Edexcel Paper 1 (Paper 1) Specimen

1. The curve \(C\) has equation

$$y = 3 x ^ { 4 } - 8 x ^ { 3 } - 3$$

1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Verify that \(C\) has a stationary point when \(x = 2\)
3. Determine the nature of this stationary point, giving a reason for your answer.

2. \begin{figure}[h] \end{figure} The shape \(A B C D O A\), as shown in Figure 1, consists of a sector \(C O D\) of a circle centre \(O\) joined to a sector \(A O B\) of a different circle, also centre \(O\). Given that arc length \(C D = 3 \mathrm {~cm} , \angle C O D = 0.4\) radians and \(A O D\) is a straight line of length 12 cm ,

1. find the length of \(O D\),
2. find the area of the shaded sector \(A O B\).

3. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 4 x + 10 y = k$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\).
2. State the range of possible values for \(k\).

4. Given that \(a\) is a positive constant and $$\int _ { a } ^ { 2 a } \frac { t + 1 } { t } \mathrm {~d} t = \ln 7$$ show that \(a = \ln k\), where \(k\) is a constant to be found.

5. A curve \(C\) has parametric equations $$x = 2 t - 1 , \quad y = 4 t - 7 + \frac { 3 } { t } , \quad t \neq 0$$ Show that the Cartesian equation of the curve \(C\) can be written in the form $$y = \frac { 2 x ^ { 2 } + a x + b } { x + 1 } , \quad x \neq - 1$$ where \(a\) and \(b\) are integers to be found.

6. A company plans to extract oil from an oil field. The daily volume of oil \(V\), measured in barrels that the company will extract from this oil field depends upon the time, \(t\) years, after the start of drilling. The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions:

• The initial daily volume of oil extracted from the oil field will be 16000 barrels.
• The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels.
• The daily volume of oil extracted will decrease over time.

The diagram below shows the graphs of two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. 1. Use model \(A\) to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.
   2. Write down a limitation of using model \(A\).
2. 1. Using an exponential model and the information given in the question, find a possible equation for model \(B\).
   2. Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.

7. Figure 2 Figure 2 shows a sketch of a triangle \(A B C\).
Given \(\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { B C } = \mathbf { i } - 9 \mathbf { j } + 3 \mathbf { k }\),
show that \(\angle B A C = 105.9 ^ { \circ }\) to one decimal place.

8. $$f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30 , \quad x > 2.5$$

1. Show that \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [3.5,4] A student takes 4 as the first approximation to \(\alpha\).
   Given \(\mathrm { f } ( 4 ) = 3.099\) and \(\mathrm { f } ^ { \prime } ( 4 ) = 16.67\) to 4 significant figures,
2. apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures.
3. Show that \(\alpha\) is the only root of \(\mathrm { f } ( x ) = 0\)

1. (a) Prove that

$$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence explain why the equation $$\tan \theta + \cot \theta = 1$$ does not have any real solutions.

10. Given that \(\theta\) is measured in radians, prove, from first principles, that the derivative of \(\sin \theta\) is \(\cos \theta\) You may assume the formula for \(\sin ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\)

11. An archer shoots an arrow. The height, \(H\) metres, of the arrow above the ground is modelled by the formula $$H = 1.8 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ where \(d\) is the horizontal distance of the arrow from the archer, measured in metres.
Given that the arrow travels in a vertical plane until it hits the ground,

1. find the horizontal distance travelled by the arrow, as given by this model.
2. With reference to the model, interpret the significance of the constant 1.8 in the formula.
3. Write \(1.8 + 0.4 d - 0.002 d ^ { 2 }\) in the form $$A - B ( d - C ) ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found. It is decided that the model should be adapted for a different archer.
   The adapted formula for this archer is $$H = 2.1 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ Hence or otherwise, find, for the adapted model
4. 1. the maximum height of the arrow above the ground.
   2. the horizontal distance, from the archer, of the arrow when it is at its maximum height.

1. In a controlled experiment, the number of microbes, \(N\), present in a culture \(T\) days after the start of the experiment were counted.
   \(N\) and \(T\) are expected to satisfy a relationship of the form

$$N = a T ^ { b } , \quad \text { where } a \text { and } b \text { are constants }$$

1. Show that this relationship can be expressed in the form $$\log _ { 10 } N = m \log _ { 10 } T + c$$ giving \(m\) and \(c\) in terms of the constants \(a\) and/or \(b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-18_1232_1046_804_513} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows the line of best fit for values of \(\log _ { 10 } N\) plotted against values of \(\log _ { 10 } T\)
2. Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
3. Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1000000 .
4. With reference to the model, interpret the value of the constant \(a\).

1. The curve \(C\) has parametric equations

$$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\)
   The line \(l\) is the normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line \(l\) intersects the curve \(C\) again at the point \(Q\).
3. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers.

14. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 5 , \quad x > 0$$ The finite region \(S\), shown shaded in Figure 4, is bounded by the curve \(C\), the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = 3\) The table below shows corresponding values of \(x\) and \(y\) with the values of \(y\) given to 4 decimal places as appropriate.

1. Use the trapezium rule, with all the values of \(y\) in the table, to obtain an estimate for the area of \(S\), giving your answer to 3 decimal places.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of \(S\).
3. Show that the exact area of \(S\) can be written in the form \(\frac { a } { b } + \ln c\), where \(a , b\) and \(c\) are integers to be found.
   (In part c, solutions based entirely on graphical or numerical methods are not acceptable.)

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 4 \sin 2 x } { \mathrm { e } ^ { \sqrt { 2 } x - 1 } } , \quad 0 \leqslant x \leqslant \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.

1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2 x = \sqrt { 2 }$$
2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation
   1. \(y = \mathrm { f } ( 2 x )\).
   2. \(y = 3 - 2 \mathrm { f } ( x )\).