Edexcel P1 (Pure Mathematics 1) 2020 January

1. Find, in simplest form,

$$\int \left( \frac { 8 x ^ { 3 } } { 3 } - \frac { 1 } { 2 \sqrt { x } } - 5 \right) \mathrm { d } x$$

2. Given \(y = 3 ^ { x }\), express each of the following in terms of \(y\). Write each expression in its simplest form.

1. \(3 ^ { 3 x }\)
2. \(\frac { 1 } { 3 ^ { x - 2 } }\)
3. \(\frac { 81 } { 9 ^ { 2 - 3 x } }\)

3. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = x ^ { 2 } + 3 x - 2\) The point \(P ( 3,16 )\) lies on the curve.

1. Find the gradient of the tangent to the curve at \(P\). The point \(Q\) with \(x\) coordinate \(3 + h\) also lies on the curve.
2. Find, in terms of \(h\), the gradient of the line \(P Q\). Write your answer in simplest form.
3. Explain briefly the relationship between the answer to (b) and the answer to (a).

4. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a house \(A B C D\) and a lawn \(A P C D A\).
\(A B C D\) is a rectangle with \(A B = 16 \mathrm {~m}\).
\(A P C O A\) is a sector of a circle centre \(O\) with radius 12 m . The point \(O\) lies on the line \(D C\), as shown in Figure 2.

1. Show that the size of angle \(A O D\) is 1.231 radians to 3 decimal places. The lawn \(A P C D A\) is shown shaded in Figure 2.
2. Find the area of the lawn, in \(\mathrm { m } ^ { 2 }\), to one decimal place.
3. Find the perimeter of the lawn, in metres, to one decimal place.

5. (a) Find, using algebra, all solutions of $$20 x ^ { 3 } - 50 x ^ { 2 } - 30 x = 0$$ (b) Hence find all real solutions of $$20 ( y + 3 ) ^ { \frac { 3 } { 2 } } - 50 ( y + 3 ) - 30 ( y + 3 ) ^ { \frac { 1 } { 2 } } = 0$$

6. The line \(l _ { 1 }\) has equation \(3 x - 4 y + 20 = 0\) The line \(l _ { 2 }\) cuts the \(x\)-axis at \(R ( 8,0 )\) and is parallel to \(l _ { 1 }\)

1. Find the equation of \(l _ { 2 }\), writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. The line \(l _ { 1 }\) cuts the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\).
   Given that \(P Q R S\) is a parallelogram, find
2. the area of \(P Q R S\),
3. the coordinates of \(S\).

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C _ { 1 }\) with equation \(y = 3 \sin x\), where \(x\) is measured in degrees. The point \(P\) and the point \(Q\) lie on \(C _ { 1 }\) and are shown in Figure 3.

1. State
   1. the coordinates of \(P\),
   2. the coordinates of \(Q\). A different curve \(C _ { 2 }\) has equation \(y = 3 \sin x + k\), where \(k\) is a constant.
      The curve \(C _ { 2 }\) has a maximum \(y\) value of 10
      The point \(R\) is the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate.
2. State the coordinates of \(R\). Figure 3

8. The straight line \(l\) has equation \(y = k ( 2 x - 1 )\), where \(k\) is a constant. The curve \(C\) has equation \(y = x ^ { 2 } + 2 x + 11\)
Find the set of values of \(k\) for which \(l\) does not cross or touch \(C\).
(6)

9. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. A curve has equation $$y = \frac { 4 x ^ { 2 } + 9 } { 2 \sqrt { x } } \quad x > 0$$ Find the \(x\) coordinate of the point on the curve at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

10. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 4 x - 3 ) ( x - 5 ) ^ { 2 }$$

1. Sketch \(C _ { 1 }\) showing the coordinates of any point where the curve touches or crosses the coordinate axes.
2. Hence or otherwise
   1. find the values of \(x\) for which \(\mathrm { f } \left( \frac { 1 } { 4 } x \right) = 0\)
   2. find the value of the constant \(p\) such that the curve with equation \(y = \mathrm { f } ( x ) + p\) passes through the origin. A second curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = \mathrm { f } ( x + 1 )\)
3. 1. Find, in simplest form, \(\mathrm { g } ( x )\). You may leave your answer in a factorised form.
   2. Hence, or otherwise, find the \(y\) intercept of curve \(C _ { 2 }\)

11. A curve has equation \(y = \mathrm { f } ( x )\), where $$f ^ { \prime \prime } ( x ) = \frac { 6 } { \sqrt { x ^ { 3 } } } + x \quad x > 0$$ The point \(P ( 4 , - 50 )\) lies on the curve.
Given that \(\mathrm { f } ^ { \prime } ( x ) = - 4\) at \(P\),

1. find the equation of the normal at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants,
   (3)
2. find \(\mathrm { f } ( x )\).
   (8)

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