Edexcel C12 (Core Mathematics 1 & 2) 2016 June

1. The first three terms in ascending powers of \(x\) in the binomial expansion of \(( 1 + p x ) ^ { 8 }\) are given by

$$1 + 12 x + q x ^ { 2 }$$ where \(p\) and \(q\) are constants.
Find the value of \(p\) and the value of \(q\).

$$\begin{gathered} 2 \log _ { 2 } y = 5 - \log _ { 2 } x \\ \log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)

3. Answer this question without a calculator, showing all your working and giving your answers in their simplest form.

1. Solve the equation $$4 ^ { 2 x + 1 } = 8 ^ { 4 x }$$
2. (a) Express $$3 \sqrt { 18 } - \sqrt { 32 }$$ in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
   (b) Hence, or otherwise, solve $$3 \sqrt { 18 } - \sqrt { 32 } = \sqrt { n }$$

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \sqrt { x + 2 } , x \geqslant - 2\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the line \(x = 6\) The table below shows corresponding values of \(x\) and \(y\) for \(y = \sqrt { x + 2 }\)

1. Complete the table above, giving the missing value of \(y\) to 4 decimal places.
2. Use the trapezium rule, with all of the values of \(y\) in the completed table, to find an approximate value for the area of \(R\), giving your answer to 3 decimal places. Use your answer to part (b) to find approximate values of
3. 1. \(\int _ { - 2 } ^ { 6 } \frac { \sqrt { x + 2 } } { 2 } \mathrm {~d} x\)
   2. \(\int _ { - 2 } ^ { 6 } ( 2 + \sqrt { x + 2 } ) \mathrm { d } x\)

5. (i) $$U _ { n + 1 } = \frac { U _ { n } } { U _ { n } - 3 } , \quad n \geqslant 1$$ Given \(U _ { 1 } = 4\), find

1. \(U _ { 2 }\)
2. \(\sum _ { n = 1 } ^ { 100 } U _ { n }\) (ii) Given $$\sum _ { r = 1 } ^ { n } ( 100 - 3 r ) < 0$$ find the least value of the positive integer \(n\).

6. (a) Show that \(\frac { x ^ { 2 } - 4 } { 2 \sqrt { } x }\) can be written in the form \(A x ^ { p } + B x ^ { q }\), where \(A , B , p\) and \(q\) are constants to be determined.
(b) Hence find $$\int \frac { x ^ { 2 } - 4 } { 2 \sqrt { x } } \mathrm {~d} x , \quad x > 0$$ giving your answer in its simplest form.

7. $$f ( x ) = 3 x ^ { 3 } + a x ^ { 2 } + b x - 10 \text {, where } a \text { and } b \text { are constants. }$$ Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\),

1. use the factor theorem to show that \(2 a + b = - 7\) Given also that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is - 36
2. find the value of \(a\) and the value of \(b\). \(\mathrm { f } ( x )\) can be written in the form $$\mathrm { f } ( x ) = ( x - 2 ) \mathrm { Q } ( x ) \text {, where } \mathrm { Q } ( x ) \text { is a quadratic function. }$$
3. 1. Find \(\mathrm { Q } ( x )\).
   2. Prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real root. You must justify your answer and show all your working.

8. In this question the angle \(\theta\) is measured in degrees throughout.

1. Show that the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta , \quad \theta \neq ( 2 n + 1 ) 90 ^ { \circ } , \quad n \in \mathbb { Z }$$ may be rewritten as $$6 \sin ^ { 2 } \theta + \sin \theta - 1 = 0$$
2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation $$\frac { 5 + \sin \theta } { 3 \cos \theta } = 2 \cos \theta$$ Give your answers to one decimal place, where appropriate.

