Edexcel P1 (Pure Mathematics 1) 2021 October

1. Find

$$\int 12 x ^ { 3 } + \frac { 1 } { 6 \sqrt { x } } - \frac { 3 } { 2 x ^ { 4 } } \mathrm {~d} x$$ giving each term in simplest form.

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).

3. (i) Solve
(ii) $$\frac { 3 } { x } > 4$$ Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only. Given that

• \(\quad l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(\quad C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define the region \(R\).

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only.
Given that

• \(l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define the region \(R\).
  

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \cos 2 x ^ { \circ } \quad 0 \leqslant x \leqslant k$$ The point \(Q\) and the point \(R ( k , 0 )\) lie on the curve and are shown in Figure 2.

1. State
   1. the coordinates of \(Q\),
   2. the value of \(k\).
2. Given that there are exactly two solutions to the equation $$\cos 2 x ^ { \circ } = p \quad \text { in the region } 0 \leqslant x \leqslant k$$ find the range of possible values for \(p\).

5. The line \(l _ { 1 }\) has equation \(3 y - 2 x = 30\) The line \(l _ { 2 }\) passes through the point \(A ( 24,0 )\) and is perpendicular to \(l _ { 1 }\)
Lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)

1. Find, using algebra and showing your working, the coordinates of \(P\). Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(B\),
2. find the area of triangle \(B P A\).

6. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 ( x + 1 ) ( x - 3 ) ^ { 2 }$$

1. Sketch a graph of \(C\). Show on your graph the coordinates of the points where \(C\) cuts or meets the coordinate axes.
2. Write \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\), where \(a , b , c\) and \(d\) are constants to be found.
3. Hence, find the equation of the tangent to \(C\) at the point where \(x = \frac { 1 } { 3 }\)

7. \begin{figure}[h] \end{figure} Figure 3 shows the design for a sign at a bird sanctuary.
The design consists of a kite \(O A B C\) joined to a sector \(O C X A\) of a circle centre \(O\).
In the design

• \(O A = O C = 0.6 \mathrm {~m}\)
• \(A B = C B = 1.4 \mathrm {~m}\)
• Angle \(O A B =\) Angle \(O C B = 2\) radians
• Angle \(A O C = \theta\) radians, as shown in Figure 3

Making your method clear,

1. show that \(\theta = 1.64\) radians to 3 significant figures,
2. find the perimeter of the sign, in metres to 2 significant figures,
3. find the area of the sign, in \(\mathrm { m } ^ { 2 }\) to 2 significant figures.

8. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation $$y = 4 + 12 x - 3 x ^ { 2 }$$ The point \(M\) is the maximum turning point on \(C\).

1. 1. Write \(4 + 12 x - 3 x ^ { 2 }\) in the form $$a + b ( x + c ) ^ { 2 }$$ where \(a , b\) and \(c\) are constants to be found.
   2. Hence, or otherwise, state the coordinates of \(M\). The line \(l _ { 1 }\) passes through \(O\) and \(M\), as shown in Figure 4.
      A line \(l _ { 2 }\) touches \(C\) and is parallel to \(l _ { 1 }\)
2. Find an equation for \(l _ { 2 }\)

9. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \sqrt { x } \quad x > 0$$ The point \(P ( 9,3 )\) lies on the curve and is shown in Figure 5.
On the next page there is a copy of Figure 5 called Diagram 1.

1. On Diagram 1, sketch and clearly label the graphs of $$y = \mathrm { f } ( 2 x ) \text { and } y = \mathrm { f } ( x ) + 3$$ Show on each graph the coordinates of the point to which \(P\) is transformed. The graph of \(y = \mathrm { f } ( 2 x )\) meets the graph of \(y = \mathrm { f } ( x ) + 3\) at the point \(Q\).
2. Show that the \(x\) coordinate of \(Q\) is the solution of $$\sqrt { x } = 3 ( \sqrt { 2 } + 1 )$$
3. Hence find, in simplest form, the coordinates of \(Q\).
   
   \includegraphics[max width=\textwidth, alt={}]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-27_599_720_274_660}
   \section*{Diagram 1} Turn over for a copy of Diagram 1 if you need to redraw your graphs. Only use this copy if you need to redraw your graphs.
   \includegraphics[max width=\textwidth, alt={}, center]{f1e1d4f5-dd27-4839-a6f3-f6906666302c-29_600_718_1991_660} Copy of Diagram 1

10. A curve has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = a x - 12 x ^ { \frac { 1 } { 3 } }\), where \(a\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) when \(x = 27\)
• the curve passes through the point \(( 1 , - 8 )\)
  1. find the value of \(a\).
  2. Hence find \(\mathrm { f } ( x )\).