Edexcel C12 (Core Mathematics 1 & 2) 2014 January

1. Find the first 3 terms in ascending powers of \(x\) of

$$\left( 2 - \frac { x } { 2 } \right) ^ { 6 }$$ giving each term in its simplest form.

2. $$\mathrm { f } ( x ) = \frac { 8 } { x ^ { 2 } } - 4 \sqrt { x } + 3 x - 1 , \quad x > 0$$ Giving your answers in their simplest form, find

1. \(\mathrm { f } ^ { \prime } ( x )\)
2. \(\int \mathrm { f } ( x ) \mathrm { d } x\)

3. $$f ( x ) = 10 x ^ { 3 } + 27 x ^ { 2 } - 13 x - 12$$

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by
   1. \(x - 2\)
   2. \(x + 3\)
2. Hence factorise \(\mathrm { f } ( x )\) completely.

4. Answer this question without the use of a calculator and show all your working.

1. Show that $$\frac { 4 } { 2 \sqrt { 2 } - \sqrt { 6 } } = 2 \sqrt { 2 } ( 2 + \sqrt { 3 } )$$
2. Show that $$\sqrt { 27 } + \sqrt { 21 } \times \sqrt { 7 } - \frac { 6 } { \sqrt { 3 } } = 8 \sqrt { 3 }$$

5. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 3
u _ { n + 1 } & = 2 - \frac { 4 } { u _ { n } } , \quad n \geqslant 1 \end{aligned}$$ Find the exact values of

1. \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
2. \(u _ { 61 }\)
3. \(\sum _ { i = 1 } ^ { 99 } u _ { i }\)

6. Given that \(a\) and \(b\) are positive constants, solve the simultaneous equations $$\begin{gathered} a b = 25
\log _ { 4 } a - \log _ { 4 } b = 3 \end{gathered}$$ Show each step of your working, giving exact values for \(a\) and \(b\).

7. (a) Show that $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ may be expressed in the form $$12 \cos ^ { 2 } x + \cos x - 1 = 0$$ (b) Hence, using trigonometry, find all the solutions in the interval \(0 \leqslant x \leqslant 360 ^ { \circ }\) of $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ Give each solution, in degrees, to 1 decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-15_106_97_2615_1784}

8. Find the range of values of \(k\) for which the quadratic equation $$k x ^ { 2 } + 8 x + 2 ( k + 7 ) = 0$$ has no real roots.

9. In the first month after opening, a mobile phone shop sold 300 phones. A model for future sales assumes that the number of phones sold will increase by \(5 \%\) per month, so that \(300 \times 1.05\) will be sold in the second month, \(300 \times 1.05 ^ { 2 }\) in the third month, and so on. Using this model, calculate

1. the number of phones sold in the 24th month,
2. the total number of phones sold over the whole 24 months. This model predicts that, in the \(N\) th month, the number of phones sold in that month exceeds 3000 for the first time.
3. Find the value of \(N\).

10. The curve \(C\) has equation \(y = \cos \left( x - \frac { \pi } { 3 } \right) , 0 \leqslant x \leqslant 2 \pi\)

1. In the space below, sketch the curve \(C\).
2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
3. Solve, for \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\), $$\cos \left( x - \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number.

11. The first three terms of an arithmetic series are \(60,4 p\) and \(2 p - 6\) respectively.

1. Show that \(p = 9\)
2. Find the value of the 20th term of this series.
3. Prove that the sum of the first \(n\) terms of this series is given by the expression $$12 n ( 6 - n )$$ \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-27_106_68_2615_1877}

12. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale Figure 1 shows the plan for a pond and platform. The platform is shown shaded in the figure and is labelled \(A B C D\). The pond and platform together form a circle of radius 22 m with centre \(O\).
\(O A\) and \(O D\) are radii of the circle. Point \(B\) lies on \(O A\) such that the length of \(O B\) is 10 m and point \(C\) lies on \(O D\) such that the length of \(O C\) is 10 m . The length of \(B C\) is 15 m . The platform is bounded by the arc \(A D\) of the circle, and the straight lines \(A B , B C\) and \(C D\). Find

1. the size of the angle \(B O C\), giving your answer in radians to 3 decimal places,
2. the perimeter of the platform to 3 significant figures,
3. the area of the platform to 3 significant figures.

13. The curve \(C\) has equation $$y = \frac { ( x - 3 ) ( 3 x - 25 ) } { x } , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in a fully simplified form.
2. Hence find the coordinates of the turning point on the curve \(C\).
3. Determine whether this turning point is a minimum or maximum, justifying your answer. The point \(P\), with \(x\) coordinate \(2 \frac { 1 } { 2 }\), lies on the curve \(C\).
4. Find the equation of the normal at \(P\), in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
   \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-35_90_72_2631_1873}

14. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale Figure 2 shows part of the line \(l\) with equation \(y = 2 x - 3\) and part of the curve \(C\) with equation \(y = x ^ { 2 } - 2 x - 15\) The line \(l\) and the curve \(C\) intersect at the points \(A\) and \(B\) as shown.

1. Use algebra to find the coordinates of \(A\) and the coordinates of \(B\). In Figure 2, the shaded region \(R\) is bounded by the line \(l\), the curve \(C\) and the positive \(x\)-axis.
2. Use integration to calculate an exact value for the area of \(R\).

15. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale The points \(X\) and \(Y\) have coordinates \(( 0,3 )\) and \(( 6,11 )\) respectively. \(X Y\) is a chord of a circle \(C\) with centre \(Z\), as shown in Figure 3.

1. Find the gradient of \(X Y\). The point \(M\) is the midpoint of \(X Y\).
2. Find an equation for the line which passes through \(Z\) and \(M\). Given that the \(y\) coordinate of \(Z\) is 10 ,
3. find the \(x\) coordinate of \(Z\),
4. find the equation of the circle \(C\), giving your answer in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$$ where \(a\), \(b\) and \(c\) are constants.