OCR MEI Paper 3 (Paper 3) Specimen

1 Express \(\frac { 2 } { x - 1 } + \frac { 5 } { 2 x + 1 }\) as a single fraction.

2 Find the first four terms of the binomial expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\). State the set of values of \(x\) for which the expansion is valid.

3 Show that points \(\mathrm { A } ( 1,4,9 ) , \mathrm { B } ( 0,11,17 )\) and \(\mathrm { C } ( 3 , - 10 , - 7 )\) are collinear.

4 Show that \(\sum _ { r = 1 } ^ { 4 } \ln \frac { r } { r + 1 } = - \ln 5\).

5 In this question you must show detailed reasoning. Fig. 5 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 10\).
The points \(( 1,0 )\) and \(( 7,0 )\) lie on the circle. The point C is the centre of the circle. \begin{figure}[h] \end{figure} Find the area of the part of the circle below the \(x\)-axis.

6 Fig. 6 shows the curve with equation \(y = x ^ { 4 } - 6 x ^ { 2 } + 4 x + 5\). \begin{figure}[h] \end{figure}

7 By finding a counter example, disprove the following statement. If \(p\) and \(q\) are non-zero real numbers with \(p < q\), then \(\frac { 1 } { p } > \frac { 1 } { q }\).

8 In Fig. 8, OAB is a thin bent rod, with \(\mathrm { OA } = 1 \mathrm {~m} , \mathrm { AB } = 2 \mathrm {~m}\) and angle \(\mathrm { OAB } = 120 ^ { \circ }\). Angles \(\theta , \phi\) and \(h\) are as shown in Fig. 8. \begin{figure}[h] \end{figure}

1. Show that \(h = \sin \theta + 2 \sin \left( \theta + 60 ^ { \circ } \right)\). The rod is free to rotate about the origin so that \(\theta\) and \(\phi\) vary. You may assume that the result for \(h\) in part (a) holds for all values of \(\theta\).
2. Find an angle \(\theta\) for which \(h = 0\).

9

1. Express \(\cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\) and \(R\) is positive and given in exact form. The function \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = \frac { 1 } { ( k + \cos \theta + 2 \sin \theta ) } , 0 \leq \theta \leq 2 \pi , k\) is a constant.
2. The maximum value of \(\mathrm { f } ( \theta )\) is \(\frac { ( 3 + \sqrt { 5 } ) } { 4 }\). Find the value of \(k\).

10 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 4 } + x ^ { 3 } - 2 x ^ { 2 } - 4 x - 2\).

1. Show that \(x = - 1\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Show that another root of \(\mathrm { f } ( x ) = 0\) lies between \(x = 1\) and \(x = 2\).
3. Show that \(\mathrm { f } ( x ) = ( x + 1 ) \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } + a x + b\) and \(a\) and \(b\) are integers to be determined.
4. Without further calculation, explain why \(\mathrm { g } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\).
5. Use the Newton-Raphson formula to show that an iteration formula for finding roots of \(\mathrm { g } ( x ) = 0\) may be written $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$ Determine the root of \(\mathrm { g } ( x ) = 0\) which lies between \(x = 1\) and \(x = 2\) correct to 4 significant figures.

11 The curve \(y = \mathrm { f } ( x )\) is defined by the function \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\) with domain \(0 \leq x \leq 4 \pi\).

1. 1. Show that the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\), when arranged in increasing order, form an arithmetic sequence.
   2. Show that the corresponding \(y\)-coordinates form a geometric sequence.
2. Would the result still hold with a larger domain? Give reasons for your answer.

12 Explain why the smaller regular hexagon in Fig. C1 has perimeter 6.

13 Show that the larger regular hexagon in Fig. C1 has perimeter \(4 \sqrt { 3 }\).

14 Show that the two values of \(b\) given on line 36 are equivalent.

15 Fig. 15 shows a unit circle and the escribed regular polygon with 12 edges. \begin{figure}[h] \end{figure}

1. Show that the perimeter of the polygon is \(24 \tan 15 ^ { \circ }\).
2. Using the formula for \(\tan ( \theta - \phi )\) show that the perimeter of the polygon is \(48 - 24 \sqrt { 3 }\).

16 On a unit circle, the inscribed regular polygon with 12 edges gives a lower bound for \(\pi\), and the escribed regular polygon with 12 edges gives an upper bound for \(\pi\). Calculate the values of these bounds for \(\pi\), giving your answers:

1. in surd form
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