Edexcel P1 (Pure Mathematics 1) 2019 January

1. Find

$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { 2 x ^ { 3 } } + 5 \right) d x$$ simplifying your answer.

1. Given

$$\frac { 3 ^ { x } } { 3 ^ { 4 y } } = 27 \sqrt { 3 }$$ find \(y\) as a simplified function of \(x\).

1. The line \(l _ { 1 }\) has equation \(3 x + 5 y - 7 = 0\)
   1. Find the gradient of \(l _ { 1 }\)
   The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the point \(( 6 , - 2 )\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

4. \begin{figure}[h] \end{figure} Figure 1 shows a line \(l _ { 1 }\) with equation \(2 y = x\) and a curve \(C\) with equation \(y = 2 x - \frac { 1 } { 8 } x ^ { 2 }\) The region \(R\), shown unshaded in Figure 1, is bounded by the line \(l _ { 1 }\), the curve \(C\) and a line \(l _ { 2 }\) Given that \(l _ { 2 }\) is parallel to the \(y\)-axis and passes through the intercept of \(C\) with the positive \(x\)-axis, identify the inequalities that define \(R\).

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

1. State the coordinates of \(P\). A copy of Figure 2, called Diagram 1, is shown at the top of the next page.
2. Sketch, on Diagram 1, the curve with equation \(y = \sin x\)
3. Hence, or otherwise, deduce the number of solutions of the equation
   1. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 20 \pi\)
   2. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 21 \pi\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-11_693_1050_301_447} \captionsetup{labelformat=empty} \caption{
      Diagram 1}\}
      \end{figure} \textbackslash section*\{Diagram 1

1. (Solutions based entirely on graphical or numerical methods are not acceptable.)

Given $$\mathrm { f } ( x ) = 2 x ^ { \frac { 5 } { 2 } } - 40 x + 8 \quad x > 0$$

1. solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\)
2. solve the equation \(\mathrm { f } ^ { \prime \prime } ( x ) = 5\)

7. \begin{figure}[h] \end{figure} Not to scale Figure 3 shows the design for a structure used to support a roof. The structure consists of four wooden beams, \(A B , B D , B C\) and \(A D\). Given \(A B = 6.5 \mathrm {~m} , B C = B D = 4.7 \mathrm {~m}\) and angle \(B A C = 35 ^ { \circ }\)

1. find, to one decimal place, the size of angle \(A C B\),
2. find, to the nearest metre, the total length of wood required to make this structure.

8. \begin{figure}[h] \end{figure} The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is shown in Figure 4. The curve \(C\)

• has a single turning point, a maximum at ( 4,9 )
• crosses the coordinate axes at only two places, \(( - 3,0 )\) and \(( 0,6 )\)
• has a single asymptote with equation \(y = 4\)
  as shown in Figure 4.
  1. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( - x )\).
  2. State the coordinates of the turning point on the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 4 } x \right)\).

Given that the line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at exactly one point,

state the possible values for \(k\). The curve \(C\) is transformed to a new curve that passes through the origin.

1. Given that the new curve has equation \(y = \mathrm { f } ( x ) - a\), state the value of the constant \(a\).
2. Write down an equation for another single transformation of \(C\) that also passes through the origin.

1. The equation

$$\frac { 3 } { x } + 5 = - 2 x + c$$ where \(c\) is a constant, has no real roots.
Find the range of possible values of \(c\).

1. A sector \(A O B\), of a circle centre \(O\), has radius \(r \mathrm {~cm}\) and angle \(\theta\) radians.

Given that the area of the sector is \(6 \mathrm {~cm} ^ { 2 }\) and that the perimeter of the sector is 10 cm ,

1. show that $$3 \theta ^ { 2 } - 13 \theta + 12 = 0$$
2. Hence find possible values of \(r\) and \(\theta\).
   □
   \includegraphics[max width=\textwidth, alt={}, center]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-21_131_19_2627_1882}

11. (a) On Diagram 1 sketch the graphs of

1. \(y = x ( 3 - x )\)
2. \(y = x ( x - 2 ) ( 5 - x )\)
   showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\) coordinates of the points of intersection of $$y = x ( 3 - x ) \text { and } y = x ( x - 2 ) ( 5 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 13 \right) = 0\) The point \(P\) lies on both curves. Given that \(P\) lies in the first quadrant,
   (c) find, using algebra and showing your working, the exact coordinates of \(P\).
   
   \includegraphics[max width=\textwidth, alt={}]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-23_824_1211_296_370}
   \section*{Diagram 1}

12. The curve with equation \(y = \mathrm { f } ( x ) , x > 0\), passes through the point \(P ( 4 , - 2 )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x \sqrt { x } - 10 x ^ { - \frac { 1 } { 2 } }$$

1. find the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found.
2. Find \(\mathrm { f } ( x )\).