Edexcel P1 (Pure Mathematics 1) 2023 January

1. A curve \(C\) has equation

$$y = 2 + 10 x ^ { \frac { 1 } { 2 } } - 2 x ^ { \frac { 3 } { 2 } } \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving your answer in simplest form.
2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(x = 2\) giving your answer in simplest form.
   (Solutions relying on calculator technology are not acceptable.)

1. The points \(P , Q\) and \(R\) have coordinates (-3, 7), (9, 11) and (12, 2) respectively.
   1. Prove that angle \(P Q R = 90 ^ { \circ }\)
   Given that the point \(S\) is such that \(P Q R S\) forms a rectangle,
2. find the coordinates of \(S\).

1. Find

$$\int \frac { 4 x ^ { 5 } + 3 } { 2 x ^ { 2 } } \mathrm {~d} x$$ giving your answer in simplest form.

1. Given that the equation
   \(k x ^ { 2 } + 6 k x + 5 = 0 \quad\) where \(k\) is a non zero constant has no real roots, find the range of possible values for \(k\).

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

Solutions relying on calculator technology are not acceptable.

1. By substituting \(p = 3 ^ { x }\), show that the equation $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$ can be rewritten in the form $$9 p ^ { 2 } + 26 p - 3 = 0$$
2. Hence solve $$3 \times 9 ^ { x } + 3 ^ { x + 2 } = 1 + 3 ^ { x - 1 }$$

6. \begin{figure}[h] \end{figure} Diagram NOT accurately drawn Figure 1 shows the plan view for the design of a stage.
The design consists of a sector \(O B C\) of a circle, with centre \(O\), joined to two congruent triangles \(O A B\) and \(O D C\). Given that

• angle \(B O C = 2.4\) radians
• area of sector \(B O C = 40 \mathrm {~m} ^ { 2 }\)
• \(A O D\) is a straight line of length 12.5 m
  1. find the radius of the sector, giving your answer, in m , to 2 decimal places,
  2. find the size of angle \(A O B\), in radians, to 2 decimal places.

Hence find

the total area of the stage, giving your answer, in \(\mathrm { m } ^ { 2 }\), to one decimal place,

the total perimeter of the stage, giving your answer, in m , to one decimal place.

1. (a) On Diagram 1, sketch a graph of the curve \(C\) with equation

$$y = \frac { 6 } { x } \quad x \neq 0$$ The curve \(C\) is transformed onto the curve with equation \(y = \frac { 6 } { x - 2 } \quad x \neq 2\)
(b) Fully describe this transformation. The curve with equation $$y = \frac { 6 } { x - 2 } \quad x \neq 2$$ and the line with equation $$y = k x + 7 \quad \text { where } k \text { is a constant }$$ intersect at exactly two points, \(P\) and \(Q\).
Given that the \(x\) coordinate of point \(P\) is - 4
(c) find the value of \(k\),
(d) find, using algebra, the coordinates of point \(Q\).
(Solutions relying entirely on calculator technology are not acceptable.) \section*{Diagram 1} Only use this copy of Diagram 1 if you need to redraw your graph.
\includegraphics[max width=\textwidth, alt={}, center]{bb21001f-fe68-4776-992d-ede1aae233d7-19_709_739_1802_664} Copy of Diagram 1
(Total for Question 7 is 10 marks)

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\) and the curve \(C\).
Given that \(l\) cuts the \(y\)-axis at - 12 and cuts the \(x\)-axis at 4 , as shown in Figure 2,

1. find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. Given that \(C\)
   • has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic expression
   • has a minimum point at \(( 7 , - 18 )\)
   • cuts the \(x\)-axis at 4 and at \(k\), where \(k\) is a constant
   • deduce the value of \(k\),
   • find \(\mathrm { f } ( x )\).
   The region \(R\) is shown shaded in Figure 2.
2. Use inequalities to define \(R\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of

• the curve with equation \(y = \tan x\)
• the straight line l with equation \(y = \pi x\)
  in the interval \(- \pi < x < \pi\)
  1. State the period of \(\tan x\)
  2. Write down the number of roots of the equation
     1. \(\tan x = ( \pi + 2 ) x\) in the interval \(- \pi < x < \pi\)
     2. \(\tan x = \pi x\) in the interval \(- 2 \pi < x < 2 \pi\)
     3. \(\tan x = \pi x\) in the interval \(- 100 \pi < x < 100 \pi\)

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )$$

1. Use the given information to state the values of \(x\) for which $$f ( x ) > 0$$
2. Expand \(( 3 x + 20 ) ( x + 6 ) ( 2 x - 3 )\), writing your answer as a polynomial in simplest form. The straight line \(l\) is the tangent to \(C\) at the point where \(C\) cuts the \(y\)-axis.
   Given that \(l\) cuts \(C\) at the point \(P\), as shown in Figure 4,
3. find, using algebra, the \(x\) coordinate of \(P\)
   (Solutions based on calculator technology are not acceptable.)

1. A curve \(C\) has equation \(y = \mathrm { f } ( x ) , \quad x > 0\)

Given that

• \(\mathrm { f } ^ { \prime \prime } ( x ) = 4 x + \frac { 1 } { \sqrt { x } }\)
• the point \(P\) has \(x\) coordinate 4 and lies on \(C\)
• the tangent to \(C\) at \(P\) has equation \(y = 3 x + 4\)
  1. find an equation of the normal to \(C\) at \(P\)
  2. find \(\mathrm { f } ( x )\), writing your answer in simplest form.