Edexcel P1 (Pure Mathematics 1) 2022 October

1. The curve \(C\) has equation

$$y = \frac { x ^ { 3 } } { 4 } - x ^ { 2 } + \frac { 17 } { x } \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(R \left( 2 , \frac { 13 } { 2 } \right)\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at the point \(R\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

1. Given that

$$( x - 5 ) ( 2 x + 1 ) ( x + 3 ) \equiv a x ^ { 3 } + b x ^ { 2 } - 32 x - 15$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\).
2. Hence find $$\int \frac { ( x - 5 ) ( 2 x + 1 ) ( x + 3 ) } { 5 \sqrt { x } } \mathrm {~d} x$$ writing each term in simplest form.

1. The share price of a company is monitored.

Exactly 3 years after monitoring began, the share price was \(\pounds 1.05\)
Exactly 5 years after monitoring began, the share price was \(\pounds 1.65\)
The share price, \(\pounds V\), of the company is modelled by the equation $$V = p t + q$$ where \(t\) is the number of years after monitoring began and \(p\) and \(q\) are constants.

1. Find the value of \(p\) and the value of \(q\). Exactly \(T\) years after monitoring began, the share price was \(\pounds 2.50\)
2. Find the value of \(T\), according to the model, giving your answer to one decimal place.

1. In this question you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

$$f ( x ) = x ^ { 2 } ( 2 x + 1 ) - 15 x$$

1. Solve $$\mathrm { f } ( x ) = 0$$
2. Hence solve $$y ^ { \frac { 4 } { 3 } } \left( 2 y ^ { \frac { 2 } { 3 } } + 1 \right) - 15 y ^ { \frac { 2 } { 3 } } = 0 \quad y > 0$$ giving your answer in simplified surd form.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\)
Given that

• \(\mathrm { f } ^ { \prime } ( \mathrm { x } ) = \frac { 12 } { \sqrt { \mathrm { x } } } + \frac { x } { 3 } - 4\)
• the point \(P ( 9,8 )\) lies on \(C\)
  1. find, in simplest form, \(\mathrm { f } ( x )\)

The line \(l\) is the normal to \(C\) at \(P\)

Find the coordinates of the point at which \(l\) crosses the \(y\)-axis.

1. (a) Given that \(k\) is a positive constant such that \(0 < k < 4\) sketch, on separate axes, the graphs of
   1. \(y = ( 2 x - k ) ( x + 4 ) ^ { 2 }\)
   2. \(y = \frac { k } { x ^ { 2 } }\)
      showing the coordinates of any points where the graphs cross or meet the coordinate axes, leaving coordinates in terms of \(k\), where appropriate.
      (b) State, with a reason, the number of roots of the equation
   $$( 2 x - k ) ( x + 4 ) ^ { 2 } = \frac { k } { x ^ { 2 } }$$

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.

1. Using Figure 1, find the range of values of \(x\) for which $$\mathrm { f } ( x ) < 6$$
2. State the largest solution of the equation $$f ( 2 x ) = 6$$
3. 1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
   2. Hence find the set of values of \(x\) for which $$f ( - x ) \geqslant 6 \text { and } x < 0$$

8. \section*{Diagram NOT to scale} \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a design for a pond.
The design consists of a sector \(A O B X\) of a circle centre \(O\) joined to a quadrilateral \(A O B C\).

• \(B C = 8 \mathrm {~m}\)
• \(O A = O B = 3 \mathrm {~m}\)
• angle \(A O B\) is \(\frac { 2 \pi } { 3 }\) radians
• angle \(B C A\) is \(\frac { \pi } { 6 }\) radians
  1. Calculate (i) the exact area of the sector \(A O B X\),
     (ii) the exact perimeter of the sector \(A O B X\).
  2. Calculate the exact area of the triangle \(A O B\).
  3. Show that the length \(A B\) is \(3 \sqrt { 3 } \mathrm {~m}\).
  4. Find the total surface area of the pond. Give your answer in \(\mathrm { m } ^ { 2 }\) correct to 2 significant figures.

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$$

1. Write \(\frac { 1 } { 2 } x ^ { 2 } - 10 x + 22\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a , b\) and \(c\) are constants to be found. The point \(M\) is the minimum turning point of \(C\), as shown in Figure 3.
2. Deduce the coordinates of \(M\) The line \(l\) is the normal to \(C\) at the point \(P\), as shown in Figure 3.
   Given that \(l\) has equation \(y = k - \frac { 1 } { 8 } x\), where \(k\) is a constant,
3. 1. find the coordinates of \(P\)
   2. find the value of \(k\) Question 9 continues on the next page \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-25_903_682_299_605} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 is a copy of Figure 3. The finite region \(R\), shown shaded in Figure 4, is bounded by \(l , C\) and the line through \(M\) parallel to the \(y\)-axis.
4. Identify the inequalities that define \(R\).