Edexcel P1 (Pure Mathematics 1) 2022 June

1. Find

$$\int \left( 10 x ^ { 5 } + 6 x ^ { 3 } - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x$$ giving your answer in simplest form.

2. In the triangle \(A B C\),

• \(A B = 21 \mathrm {~cm}\)
• \(B C = 13 \mathrm {~cm}\)
• angle \(B A C = 25 ^ { \circ }\)
• angle \(A C B = x ^ { \circ }\)
  1. Use the sine rule to find the value of \(\sin x ^ { \circ }\), giving your answer to 4 decimal places.

Given also that \(A B\) is the longest side of the triangle,

find the value of \(x\), giving your answer to 2 decimal places.

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

1. Show that \(\frac { \sqrt { 180 } - \sqrt { 80 } } { \sqrt { 5 } }\) is an integer and find its value.
2. Simplify $$\frac { 4 \sqrt { 5 } - 5 } { 7 - 3 \sqrt { 5 } }$$ giving your answer in the form \(a + b \sqrt { 5 }\) where \(a\) and \(b\) are rational numbers.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\)
The curve has a minimum at \(P ( - 1,0 )\) and a maximum at \(Q \left( \frac { 3 } { 2 } , 2 \right)\)
The line with equation \(y = 1\) is the only asymptote to the curve. On separate diagrams sketch the curves with equation

1. \(y = \mathrm { f } ( x ) - 2\)
2. \(y = \mathrm { f } ( - x )\) On each sketch you must clearly state
   • the coordinates of the maximum and minimum points
   • the equation of the asymptote

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

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• the maximum turning point on \(C\) has coordinates \(( - 2,12 )\)
• \(C\) cuts the negative \(x\)-axis at - 5
  1. find \(\mathrm { f } ( x )\)

The line \(l _ { 1 }\) has equation \(y = \frac { 4 } { 5 } x\) Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(( - 5,0 )\)

find an equation for \(l _ { 2 }\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
(3) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-10_983_712_1126_616} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve \(C\) and the lines \(l _ { 1 }\) and \(l _ { 2 }\)

Define the region \(R\), shown shaded in Figure 2, using inequalities.

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

1. Given that $$2 x y - 3 x ^ { 2 } = 50$$ and $$y - x ^ { 3 } + 6 x = 0$$ show that $$2 x ^ { 4 } - 15 x ^ { 2 } - 50 = 0$$
2. Hence solve the simultaneous equations $$\begin{aligned} 2 x y - 3 x ^ { 2 } & = 50
   y - x ^ { 3 } + 6 x & = 0 \end{aligned}$$ Give your answers in fully simplified surd form.
   \includegraphics[max width=\textwidth, alt={}, center]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-14_2257_52_312_1982}

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

• \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \sqrt { x } } + \frac { A } { x ^ { 2 } } + 3\), where \(A\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) when \(x = 4\)
  1. find the value of \(A\).

Given also that

• \(\mathrm { f } ( x ) = 8 \sqrt { 3 }\), when \(x = 12\)
• find \(\mathrm { f } ( x )\), giving each term in simplest form.
  

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the outline of the face of a ceiling fan viewed from below.
The fan consists of three identical sections congruent to \(O A B C D O\), shown in Figure 3, where

• \(O A B O\) is a sector of a circle with centre \(O\) and radius 9 cm
• \(O B C D O\) is a sector of a circle with centre \(O\) and radius 84 cm
• angle \(A O D = \frac { 2 \pi } { 3 }\) radians

Given that the length of the arc \(A B\) is 15 cm ,

1. show that the length of the arc \(C D\) is 35.9 cm to one decimal place. The face of the fan is modelled to be a flat surface.
   Find, according to the model,
2. the perimeter of the face of the fan, giving your answer to the nearest cm,
3. the surface area of the face of the fan. Give your answer to 3 significant figures and make your units clear.

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the graph of the curve with equation \(y = \sin x\) Given that \(\sin \alpha = p\), where \(0 < \alpha < 90 ^ { \circ }\)

1. state, in terms of \(p\), the value of
   1. \(2 \sin \left( 180 ^ { \circ } - \alpha \right)\)
   2. \(\sin \left( \alpha - 180 ^ { \circ } \right)\)
   3. \(3 + \sin \left( 180 ^ { \circ } + \alpha \right)\) A copy of Figure 4, labelled Diagram 1, is shown on page 27. On Diagram 1,
2. sketch the graph of \(y = \sin 2 x\)
3. Hence find, in terms of \(\alpha\), the \(x\) coordinates of any points in the interval \(0 < x < 180 ^ { \circ }\) where $$\sin 2 x = p$$
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-27_433_1331_296_310}
   \section*{Diagram 1}

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation $$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The line \(l\), shown in Figure 5, is the normal to \(C\) at the point \(A\) with \(x\) coordinate \(- \frac { 7 } { 2 }\) Given that \(l\) is also a tangent to \(C\) at the point \(B\),
2. show that the \(x\) coordinate of the point \(B\) is a solution of the equation $$12 x ^ { 2 } + 4 x - 33 = 0$$
3. Hence find the \(x\) coordinate of \(B\), justifying your answer. Given that the \(y\) intercept of \(l\) is - 1
4. find the value of \(k\).
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}