Edexcel P1 (Pure Mathematics 1) 2019 June

1. The curve \(C\) has equation \(y = \frac { 1 } { 8 } x ^ { 3 } - \frac { 24 } { \sqrt { x } } + 1\)
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in its simplest form.
      (3)
   The point \(P ( 4 , - 3 )\) lies on \(C\).
2. Find the equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

1. Answer this question showing each stage of your working.
   1. Simplify \(\frac { 1 } { 4 - 2 \sqrt { 2 } }\)
      giving your answer in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational numbers.
   2. Hence, or otherwise, solve the equation
   $$4 x = 2 \sqrt { 2 } x + 20 \sqrt { 2 }$$ giving your answer in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are rational numbers.

3. \begin{figure}[h] \end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 29 m ,

1. show that \(x > 1.5 \mathrm {~m}\) Given also that the area of the garden is less than \(72 \mathrm {~m} ^ { 2 }\),
2. form and solve a quadratic inequality in \(x\).
3. Hence state the range of possible values of \(x\).
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1. Find

$$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } d x$$ giving the answer in its simplest form. $$\int \frac { 4 x ^ { 2 } + 1 } { 2 \sqrt { x } } \mathrm {~d} x$$ giving the answer in its simplest form.

1. (a) Find, using algebra, all real solutions of

$$2 x ^ { 3 } + 3 x ^ { 2 } - 35 x = 0$$ (b) Hence find all real solutions of $$2 ( y - 5 ) ^ { 6 } + 3 ( y - 5 ) ^ { 4 } - 35 ( y - 5 ) ^ { 2 } = 0$$

1. The line with equation \(y = 4 x + c\), where \(c\) is a constant, meets the curve with equation \(y = x ( x - 3 )\) at only one point.
   1. Find the value of \(c\).
   2. Hence find the coordinates of the point of intersection.

7. \begin{figure}[h] \end{figure} The shape \(A B C D A\) consists of a sector \(A B C O A\) of a circle, centre \(O\), joined to a triangle \(A O D\), as shown in Figure 2. The point \(D\) lies on \(O C\).
The radius of the circle is 6 cm , length \(A D\) is 5 cm and angle \(A O D\) is 0.7 radians.

1. Find the area of the sector \(A B C O A\), giving your answer to one decimal place. Given angle \(A D O\) is obtuse,
2. find the size of angle \(A D O\), giving your answer to 3 decimal places.
3. Hence find the perimeter of shape \(A B C D A\), giving your answer to one decimal place.
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1. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , \quad x > 0\), passes through the point \(P ( 4,1 )\).

Given that \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x } - 2 - \frac { 8 } { 3 x ^ { 2 } }\)

1. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
   (4)
2. Find \(\mathrm { f } ( x )\).
   (5)
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9. \begin{figure}[h] \end{figure} Figure 3 shows a plot of the curve with equation \(y = \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)

1. State the coordinates of the minimum point on the curve with equation $$y = 4 \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$$ A copy of Figure 3, called Diagram 1, is shown on the next page.
2. On Diagram 1, sketch and label the curves
   1. \(y = 1 + \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
   2. \(y = \tan \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
3. Hence find the number of solutions of the equation
   1. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 2160 ^ { \circ }\)
   2. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 1980 ^ { \circ }\)
      \includegraphics[max width=\textwidth, alt={}]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-25_746_808_577_575}
      \section*{Diagram 1}

1. A curve has equation \(y = \mathrm { f } ( x )\), where

$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$ The curve touches the \(x\)-axis at the point \(P\) and crosses the \(x\)-axis at the point \(Q\).

1. State the coordinates of the point \(P\).
2. Find \(f ^ { \prime } ( x )\).
3. Hence show that the equation of the tangent to the curve at the point where \(x = \frac { 5 } { 2 }\) can be expressed in the form \(y = k\), where \(k\) is a constant to be found. The curve with equation \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, passes through the origin \(O\).
4. State the possible values of \(a\).
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