Edexcel P1 (Pure Mathematics 1) 2024 January

1. Find

$$\int ( 2 x - 5 ) ( 3 x + 2 ) ( 2 x + 5 ) \mathrm { d } x$$ writing your answer in simplest form.

1. The triangle \(A B C\) is such that

• \(A B = 15 \mathrm {~cm}\)
• \(A C = 25 \mathrm {~cm}\)
• angle \(B A C = \theta ^ { \circ }\)
• area triangle \(A B C = 100 \mathrm {~cm} ^ { 2 }\)
  1. Find the value of \(\sin \theta ^ { \circ }\)

Given that \(\theta > 90\)

find the length of \(B C\), in cm , to 3 significant figures.

1. The curve \(C\) has equation

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) writing your answer in simplest form. The point \(P ( 2,4 )\) lies on \(C\).
2. Find an equation for the tangent to \(C\) at \(P\) writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 2 ^ { x }\), show that the equation $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ can be written in the form $$4 p ^ { 2 } - 33 p + 8 = 0$$
2. Hence solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$

5. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The straight line \(l _ { 1 }\), shown in Figure 1, passes through the points \(P ( - 2,9 )\) and \(Q ( 10,6 )\).

1. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The straight line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\)
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\) as shown in Figure 1.
2. Find the coordinates of \(R\)
3. Find the exact area of triangle \(O P Q\).

6. \begin{figure}[h] \end{figure} Figure 2 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = 5 \cos x$$ with \(x\) being measured in degrees.
The point \(P\), shown in Figure 2, is a minimum point on \(C _ { 1 }\)

1. State the coordinates of \(P\) The point \(Q\) lies on a different curve \(C _ { 2 }\)
   Given that point \(Q\)
   • is a maximum point on the curve
   • is the maximum point with the smallest \(x\) coordinate, \(x > 0\)
   • find the coordinates of \(Q\) when
     1. \(C _ { 2 }\) has equation \(y = 5 \cos x - 2\)
     2. \(C _ { 2 }\) has equation \(y = - 5 \cos x\)

1. (a) Sketch the graph of the curve \(C\) with equation

$$y = \frac { 4 } { x - k }$$ where \(k\) is a positive constant.
Show on your sketch

• the coordinates of any points where \(C\) cuts the coordinate axes
• the equation of the vertical asymptote to \(C\)

Given that the straight line with equation \(y = 9 - x\) does not cross or touch \(C\)
(b) find the range of values of \(k\).

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the plan view of a platform.
The plan view of the platform consists of a sector \(D O C\) of a circle centre \(O\) joined to a sector \(A O B E A\) of a different circle, also with centre \(O\). Given that

• angle \(A O B = 0.8\) radians
• arc length \(C D = 9 \mathrm {~m}\)
• \(D A : A O = 3 : 5\)
  1. show that \(A O = 7.03 \mathrm {~m}\) to 3 significant figures.
  2. Find the perimeter of the platform, in m , to 3 significant figures.
  3. Find the total area of the platform, giving your answer in \(\mathrm { m } ^ { 2 }\) to the nearest whole number.

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

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
• \(C _ { 1 }\) passes through the origin
  1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
  2. find an expression for \(\mathrm { f } ( x )\).

The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\)
Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\)
Given that the \(x\) coordinate of the point \(P\) is negative,

using algebra and showing all stages of your working, find the coordinates of \(P\)

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

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

• the point \(P ( 2,8 \sqrt { 2 } )\) lies on \(C\)
• \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + \frac { k } { x ^ { 2 } }\) where \(k\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(P\)
  1. find the exact value of \(k\),
  2. find \(\mathrm { f } ( x )\), giving your answer in simplest form.