Edexcel P1 (Pure Mathematics 1) 2022 January

1. Find

$$\int \left( \frac { 8 x ^ { 3 } } { 5 } - \frac { 2 } { 3 x ^ { 4 } } - 1 \right) d x$$ giving each term in simplest form.

2. $$f ( x ) = 11 - 4 x - 2 x ^ { 2 }$$

1. Express \(\mathrm { f } ( x )\) in the form $$a + b ( x + c ) ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.
2. Sketch the graph of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), showing clearly the coordinates of the point where the curve crosses the \(y\)-axis.
3. Write down the equation of the line of symmetry of \(C\).

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

Solutions relying on calculator technology are not acceptable.

1. $$f ( x ) = ( x + \sqrt { 2 } ) ^ { 2 } + ( 3 x - 5 \sqrt { 8 } ) ^ { 2 }$$ Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 2 } + b x \sqrt { 2 } + c\) where \(a , b\) and \(c\) are integers to be found.
2. Solve the equation $$\sqrt { 3 } ( 4 y - 3 \sqrt { 3 } ) = 5 y + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are simplified fractions to be found.

4. \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.} Figure 1 shows a line \(l\) with equation \(x + y = 6\) and a curve \(C\) with equation \(y = 6 x - 2 x ^ { 2 } + 1\) The line \(l\) intersects the curve \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.

1. Find, using algebra, the coordinates of \(P\) and the coordinates of \(Q\). The region \(R\), shown shaded in Figure 1, is bounded by \(C , l\) and the \(x\)-axis.
2. Use inequalities to define the region \(R\).
   

   VIIV SIHI NI IIIIM IONOO

   VIIIV SIHI NI JIIIM I ON OO

   VIAV SIHI NI III HM ION OC

5. \begin{figure}[h] \end{figure} Figure 2 shows a plan view of a semicircular garden \(A B C D E O A\) The semicircle has

• centre \(O\)
• diameter \(A O E\)
• radius 3 m

The straight line \(B D\) is parallel to \(A E\) and angle \(B O A\) is 0.7 radians.

1. Show that, to 4 significant figures, angle \(B O D\) is 1.742 radians. The flowerbed \(R\), shown shaded in Figure 2, is bounded by \(B D\) and the arc \(B C D\).
2. Find the area of the flowerbed, giving your answer in square metres to one decimal place.
3. Find the perimeter of the flowerbed, giving your answer in metres to one decimal place.

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

• \(\mathrm { f } ^ { \prime } ( x ) = \frac { ( x + 3 ) ^ { 2 } } { x \sqrt { x } }\)
• the point \(P ( 4,20 )\) lies on \(C\)
  1. (i) find the value of the gradient at \(P\)
     (ii) Hence find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers to be found.
  2. Find \(\mathrm { f } ( x )\), simplifying your answer.
     

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 4 ) ( x - 2 ) ( 2 x - 9 )$$ Given that the curve with equation \(y = \mathrm { f } ( x ) - p\) passes through the point with coordinates \(( 0,50 )\)

1. find the value of the constant \(p\). Given that the curve with equation \(y = \mathrm { f } ( x + q )\) passes through the origin,
2. write down the possible values of the constant \(q\).
3. Find \(\mathrm { f } ^ { \prime } ( x )\).
4. Hence find the range of values of \(x\) for which the gradient of the curve with equation \(y = \mathrm { f } ( x )\) is less than - 18
   \includegraphics[max width=\textwidth, alt={}, center]{6c320b71-8793-461a-a078-e4f64c144a3a-23_68_37_2617_1914}

8. The line \(l _ { 1 }\) has equation $$2 x - 5 y + 7 = 0$$

1. Find the gradient of \(l _ { 1 }\) Given that
   • the point \(A\) has coordinates \(( 6 , - 2 )\)
   • the line \(l _ { 2 }\) passes through \(A\) and is perpendicular to \(l _ { 1 }\)
   • find the equation of \(l _ { 2 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(M\).
2. Using algebra and showing all your working, find the coordinates of \(M\).
   (Solutions relying on calculator technology are not acceptable.) Given that the diagonals of a square \(A B C D\) meet at \(M\),
3. find the coordinates of the point \(C\).

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation $$y = A \cos ( x - 30 ) ^ { \circ }$$ where \(A\) is a constant. The point \(P\) is a minimum point on the curve and has coordinates \(( 30 , - 3 )\) as shown in Figure 4.

1. Write down the value of \(A\). The point \(Q\) is shown in Figure 4 and is a maximum point.
2. Find the coordinates of \(Q\).

10. The curve \(C\) has equation $$y = \frac { 1 } { x ^ { 2 } } - 9$$

1. Sketch the graph of \(C\). On your sketch
   • show the coordinates of any points of intersection with the coordinate axes
   • state clearly the equations of any asymptotes
   The curve \(D\) has equation \(y = k x ^ { 2 }\) where \(k\) is a constant. Given that \(C\) meets \(D\) at 4 distinct points,
2. find the range of possible values for \(k\).