Edexcel Paper 1 (Paper 1) Specimen

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { x } { 1 + \sqrt { x } } , x \geqslant 0\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, 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 \(y\) for \(y = \frac { x } { 1 + \sqrt { } x }\)

1. Use the trapezium rule, with all the values of \(y\) in the table, to find an estimate for the area of \(R\), giving your answer to 3 decimal places.
2. Explain how the trapezium rule can be used to give a better approximation for the area of \(R\).
3. Giving your answer to 3 decimal places in each case, use your answer to part (a) to deduce an estimate for
   1. \(\int _ { 1 } ^ { 3 } \frac { 5 x } { 1 + \sqrt { x } } \mathrm {~d} x\)
   2. \(\int _ { 1 } ^ { 3 } \left( 6 + \frac { x } { 1 + \sqrt { x } } \right) \mathrm { d } x\)

1. (a) Show that the binomial expansion of

$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.

1. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by

$$\begin{aligned} a _ { 1 } & = 3
a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
   the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
   and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
   Given that \(A B C D\) is a parallelogram,
   1. find the position vector of point \(D\).
   The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
   Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
2. find the position vector of \(X\).

1. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 48\), where \(a\) is a constant

Given that \(\mathrm { f } ( - 6 ) = 0\)

1. 1. show that \(a = 4\)
   2. express \(\mathrm { f } ( x )\) as a product of two algebraic factors. Given that \(2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3\)
2. show that \(x ^ { 3 } + 4 x ^ { 2 } - 4 x + 48 = 0\)
3. hence explain why $$2 \log _ { 2 } ( x + 2 ) + \log _ { 2 } x - \log _ { 2 } ( x - 6 ) = 3$$ has no real roots.

6. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 2 shows the entrance to a road tunnel. The maximum height of the tunnel is measured as 5 metres and the width of the base of the tunnel is measured as 6 metres. Figure 3 shows a quadratic curve \(B C A\) used to model this entrance.
The points \(A , O , B\) and \(C\) are assumed to lie in the same vertical plane and the ground \(A O B\) is assumed to be horizontal.

1. Find an equation for curve \(B C A\). A coach has height 4.1 m and width 2.4 m .
2. Determine whether or not it is possible for the coach to enter the tunnel.
3. Suggest a reason why this model may not be suitable to determine whether or not the coach can pass through the tunnel.

7. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = 2 \mathrm { e } ^ { 2 x } - x \mathrm { e } ^ { 2 x } , \quad x \in \mathbb { R }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(p \mathrm { e } ^ { 4 } + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. There were 2100 tonnes of wheat harvested on a farm during 2017.

The mass of wheat harvested during each subsequent year is expected to increase by \(1.2 \%\) per year.

1. Find the total mass of wheat expected to be harvested from 2017 to 2030 inclusive, giving your answer to 3 significant figures. Each year it costs
   • £5.15 per tonne to harvest the first 2000 tonnes of wheat
   • £6.45 per tonne to harvest wheat in excess of 2000 tonnes
   • Use this information to find the expected cost of harvesting the wheat from 2017 to 2030 inclusive. Give your answer to the nearest \(\pounds 1000\)

1. The curve \(C\) has equation

$$y = 2 x ^ { 3 } + 5$$ The curve \(C\) passes through the point \(P ( 1,7 )\).
Use differentiation from first principles to find the value of the gradient of the tangent to \(C\) at \(P\).

1. The function f is defined by

$$f : x \mapsto \frac { 3 x - 5 } { x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - 1$$

1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
2. Show that $$\mathrm { ff } ( x ) = \frac { x + a } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq \pm 1$$ where \(a\) is an integer to be found. The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 3 x , \quad x \in \mathbb { R } , 0 \leqslant x \leqslant 5$$
3. Find the value of \(\mathrm { fg } ( 2 )\).
4. Find the range of g.
5. Explain why the function \(g\) does not have an inverse.

11. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f ^ { \prime } ( x ) = k - 4 x - 3 x ^ { 2 }$$ where \(k\) is constant,

1. show that \(C\) has a point of inflection at \(x = - \frac { 2 } { 3 }\) Given also that the distance \(A B = 4 \sqrt { 2 }\)
2. find, showing your working, the integer value of \(k\).

1. Show that

$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin 2 \theta } { 1 + \cos \theta } d \theta = 2 - 2 \ln 2$$

13. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places. Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula $$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017. Use Tom's model to
(b) find the depth of water at 00:00 hours on 18th October 2017,
(c) find the maximum depth of water,
(d) find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula $$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$ where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
(e) (i) explain why Jolene's model is not correct,
(ii) hence find a suitable model for \(H\) in terms of \(x\).

14. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) , y = 2 \sin t , \quad 0 < t \leqslant 2 \pi$$ Show that a Cartesian equation of \(C\) can be written in the form $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be found.