Edexcel C12 (Core Mathematics 1 & 2) 2014 June

1. \begin{figure}[h] \end{figure} Figure 1 shows the position of three stationary fishing boats \(A , B\) and \(C\), which are assumed to be in the same horizontal plane. Boat \(A\) is 10 km due north of boat \(B\). Boat \(C\) is 8 km on a bearing of \(065 ^ { \circ }\) from boat \(B\).

1. Find the distance of boat \(C\) from boat \(A\), giving your answer to the nearest 10 metres.
2. Find the bearing of boat \(C\) from boat \(A\), giving your answer to one decimal place.

2. Without using your calculator, solve $$x \sqrt { 27 } + 21 = \frac { 6 x } { \sqrt { 3 } }$$ Write your answer in the form \(a \sqrt { b }\) where \(a\) and \(b\) are integers. You must show all stages of your working.

3. Solve, giving each answer to 3 significant figures, the equations

1. \(4 ^ { a } = 20\)
2. \(3 + 2 \log _ { 2 } b = \log _ { 2 } ( 30 b )\)
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \frac { 16 } { x } , \quad x > 0$$ The curve has a minimum turning point at \(A\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Hence find the coordinates of \(A\).
3. Use your answer to part (b) to write down the turning point of the curve with equation
   1. \(y = \mathrm { f } ( x + 1 )\),
   2. \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).

5. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 3 shows the points \(P , Q\) and \(R\). Points \(P\) and \(Q\) have coordinates ( \(- 1,4\) ) and ( 4,7 ) respectively.

1. Find an equation for the straight line passing through points \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The point \(R\) has coordinates ( \(p , - 3\) ), where \(p\) is a positive constant. Given that angle \(Q P R = 90 ^ { \circ }\),
2. find the value of \(p\).

6. (a) Show that $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } \equiv 1 - \tan ^ { 2 } x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$\frac { \cos ^ { 2 } x - \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } + 2 = 0$$ Give your answers in terms of \(\pi\).

7. (i) A curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,3 )\). Given that $$f ^ { \prime } ( x ) = \frac { 4 } { x ^ { 3 } } + 2 x - 1$$ find the value of \(\mathrm { f } ( 1 )\).
(ii) Given that $$\int _ { 1 } ^ { 4 } ( 3 \sqrt { x } + A ) \mathrm { d } x = 21$$ find the exact value of the constant \(A\).

8. Given that $$1 + 12 x + 70 x ^ { 2 } + \ldots$$ is the binomial expansion, in ascending powers of \(x\) of \(( 1 + b x ) ^ { n }\), where \(n \in \mathbb { N }\) and \(b\) is a constant,

1. show that \(n b = 12\)
2. find the values of the constants \(b\) and \(n\).

9. (i) Find the value of \(\sum _ { r = 1 } ^ { 20 } ( 3 + 5 r )\)
(ii) Given that \(\sum _ { r = 0 } ^ { \infty } \frac { a } { 4 ^ { r } } = 16\), find the value of the constant \(a\).

10. The equation $$k x ^ { 2 } + 4 x + k = 2 , \text { where } k \text { is a constant, }$$ has two distinct real solutions for \(x\).

1. Show that \(k\) satisfies $$k ^ { 2 } - 2 k - 4 < 0$$
2. Hence find the set of all possible values of \(k\).

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the circle \(C\) with centre \(Q\) and equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 2 y + 5 = 0$$

1. Find
   1. the coordinates of \(Q\),
   2. the exact value of the radius of \(C\). The tangents to \(C\) from the point \(T ( 8,4 )\) meet \(C\) at the points \(M\) and \(N\), as shown in Figure 4.
2. Show that the obtuse angle \(M Q N\) is 2.498 radians to 3 decimal places. The region \(R\), shown shaded in Figure 4, is bounded by the tangent \(T N\), the minor arc \(N M\), and the tangent \(M T\).
3. Find the area of region \(R\).

12. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 } C\) touches the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).

1. Show that the coordinates of \(A\) are \(( 3,0 )\).
2. Show that the equation of the tangent to \(C\) at the point \(A\) is \(y = - 3 x + 9\) The tangent to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 5.
3. Use algebra to find the \(x\) coordinate of \(B\). The region \(R\), shown shaded in Figure 5, is bounded by the curve \(C\) and the tangent to \(C\) at \(A\).
4. Find, by using calculus, the area of region \(R\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

13. The height of sea water, \(h\) metres, on a harbour wall at time \(t\) hours after midnight is given by $$h = 3.7 + 2.5 \cos ( 30 t - 40 ) ^ { \circ } , \quad 0 \leqslant t < 24$$

1. Calculate the maximum value of \(h\) and the exact time of day when this maximum first occurs. Fishing boats cannot enter the harbour if \(h\) is less than 3
2. Find the times during the morning between which fishing boats cannot enter the harbour.
   Give these times to the nearest minute.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

14. \begin{figure}[h] \end{figure} Figure 6 shows a solid triangular prism \(A B C D E F\) in which \(A B = 2 x \mathrm {~cm}\) and \(C D = l \mathrm {~cm}\). The cross section \(A B C\) is an equilateral triangle. The rectangle \(B C D F\) is horizontal and the triangles \(A B C\) and \(D E F\) are vertical.
The total surface area of the prism is \(S \mathrm {~cm} ^ { 2 }\) and the volume of the prism is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(S = 2 x ^ { 2 } \sqrt { 3 } + 6 x l\) Given that \(S = 960\),
2. show that \(V = 160 x \sqrt { 3 } - x ^ { 3 }\)
3. Use calculus to find the maximum value of \(V\), giving your answer to the nearest integer.
4. Justify that the value of \(V\) found in part (c) is a maximum.
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