Edexcel P1 (Pure Mathematics 1) 2021 June

1. The curve \(C\) has equation

$$y = \frac { x ^ { 2 } } { 3 } + \frac { 4 } { \sqrt { x } } + \frac { 8 } { 3 x } - 5 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(P ( 4,3 )\) lies on \(C\).
2. Find the equation of the normal to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

2. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = a x ^ { 3 } + ( 6 a + 8 ) x ^ { 2 } - a ^ { 2 } x$$ where \(a\) is a positive constant. Given \(\mathrm { f } ( - 1 ) = 32\)

1. 1. show that the only possible value for \(a\) is 3
   2. Using \(a = 3\) solve the equation $$\mathrm { f } ( x ) = 0$$
2. Hence find all real solutions of
   1. \(3 y + 26 y ^ { \frac { 2 } { 3 } } - 9 y ^ { \frac { 1 } { 3 } } = 0\)
   2. \(3 \left( 9 ^ { 3 z } \right) + 26 \left( 9 ^ { 2 z } \right) - 9 \left( 9 ^ { z } \right) = 0\)

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

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows the plan view of a flower bed.
The flowerbed is in the shape of a triangle \(A B C\) with

• \(A B = p\) metres
• \(A C = q\) metres
• \(B C = 2 \sqrt { 2 }\) metres
• angle \(B A C = 60 ^ { \circ }\)
  1. Show that

$$p ^ { 2 } + q ^ { 2 } - p q = 8$$ Given that side \(A C\) is 2 metres longer than side \(A B\), use algebra to find

1. the exact value of \(p\),
2. the exact value of \(q\). Using the answers to part (b),

calculate the exact area of the flower bed.

4. Find $$\int \frac { ( 3 \sqrt { x } + 2 ) ( x - 5 ) } { 4 \sqrt { x } } d x$$ writing each term in simplest form.

5. \begin{figure}[h] \end{figure} The share value of two companies, company \(A\) and company \(B\), has been monitored over a 15-year period. The share value \(P _ { A }\) of company \(\boldsymbol { A }\), in millions of pounds, is modelled by the equation $$P _ { A } = 53 - 0.4 ( t - 8 ) ^ { 2 } \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. The share value \(P _ { B }\) of company \(B\), in millions of pounds, is modelled by the equation $$P _ { B } = - 1.6 t + 44.2 \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. Figure 2 shows a graph of both models. Use the equations of one or both models to answer parts (a) to (d).

1. Find the difference between the share value of company \(\boldsymbol { A }\) and the share value of company \(\boldsymbol { B }\) at the point monitoring began.
2. State the maximum share value of company \(\boldsymbol { A }\) during the 15-year period.
3. Find, using algebra and showing your working, the times during this 15-year period when the share value of company \(\boldsymbol { A }\) was greater than the share value of company \(\boldsymbol { B }\).
4. Explain why the model for company \(\boldsymbol { A }\) should not be used to predict its share value when \(t = 20\)
   \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-17_2644_1838_121_116}

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

• \(C\) passes through the point \(P ( 8,2 )\)
• \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 3 x ^ { 2 } } + 3 - 2 ( \sqrt [ 3 ] { x } )\)
  1. find the equation of the tangent to \(C\) at \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
     (3)
  2. Find, in simplest form, \(\mathrm { f } ( x )\).
     \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-21_2647_1840_118_111}

7. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(4 y + 3 x = 48\)
The line \(l _ { 1 }\) cuts the \(y\)-axis at the point \(C\), as shown in Figure 3.

1. State the \(y\) coordinate of \(C\). The point \(D ( 8,6 )\) lies on \(l _ { 1 }\)
   The line \(l _ { 2 }\) passes through \(D\) and is perpendicular to \(l _ { 1 }\) The line \(l _ { 2 }\) cuts the \(y\)-axis at the point \(E\) as shown in Figure 3.
2. Show that the \(y\) coordinate of \(E\) is \(- \frac { 14 } { 3 }\) A sector \(B C E\) of a circle with centre \(C\) is also shown in Figure 3. Given that angle \(B C E\) is 1.8 radians,
3. find the length of arc \(B E\). The region \(C B E D\), shown shaded in Figure 3, consists of the sector \(B C E\) joined to the triangle \(C D E\).
4. Calculate the exact area of the region \(C B E D\).

8. The curve \(C _ { 1 }\) has equation $$y = 3 x ^ { 2 } + 6 x + 9$$

1. Write \(3 x ^ { 2 } + 6 x + 9\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The point \(P\) is the minimum point of \(C _ { 1 }\)
2. Deduce the coordinates of \(P\). A different curve \(C _ { 2 }\) has equation $$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$ where \(A\), \(B\), \(C\) and \(D\) are constants. Given that \(C _ { 2 }\)
   • passes through \(P\)
   • intersects the \(x\)-axis at \(- 4 , - 2\) and 3
   • find, making your method clear, the values of \(A , B , C\) and \(D\).
     \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-27_2644_1840_118_111}
   \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-29_2646_1838_121_116}

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$y = \tan x \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ The line \(l\), shown in Figure 4, is an asymptote to \(y = \tan x\)

1. State an equation for \(l\). A copy of Figure 4, labelled Diagram 1, is shown on the next page.
2. 1. On Diagram 1, sketch the curve with equation $$y = \frac { 1 } { x } + 1 \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ stating the equation of the horizontal asymptote of this curve.
   2. Hence, giving a reason, state the number of solutions of the equation
3. State the number of solutions of the equation \(\tan x = \frac { 1 } { x } + 1\) in the region
   1. \(0 \leqslant x \leqslant 40 \pi\)
   2. \(- 10 \pi \leqslant x \leqslant \frac { 5 } { 2 } \pi\) $$\begin{aligned} & \qquad \tan x = \frac { 1 } { x } + 1
      & \text { in the region } - 2 \pi \leqslant x \leqslant 2 \pi \end{aligned}$$" \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-31_725_1047_1078_447} \captionsetup{labelformat=empty} \caption{Diagram 1}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-32_2644_1837_118_114}