Given

• \(O A = O B = 5 \mathrm {~km}\)
• angle \(A O B = 1.2\) radians
  1. find the perimeter of the plot of land.
     (2)

\begin{figure}[h] \end{figure} A point \(P\) lies on \(O B\) such that the line \(A P\) divides the plot of land into two regions \(R _ { 1 }\) and \(R _ { 2 }\) as shown in Figure 2. Given that $$\text { area of } R _ { 1 } = 3 \times \text { area of } R _ { 2 }$$

show that the area of \(R _ { 2 } = 3.75 \mathrm {~km} ^ { 2 }\)

Find the length of \(A P\), giving your answer to the nearest 100 m .

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.}

1. Sketch the curve \(C\) with equation $$y = \frac { 1 } { 2 - x } \quad x \neq 2$$ State on your sketch
   • the equation of the vertical asymptote
   • the coordinates of the intersection of \(C\) with the \(y\)-axis
   The straight line \(l\) has equation \(y = k x - 4\), where \(k\) is a constant.
   Given that \(l\) cuts \(C\) at least once,
2. 1. show that $$k ^ { 2 } - 5 k + 4 \geqslant 0$$
   2. find the range of possible values for \(k\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = - 4 \cos x$$ where \(x\) is measured in radians.
Points \(P\) and \(Q\) lie on the curve and are shown in Figure 3.

1. State
   1. the coordinates of \(P\)
   2. the coordinates of \(Q\) The curve \(C _ { 2 }\) has equation \(y = - 4 \cos x + k\) where \(x\) is measured in radians and \(k\) is a constant. Given that \(C _ { 2 }\) has a maximum \(y\) value of 11
2. 1. state the value of \(k\)
   2. state the coordinates of the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate. On the opposite page there is a copy of Figure 3 labelled Diagram 1.
3. Using Diagram 1, state the number of solutions of the equation $$- 4 \cos x = 5 - \frac { 10 } { \pi } x$$ giving a reason for your answer. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-23_860_1016_1676_529} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\).

The point \(P\) with \(x\) coordinate 3 lies on \(C\) \section*{Given}

• \(\mathrm { f } ^ { \prime } ( x ) = 4 x ^ { 2 } + k x + 3\) where \(k\) is a constant
• the normal to \(C\) at \(P\) has equation \(y = - \frac { 1 } { 24 } x + 5\)
  1. show that \(k = - 5\)
  2. Hence find \(\mathrm { f } ( x )\).

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$

1. Deduce the range of values of \(x\) for which \(\mathrm { f } ( x ) \geqslant 0\)
2. Find \(\mathrm { f } ^ { \prime } ( x )\) giving your answer in simplest form. The point \(R ( - 4,84 )\) lies on \(C\).
   Given that the tangent to \(C\) at the point \(P\) is parallel to the tangent to \(C\) at the point \(R\) (c) find the \(x\) coordinate of \(P\).
   (d) Find the point to which \(R\) is transformed when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation,
   1. \(y = \mathrm { f } ( x - 3 )\)
   2. \(y = 4 \mathrm { f } ( x )\)