Edexcel P1 (Pure Mathematics 1) 2020 October

1. Given that

$$\left( 3 p q ^ { 2 } \right) ^ { 4 } \times 2 p \sqrt { q ^ { 8 } } \equiv a p ^ { b } q ^ { c }$$ find the values of the constants \(a , b\) and \(c\).

2. $$f ( x ) = 3 + 12 x - 2 x ^ { 2 }$$

1. Express \(\mathrm { f } ( x )\) in the form
   2. \(\mathrm { f } ( x ) = 3 + 12 x - 2 x ^ { 2 }\)
2. Express \(\mathrm { f } ( x )\) in the form $$\begin{aligned} & \qquad a - b ( x + c ) ^ { 2 }
   & \text { where } a , b \text { and } c \text { are integers to be found. }
   & \text { he curve with equation } y = \mathrm { f } ( x ) - 7 \text { crosses the } x \text {-axis at the points } P \text { and } Q \text { and crosses }
   & \text { te } y \text {-axis at the point } R \text {. }
   & \text { F) Find the area of the triangle } P Q R \text {, giving your answer in the form } m \sqrt { n } \text { where } m \text { and }
   & n \text { are integers to be found. } \end{aligned}$$ \(\_\_\_\_\) "

3. \begin{figure}[h] \end{figure} Figure 1 shows the design for a badge.
The design consists of two congruent triangles, \(A O C\) and \(B O C\), joined to a sector \(A O B\) of a circle centre \(O\).

• Angle \(A O B = \alpha\)
• \(A O = O B = 3 \mathrm {~cm}\)
• \(O C = 5 \mathrm {~cm}\)

Given that the area of sector \(A O B\) is \(7.2 \mathrm {~cm} ^ { 2 }\)

1. show that \(\alpha = 1.6\) radians.
2. Hence find
   1. the area of the badge, giving your answer in \(\mathrm { cm } ^ { 2 }\) to 2 significant figures,
   2. the perimeter of the badge, giving your answer in cm to one decimal place.

      VIXV SIHIANI III IM IONOO

      VIAV SIHI NI JYHAM ION OO

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4. Use algebra to solve the simultaneous equations $$\begin{array} { r } y - 3 x = 4
x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 4 \end{array}$$ You must show all stages of your working.

5. (i) \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the points \(( - 5,0 )\) and \(( 0 , - 3 )\) and touches the \(x\)-axis at the point \(( 2,0 )\). On separate diagrams sketch the curve with equation

1. \(y = \mathrm { f } ( x + 2 )\)
2. \(y = \mathrm { f } ( - x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.
   (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_415_814_1548_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve with equation $$y = k \cos \left( x + \frac { \pi } { 6 } \right) \quad 0 \leqslant x \leqslant 2 \pi$$ where \(k\) is a constant.
   The curve meets the \(y\)-axis at the point \(( 0 , \sqrt { 3 } )\) and passes through the points \(( p , 0 )\) and ( \(q , 0\) ). Find
3. the value of \(k\),
4. the exact value of \(p\) and the exact value of \(q\).

6. The point \(A\) has coordinates \(( - 4,11 )\) and the point \(B\) has coordinates \(( 8,2 )\).

1. Find the gradient of the line \(A B\), giving your answer as a fully simplified fraction. The point \(M\) is the midpoint of \(A B\). The line \(l\) passes through \(M\) and is perpendicular to \(A B\).
2. Find an equation for \(l\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found. The point \(C\) lies on \(l\) such that the area of triangle \(A B C\) is 37.5 square units.
3. Find the two possible pairs of coordinates of point \(C\).

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

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7. The curve \(C\) has equation $$y = \frac { 1 } { 2 - x }$$

1. Sketch the graph of \(C\). On your sketch you should show the coordinates of any points of intersection with the coordinate axes and state clearly the equations of any asymptotes. The line \(l\) has equation \(y = 4 x + k\), where \(k\) is a constant. Given that \(l\) meets \(C\) at two distinct points,
2. show that $$k ^ { 2 } + 16 k + 48 > 0$$
3. Hence find the range of possible values for \(k\).

8. The curve \(C\) has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line \(l _ { 1 }\) is the tangent to \(C\) at the point where \(x = 6\)
2. 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 line \(l _ { 2 }\) is the tangent to \(C\) at the point where \(x = \alpha\)
   Given that \(l _ { 1 }\) and \(l _ { 2 }\) are parallel and distinct,
3. find the value of \(\alpha\)
   

9. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 9,10 )\). Given that $$f ^ { \prime } ( x ) = 27 x ^ { 2 } - \frac { 21 x ^ { 3 } - 5 x } { 2 \sqrt { x } } \quad x > 0$$ find \(\mathrm { f } ( x )\), fully simplifying each term.