Edexcel C12 (Core Mathematics 1 & 2) 2018 January

1. Given that

$$y = \frac { 2 x ^ { \frac { 2 } { 3 } } + 3 } { 6 } , \quad x > 0$$ find, in the simplest form,

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\)

2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 1
u _ { n + 1 } & = 2 - 3 u _ { n } \quad n \geqslant 1 \end{aligned}$$

1. Find the value of \(u _ { 2 }\) and the value of \(u _ { 3 }\)
2. Calculate the value of \(\sum _ { r = 1 } ^ { 4 } \left( r - u _ { r } \right)\)
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3. Simplify fully

1. \(\left( 3 x ^ { \frac { 1 } { 2 } } \right) ^ { 4 }\)
2. \(\frac { 2 y ^ { 7 } \times ( 4 y ) ^ { - 2 } } { 3 y }\)

4. The equation
\(( p - 2 ) x ^ { 2 } + 8 x + ( p + 4 ) = 0 , \quad\) where \(p\) is a constant has no real roots.

1. Show that \(p\) satisfies \(p ^ { 2 } + 2 p - 24 > 0\)
2. Hence find the set of possible values of \(p\).

5. (In this question, solutions based entirely on graphical or numerical methods are not acceptable.)

1. Solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$5 \sin 3 \theta - 7 \cos 3 \theta = 0$$ Give each solution, in radians, to 3 significant figures.
2. Solve, for \(0 < x < 360 ^ { \circ }\) $$9 \cos ^ { 2 } x + 5 \cos x = 3 \sin ^ { 2 } x$$ Give each solution, in degrees, to one decimal place.

6. $$f ( x ) = a x ^ { 3 } - 8 x ^ { 2 } + b x + 6$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ) there is no remainder. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\) the remainder is - 12

1. Find the value of \(a\) and the value of \(b\).
2. Factorise \(\mathrm { f } ( x )\) completely.

7. \begin{figure}[h] \end{figure} Figure 1 shows a rectangular sheet of metal of negligible thickness, which measures 25 cm by 15 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open cuboid box, as shown in Figure 2.

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the box is given by $$V = 4 x ^ { 3 } - 80 x ^ { 2 } + 375 x$$
2. Use calculus to find the value of \(x\), to 3 significant figures, for which the volume of the box is a maximum.
3. Justify that this value of \(x\) gives a maximum value for \(V\).
4. Find, to 3 significant figures, the maximum volume of the box.
   \section*{8.} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6f9ace43-747b-419f-a9d1-d30165d77379-22_670_1004_292_392} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve crosses the \(y\)-axis at the point \(( 0,5 )\) and crosses the \(x\)-axis at the point \(( 6,0 )\). The curve has a minimum point at \(( 1,3 )\) and a maximum point at \(( 4,7 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( - x )\)
2. \(y = \mathrm { f } ( 2 x )\) On each diagram, show clearly the coordinates of any points of intersection of the curve with the two coordinate axes and the coordinates of the stationary points.

1. The first term of a geometric series is 20 and the common ratio is 0.9
   1. Find the value of the fifth term.
   2. Find the sum of the first 8 terms, giving your answer to one decimal place.
   Given that \(S _ { \infty } - S _ { N } < 0.04\), where \(S _ { N }\) is the sum of the first \(N\) terms of this series, (c) show that \(0.9 ^ { N } < 0.0002\)
2. Hence find the smallest possible value of \(N\).

10. (i) Use the laws of logarithms to solve the equation $$3 \log _ { 8 } 2 + \log _ { 8 } ( 7 - x ) = 2 + \log _ { 8 } x$$ (ii) Using algebra, find, in terms of logarithms, the exact value of \(y\) for which $$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$

11. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x - 10 y + 16 = 0$$ The centre of \(C\) is at the point \(T\).

1. Find
   1. the coordinates of the point \(T\),
   2. the radius of the circle \(C\). The point \(M\) has coordinates \(( 20,12 )\).
2. Find the exact length of the line \(M T\). Point \(P\) lies on the circle \(C\) such that the tangent at \(P\) passes through the point \(M\).
3. Find the exact area of triangle \(M T P\), giving your answer as a simplified surd.

1. The line \(l _ { 1 }\) has equation \(x + 3 y - 11 = 0\)

The point \(A\) and the point \(B\) lie on \(l _ { 1 }\)
Given that \(A\) has coordinates ( \(- 1 , p\) ) and \(B\) has coordinates ( \(q , 2\) ), where \(p\) and \(q\) are integers,

1. find the value of \(p\) and the value of \(q\),
2. find the length of \(A B\), giving your answer as a simplified surd. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\).
3. Find an equation for \(l _ { 2 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

13. \begin{figure}[h] \end{figure} Figure 4 shows the position of two stationary boats, \(A\) and \(B\), and a port \(P\) which are assumed to be in the same horizontal plane. Boat \(A\) is 8.7 km on a bearing of \(314 ^ { \circ }\) from port \(P\).
Boat \(B\) is 3.5 km on a bearing of \(052 ^ { \circ }\) from port \(P\).

1. Show that angle \(A P B\) is \(98 ^ { \circ }\)
2. Find the distance of boat \(B\) from boat \(A\), giving your answer to one decimal place.
3. Find the bearing of boat \(B\) from boat \(A\), giving your answer to the nearest degree.

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the line \(l\) with equation \(y = 8 - x\) and part of the curve \(C\) with equation \(y = 14 + 3 x - 2 x ^ { 2 }\) The line \(l\) and the curve \(C\) intersect at the point \(A\) and the point \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the coordinate axes, the line \(l\), and the curve \(C\).
2. Use algebraic integration to calculate the exact area of \(R\).

15. The binomial expansion, in ascending powers of \(x\), of \(( 1 + k x ) ^ { n }\) is $$1 + 36 x + 126 k x ^ { 2 } + \ldots$$ where \(k\) is a non-zero constant and \(n\) is a positive integer.

1. Show that \(n k ( n - 1 ) = 252\)
2. Find the value of \(k\) and the value of \(n\).
3. Using the values of \(k\) and \(n\), find the coefficient of \(x ^ { 3 }\) in the binomial expansion of \(( 1 + k x ) ^ { n }\)