Edexcel P1 (Pure Mathematics 1) 2023 June

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

Solve the inequality $$4 x ^ { 2 } - 3 x + 7 \geq 4 x + 9$$

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

A rectangular sports pitch has length \(x\) metres and width \(y\) metres, where \(x > y\) Given that the perimeter of the pitch is 350 m ,

1. write down an equation linking \(x\) and \(y\) Given also that the area of the pitch is \(7350 \mathrm {~m} ^ { 2 }\)
2. write down a second equation linking \(x\) and \(y\)
3. hence find the value of \(x\) and the value of \(y\)

1. (a) Express \(3 x ^ { 2 } + 12 x + 13\) in the form

$$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
(b) Hence sketch the curve with equation \(y = 3 x ^ { 2 } + 12 x + 13\) On your sketch show clearly

• the coordinates of the \(y\) intercept
• the coordinates of the turning point of the curve

1. In this question you must show all stages of your working.
   1. Write
   $$y = \frac { 5 x ^ { 2 } + \sqrt { x ^ { 3 } } } { \sqrt [ 3 ] { 8 x } }$$ in the form $$y = A x ^ { p } + B x ^ { q }$$ where \(A , B , p\) and \(q\) are constants to be found.
2. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each coefficient in simplest form.

5. \begin{figure}[h] \end{figure} Figure 1 shows the plan for a garden.
In the plan

• \(O A\) and \(C D\) are perpendicular to \(O D\)
• \(A B\) is an arc of the circle with centre \(O\) and radius 4 metres
• \(\quad B C\) is parallel to \(O D\)
• \(O D\) is 6 metres, \(O A\) is 4 metres and \(C D\) is 1.5 metres
  1. Show that angle \(A O B\) is 1.186 radians to 4 significant figures.
  2. Find the perimeter of the garden, giving your answer in metres to 3 significant figures.
  3. Find the area of the garden, giving your answer in square metres to 3 significant figures.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
   1. Expand and simplify
   $$\left( r - \frac { 1 } { r } \right) ^ { 2 }$$
2. Express \(\frac { 1 } { 3 + 2 \sqrt { 2 } }\) in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are integers.
3. Use the results of parts (a) and (b), or otherwise, to show that $$\sqrt { 3 + 2 \sqrt { 2 } } - \frac { 1 } { \sqrt { 3 + 2 \sqrt { 2 } } } = 2$$

7. \begin{figure}[h] \end{figure} The region \(R _ { 1 }\), shown shaded in Figure 2, is defined by the inequalities $$0 \leqslant y \leqslant 2 \quad y \leqslant 10 - 2 x \quad y \leqslant k x$$ where \(k\) is a constant.
The line \(x = a\), where \(a\) is a constant, passes through the intersection of the lines \(y = 2\) and \(y = k x\)
Given that the area of \(R _ { 1 }\) is \(\frac { 27 } { 4 }\) square units,

1. find
   1. the value of \(a\)
   2. the value of \(k\)
2. Define the region \(R _ { 2 }\), also shown shaded in Figure 2, using inequalities.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Find the equation of the tangent to the curve with equation $$y = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$ at the point \(P ( 4,12 )\)
   Give your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers. The curve with equation \(y = \mathrm { f } ( x )\) also passes through the point \(P ( 4,12 )\)
   Given that $$f ^ { \prime } ( x ) = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$
2. find \(\mathrm { f } ( x )\) giving the coefficients in simplest form.

\begin{figure}[h] \end{figure} Figure 3 shows part of the graph of the trigonometric function with equation \(y = \mathrm { f } ( x )\)

1. Write down an expression for \(\mathrm { f } ( x )\) On a separate diagram,
2. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { f } \left( x + \frac { \pi } { 4 } \right)\) Show clearly the coordinates of all the points where the curve intersects the coordinate axes.
   (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_378_1251_1617_408} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows part of the graph of the trigonometric function with equation \(y = \mathrm { g } ( x )\)
3. Write down an expression for \(\mathrm { g } ( x )\) On a separate diagram,
4. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { g } ( x ) - 2\) Show clearly the coordinates of the \(y\) intercept.

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the quadratic curve \(C\) with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line \(l _ { 1 }\) also shown in Figure 5,

• has gradient \(\frac { 1 } { 2 }\)
• intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
  1. (i) Write down an equation for \(l _ { 1 }\)
     (ii) Find, in terms of \(b\), the coordinates of \(P\)

Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and intersects \(C\) on the positive \(x\)-axis,

find, in terms of \(b\), an equation for \(l _ { 2 }\) Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)

show that another equation for \(l _ { 2 }\) is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$

Hence, or otherwise, find the value of \(b\)