OCR C2 (Core Mathematics 2)

1. (i) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
   (ii) Hence state how many solutions exist to the equation

$$\sin 2 x = \tan \frac { x } { 2 } ,$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.

2.
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-1_588_513_813_593} The diagram shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).

3. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { n + 1 } = \left( u _ { n } \right) ^ { 2 } - 1 , \quad n \geq 1 .$$ Given that \(u _ { 1 } = k\), where \(k\) is a constant,

1. find expressions for \(u _ { 2 }\) and \(u _ { 3 }\) in terms of \(k\). Given also that \(u _ { 2 } + u _ { 3 } = 11\),
2. find the possible values of \(k\).

4.
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-2_465_844_246_516} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } + 1 }\).
The shaded region \(R\) is bounded by the curve, the coordinate axes and the line \(x = 2\).

1. Use the trapezium rule with four strips of equal width to estimate the area of \(R\). The cross-section of a support for a bookshelf is modelled by \(R\) with 1 unit on each axis representing 8 cm . Given that the support is 2 cm thick,
2. find an estimate for the volume of the support.

5. (i) Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$ (ii) Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.

6. (i) Evaluate $$\int _ { 2 } ^ { 4 } \left( 2 - \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x$$ (ii) Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } + 1$$ and that \(y = 3\) when \(x = 0\), find the value of \(y\) when \(x = 2\).

7.
\includegraphics[max width=\textwidth, alt={}, center]{e5d62032-84ad-4e0b-9b72-ccfd8f4dbac8-3_499_721_248_552} The diagram shows part of the curve \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { 1 - 8 x ^ { 3 } } { x ^ { 2 } } , x \neq 0\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).
3. Find the area of the shaded region bounded by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = 2\).

8. A store begins to stock a new range of DVD players and achieves sales of \(\pounds 1500\) of these products during the first month. In a model it is assumed that sales will decrease by \(\pounds x\) in each subsequent month, forming an arithmetic sequence. Given that sales total \(\pounds 8100\) during the first six months, use the model to

1. find the value of \(x\),
2. find the expected value of sales in the eighth month,
3. show that the expected total of sales in pounds during the first \(n\) months is given by \(k n ( 51 - n )\), where \(k\) is an integer to be found.
4. Explain why this model cannot be valid over a long period of time.

9. \(f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + x + 2\).

1. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Fully factorise \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 0\).
4. Find, in terms of \(\pi\), the values of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) for which $$2 \sin ^ { 3 } \theta - 5 \sin ^ { 2 } \theta + \sin \theta + 2 = 0$$