Edexcel P1 (Pure Mathematics 1) 2019 October

1. \begin{figure}[h] \end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The angle \(A O B\) is 1.25 radians. Given that the area of the sector \(A O B\) is \(15 \mathrm {~cm} ^ { 2 }\)

1. find the exact value of \(r\),
2. find the exact length of the perimeter of the sector. Write your answer in simplest form.
   

2. A tree was planted in the ground. Exactly 2 years after it was planted, the height of the tree was 1.85 m . Exactly 7 years after it was planted, the height of the tree was 3.45 m . Given that the height, \(H\) metres, of the tree, \(t\) years after it was planted in the ground, can be modelled by the equation $$H = a t + b$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\).
2. State, according to the model, the height of the tree when it was planted.
   

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation \(y = x ^ { 2 } - 5 x + 13\) The point \(M\) is the minimum point of \(C\). The straight line \(l\) passes through the origin \(O\) and intersects \(C\) at the points \(M\) and \(N\) as shown. Find, showing your working,

1. the coordinates of \(M\),
2. the coordinates of \(N\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_531_561_1793_680} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows the curve \(C\) and the line \(l\). The finite region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
3. Use inequalities to define the region \(R\).

4. A parallelogram \(A B C D\) has area \(40 \mathrm {~cm} ^ { 2 }\) Given that \(A B\) has length \(10 \mathrm {~cm} , B C\) has length 6 cm and angle \(D A B\) is obtuse, find

1. the size of angle \(D A B\), in degrees, to 2 decimal places,
2. the length of diagonal \(B D\), in cm , to one decimal place.

5. A curve has equation $$y = \frac { x ^ { 3 } } { 6 } + 4 \sqrt { x } - 15 \quad x \geqslant 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in simplest form. The point \(P \left( 4 , \frac { 11 } { 3 } \right)\) lies on the curve.
2. Find the equation of the normal to the curve at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

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6. The curve \(C\) has equation \(y = \frac { 4 } { x } + k\), where \(k\) is a positive constant.

1. Sketch a graph of \(C\), stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the \(x\)-axis. The line with equation \(y = 10 - 2 x\) is a tangent to \(C\).
2. Find the possible values for \(k\).
   \(\_\_\_\_\) -

7. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation \(y = 2 x ^ { 2 } + 5\) The point \(P ( 2,13 )\) lies on the curve.

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

8. Solve, using algebra, the equation $$x - 6 x ^ { \frac { 1 } { 2 } } + 4 = 0$$ Fully simplify your answers, writing them in the form \(a + b \sqrt { c }\), where \(a , b\) and \(c\) are integers to be found.
(5)

9. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \sin \left( \frac { x } { 12 } \right)\), where \(x\) is measured in radians. The point \(M\) shown in Figure 5 is a minimum point on \(C\).

1. State the period of \(C\).
2. State the coordinates of \(M\). The smallest positive solution of the equation \(\sin \left( \frac { x } { 12 } \right) = k\), where \(k\) is a constant, is \(\alpha\). Find, in terms of \(\alpha\),
3. 1. the negative solution of the equation \(\sin \left( \frac { x } { 12 } \right) = k\) that is closest to zero,
   2. the smallest positive solution of the equation \(\cos \left( \frac { x } { 12 } \right) = k\).

10. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\) The curve crosses the \(y\)-axis at the point \(P\), as shown.
2. Expand \(\mathrm { f } ( x )\) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
3. Hence, or otherwise, find
   1. the coordinates of \(P\),
   2. the gradient of the curve at \(P\). The curve with equation \(y = \mathrm { f } ( x )\) is translated two units in the positive \(x\) direction to a curve with equation \(y = \mathrm { g } ( x )\).
4. 1. Find \(\mathrm { g } ( x )\), giving your answer in a simplified factorised form.
   2. Hence state the \(y\) intercept of the curve with equation \(y = \mathrm { g } ( x )\).

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

The point \(P \left( 4 , \frac { 32 } { 3 } \right)\) lies on the curve.
Given that

• \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 4 } { \sqrt { x } } - 3\)
• \(\quad \mathrm { f } ^ { \prime } ( x ) = 5\) at \(P\)
  find
  1. the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found,
  2. \(\mathrm { f } ( x )\).