Edexcel C12 (Core Mathematics 1 & 2) 2017 June

1. An arithmetic sequence has first term 6 and common difference 10 Find
   1. the 15th term of the sequence,
   2. the sum of the first 20 terms of the sequence.
      

1. Simplify the following expressions fully.
   1. \(\left( \frac { 1 } { 9 } x ^ { 4 } \right) ^ { 0.5 }\)
   2. \(\left( \frac { x } { \sqrt { 2 } } \right) ^ { - 2 }\)
   3. \(x \sqrt { 3 } \div \sqrt { \frac { 48 } { x ^ { 4 } } }\)

1. The line \(l _ { 1 }\) has equation \(2 x + 3 y = 6\)

The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(( 3 , - 5 )\).
Find the equation for the line \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

4. The curve \(C\) has equation \(y = 4 x \sqrt { x } + \frac { 48 } { \sqrt { x } } - \sqrt { 8 } , x > 0\)

1. Find, simplifying each term,
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Use part (a) to find the exact coordinates of the stationary point of \(C\).
3. Determine whether the stationary point of \(C\) is a maximum or minimum, giving a reason for your answer.

5. $$f ( x ) = - 4 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 3$$

1. Use the remainder theorem to find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 1\) ).
2. Use the factor theorem to show that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).
3. Hence fully factorise \(\mathrm { f } ( x )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08b1be3e-2d9a-4832-b230-d5519540f494-12_581_636_731_657} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
4. Use your answer to part (c) and the sketch to deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\)

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of a design for a triangular garden \(A B C\). The garden has sides \(B A\) with length \(10 \mathrm {~m} , B C\) with length 6 m and \(C A\) with length 12 m . The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 10 m . A flowerbed \(B C D\) is shown shaded in Figure 2.

1. Find the size of angle \(B A C\), in radians, to 4 decimal places.
2. Find the perimeter of the flowerbed \(B C D\), in m , to 2 decimal places.
3. Find the area of the flowerbed \(B C D\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at the point \(( 0,8 )\). The line with equation \(y = 10\) is the only asymptote to the curve.
The curve has a single turning point, a minimum point at \(( 2,5 )\), as shown in Figure 3.

1. State the coordinates of the minimum point of the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 4 } x \right)\)
2. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( x ) - 3\) The curve with equation \(y = \mathrm { f } ( x )\) meets the line with equation \(y = k\), where \(k\) is a constant, at two distinct points.
3. State the set of possible values for \(k\).
4. Sketch the curve with equation \(y = - \mathrm { f } ( x )\). On your sketch, show clearly the coordinates of the turning point, the coordinates of the intersection with the \(y\)-axis and the equation of the asymptote. \section*{\textbackslash section*\{D\}}

8. (a) Find \(\int \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x\), simplifying each term. Given that \(b\) is a constant and $$\int _ { b } ^ { 4 } \left( 3 x ^ { 2 } + 4 x - 15 \right) \mathrm { d } x = 36$$ (b) show that \(b ^ { 3 } + 2 b ^ { 2 } - 15 b = 0\)
(c) Hence find the possible values of \(b\).

9. (i) Find the exact value of \(x\) for which $$2 \log _ { 10 } ( x - 2 ) - \log _ { 10 } ( x + 5 ) = 0$$ (ii) Given $$\log _ { p } ( 4 y + 1 ) - \log _ { p } ( 2 y - 2 ) = 1 \quad p > 2 , y > 1$$ express \(y\) in terms of \(p\).

1. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 2 - \frac { x } { 8 } \right) ^ { 10 }$$ giving each term in its simplest form. $$\mathrm { f } ( x ) = \left( 2 - \frac { x } { 8 } \right) ^ { 10 } ( a + b x ) , \text { where } a \text { and } b \text { are constants }$$ Given that the first two terms, in ascending powers of \(x\) in the series expansion of \(\mathrm { f } ( x )\), are 256 and \(352 x\),
(b) find the value of \(a\),
(c) find the value of \(b\).

11. Wheat is to be grown on a farm. A model predicts that the mass of wheat harvested on the farm will increase by \(1.5 \%\) per year, so that the mass of wheat harvested each year forms a geometric sequence. Given that the mass of wheat harvested during year one is 6000 tonnes,

1. show that, according to the model, the mass of wheat harvested on the farm during year 4 will be approximately 6274 tonnes. During year \(N\), according to the model, there is predicted to be more than 8000 tonnes of wheat harvested on the farm.
2. Find the smallest possible value of \(N\). It costs \(\pounds 5\) per tonne to harvest the wheat.
3. Assuming the model, find the total amount that it would cost to harvest the wheat from year one to year 10 inclusive. Give your answer to the nearest \(\pounds 1000\).

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 9 x ^ { 2 } + 26 x - 18$$ The point \(P ( 4,6 )\) lies on \(C\).

1. Use calculus to show that the normal to \(C\) at the point \(P\) has equation $$2 y + x = 16$$ The region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(x\)-axis and the normal to \(C\) at \(P\).
2. Show that \(C\) cuts the \(x\)-axis at \(( 1,0 )\)
3. Showing all your working, use calculus to find the exact area of \(R\).

13. (a) Show that the equation $$5 \cos x + 1 = \sin x \tan x$$ can be written in the form $$6 \cos ^ { 2 } x + \cos x - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 \cos 2 \theta + 1 = \sin 2 \theta \tan 2 \theta$$ giving your answers, where appropriate, to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the circle \(C _ { 1 }\)
The points \(A ( 1,4 )\) and \(B ( 7,8 )\) lie on \(C _ { 1 }\)
Given that \(A B\) is a diameter of the circle \(C _ { 1 }\)

1. find the coordinates for the centre of \(C _ { 1 }\)
2. find the exact radius of \(C _ { 1 }\) simplifying your answer. Two distinct circles \(C _ { 2 }\) and \(C _ { 3 }\) each have centre \(( 0,0 )\).
   Given that each of these circles touch circle \(C _ { 1 }\)
3. find the equation of circle \(C _ { 2 }\) and the equation of circle \(C _ { 3 }\)

15. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 4 + 1.5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight, and \(\frac { \pi t } { 6 }\) is measured in radians.

1. Show that the height of the water at 1 a.m. is 4.75 metres.
2. Find the height of the water at 2 p.m.
3. Find, to the nearest minute, the first two times when the height of the water is 3 metres.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)