1. The first term of a geometric series is 6 and the common ratio is 0.92

For this series, find

1. 1. the \(25 ^ { \text {th } }\) term, giving your answer to 2 significant figures,
   2. the sum to infinity. The sum to \(n\) terms of this series is greater than 72
2. Calculate the smallest possible value of \(n\).
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10. The curve \(C\) has equation \(y = \sin \left( x + \frac { \pi } { 4 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\)

1. On the axes below, sketch the curve \(C\).
2. Write down the exact coordinates of all the points at which the curve \(C\) meets or intersects the \(x\)-axis and the \(y\)-axis.
3. Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-14_677_1031_1446_445}

11. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows the design for a sail \(A P B C A\). The curved edge \(A P B\) of the sail is an arc of a circle centre \(O\) and radius \(r \mathrm {~m}\). The straight edge \(A C B\) is a chord of the circle. The height \(A B\) of the sail is 2.4 m . The maximum width \(C P\) of the sail is 0.4 m .

1. Show that \(r = 2\)
2. Show, to 4 decimal places, that angle \(A O B = 1.2870\) radians.
3. Hence calculate the area of the sail, giving your answer, in \(\mathrm { m } ^ { 2 }\), to 3 decimal places.

12. \begin{figure}[h] \end{figure} Figure 3 shows a circle \(C\) \(C\) touches the \(y\)-axis and has centre at the point ( \(a , 0\) ) where \(a\) is a positive constant.

1. Write down an equation for \(C\) in terms of \(a\) Given that the point \(P ( 4 , - 3 )\) lies on \(C\),
2. find the value of \(a\)

1. (a) Show that the equation

$$2 \log _ { 2 } y = 5 - \log _ { 2 } x \quad x > 0 , y > 0$$ may be written in the form \(y ^ { 2 } = \frac { k } { x }\) where \(k\) is a constant to be found.
(b) Hence, or otherwise, solve the simultaneous equations $$\begin{gathered} 2 \log _ { 2 } y = 5 - \log _ { 2 } x \\ \log _ { x } y = - 3 \end{gathered}$$ for \(x > 0 , y > 0\)

14. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph of \(y = g ( x ) , - 3 \leqslant x \leqslant 4\) and part of the line \(l\) with equation \(y = \frac { 1 } { 2 } x\) The graph of \(y = \mathrm { g } ( x )\) consists of three line segments, from \(P ( - 3,4 )\) to \(Q ( 0,4 )\), from \(Q ( 0,4 )\) to \(R ( 2,0 )\) and from \(R ( 2,0 )\) to \(S ( 4,10 )\). The line \(l\) intersects \(y = \mathrm { g } ( x )\) at the points \(A\) and \(B\) as shown in Figure 4.

1. Use algebra to find the \(x\) coordinate of the point \(A\) and the \(x\) coordinate of the point \(B\). Show each step of your working and give your answers as exact fractions.
2. Sketch the graph with equation $$y = \frac { 3 } { 2 } g ( x ) , \quad - 3 \leqslant x \leqslant 4$$ On your sketch show the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.

15. \begin{figure}[h] \end{figure} Figure 5 shows a design for a water barrel.
It is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and radius \(r \mathrm {~cm}\). The barrel has a base but has no lid, is open at the top and is made of material of negligible thickness. The barrel is designed to hold \(60000 \mathrm {~cm} ^ { 3 }\) of water when full.

1. Show that the total external surface area, \(S \mathrm {~cm} ^ { 2 }\), of the barrel is given by the formula $$S = \pi r ^ { 2 } + \frac { 120000 } { r }$$
2. Use calculus to find the minimum value of \(S\), giving your answer to 3 significant figures.
3. Justify that the value of \(S\) you found in part (b) is a minimum.

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve \(C\) with equation $$y = x ( x - 1 ) ( x - 2 )$$ The point \(P\) lies on \(C\) and has \(x\) coordinate \(\frac { 1 } { 2 }\) The line \(l\), as shown on Figure 6, is the tangent to \(C\) at \(P\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Use part (a) to find an equation for \(l\) in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The finite region \(R\), shown shaded in Figure 6, is bounded by the line \(l\), the curve \(C\) and the \(x\)-axis. The line \(l\) meets the curve again at the point \(( 2,0 )\)
3. Use integration to find the exact area of the shaded region \(R\).