Edexcel P1 (Pure Mathematics 1) 2024 June

1. Find

$$\int \left( 10 x ^ { 4 } - \frac { 3 } { 2 x ^ { 3 } } - 7 \right) \mathrm { d } x$$ giving each term in simplest form.

1. 1. Given that \(m = 2 ^ { n }\), express each of the following in simplest form in terms of \(m\).
      1. \(2 ^ { n + 3 }\)
   2. \(16 ^ { 3 n }\)
      (ii) In this question you must show all stages of your working.
   Solutions relying on calculator technology are not acceptable. Solve the equation $$x \sqrt { 3 } - 3 = x + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are integers.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the points \(( - 1,0 )\) and \(( 0,2 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( \mathrm { x } + 3 )\)
2. \(y = \mathrm { f } ( - 3 x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.

1. The curve \(C _ { 1 }\) has equation

$$y = x ^ { 2 } + k x - 9$$ and the curve \(C _ { 2 }\) has equation $$y = - 3 x ^ { 2 } - 5 x + k$$ where \(k\) is a constant.
Given that \(C _ { 1 }\) and \(C _ { 2 }\) meet at a single point \(P\)

1. show that $$k ^ { 2 } + 26 k + 169 = 0$$
2. Hence find the coordinates of \(P\)

5. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\)
Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)

1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

6. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.} Figure 3 shows

• the line \(l\) with equation \(y - 5 x = 75\)
• the curve \(C\) with equation \(y = 2 x ^ { 2 } + x - 21\)

The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\), as shown in Figure 3 .

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).

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

$$f ( x ) = 2 x ^ { 3 } - k x ^ { 2 } + 14 x + 24$$ and \(k\) is a constant.

1. Find, in simplest form,
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { f } ^ { \prime \prime } ( x )\) The curve with equation \(y = \mathrm { f } ^ { \prime } ( x )\) intersects the curve with equation \(y = \mathrm { f } ^ { \prime \prime } ( x )\) at the points \(A\) and \(B\). Given that the \(x\) coordinate of \(A\) is 5
2. find the value of \(k\).
3. Hence find the coordinates of \(B\).

1. The curve \(C _ { 1 }\) has equation

$$y = x \left( 4 - x ^ { 2 } \right)$$

1. Sketch the graph of \(C _ { 1 }\) showing the coordinates of any points of intersection with the coordinate axes. The curve \(C _ { 2 }\) has equation \(y = \frac { A } { x }\) where \(A\) is a constant.
2. Show that the \(x\) coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) satisfy the equation $$x ^ { 4 } - 4 x ^ { 2 } + A = 0$$
3. Hence find the range of possible values of \(A\) for which \(C _ { 1 }\) meets \(C _ { 2 }\) at 4 distinct points.

1. Given that

• the point \(A\) has coordinates \(( 4,2 )\)
• the point \(B\) has coordinates \(( 15,7 )\)
• the line \(l _ { 1 }\) passes through \(A\) and \(B\)
  1. find an equation for \(l _ { 1 }\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found.

The line \(l _ { 2 }\) passes through \(A\) and is parallel to the \(x\)-axis.
The point \(C\) lies on \(l _ { 2 }\) so that the length of \(B C\) is \(5 \sqrt { 5 }\)

Find both possible pairs of coordinates of the point \(C\).

Hence find the minimum possible area of triangle \(A B C\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\)

Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 6 x - \frac { ( 2 x - 1 ) ( 3 x + 2 ) } { 2 \sqrt { x } }\)
• the point \(P ( 4,12 )\) lies on \(C\)
  1. find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found,
  2. find \(\mathrm { f } ( x )\), giving each term in simplest form.

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 12 \sin x$$ where \(x\) is measured in radians.
The point \(P\) shown in Figure 4 is a maximum point on \(C _ { 1 }\)

1. Find the coordinates of \(P\). The curve \(C _ { 2 }\) has equation $$y = 12 \sin x + k$$ where \(k\) is a constant.
   Given that the maximum value of \(y\) on \(C _ { 2 }\) is 3
2. find the coordinates of the minimum point on \(C _ { 2 }\) which has the smallest positive \(x\) coordinate. The curve \(C _ { 3 }\) has equation $$y = 12 \sin ( x + B )$$ where \(B\) is a positive constant.
   Given that \(\left( \frac { \pi } { 4 } , A \right)\), where \(A\) is a constant, is the minimum point on \(C _ { 3 }\) which has the smallest positive \(x\) coordinate,
3. find
   1. the value of \(A\),
   2. the smallest possible value of \(B\).