OCR MEI C4 (Core Mathematics 4)

1 Solve the equation. $$\frac { 8 } { x } - \frac { 9 } { x + 1 } = 1$$

2 Solve the equation \(3 \operatorname { cosec } ^ { 2 } x = 2 \cot ^ { 2 } x + 3\) for values of \(x\) in the range \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3 The curve \(y ^ { 2 } = x - 1\) for \(1 \leq x \leq 3\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed.

4 A curve is given by the parametric equations \(x = t ^ { 2 } , y = 3 t\) for all values of \(t\). Find the equation of the tangent to the curve at the point where \(t = - 2\).

5

1. Express \(\frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence find \(\int _ { 2 } ^ { 3 } \frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) } \mathrm { d } x\).

6 The function \(\mathrm { f } ( \theta ) = 3 \sin \theta + 4 \cos \theta\) is to be expressed in the form \(r \sin ( \theta + \alpha )\) where \(r > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).

1. Find the values of \(r\) and \(\alpha\).
2. Write down the maximum and minimum value of \(\mathrm { f } ( \theta )\).
3. Solve the equation \(\mathrm { f } ( \theta ) = 1\) for \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).

7

1. Show that \(\frac { 1 } { \sqrt { 25 - x } } = \frac { 1 } { 5 } \left( 1 - \frac { x } { 25 } \right) ^ { - \frac { 1 } { 2 } }\).
2. Hence expand \(\frac { 1 } { \sqrt { 25 - x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
3. Write down the range of values of \(x\) for which the expansion is valid.

8 The new price of a particular make of car is \(\pounds 10000\). When its age is \(t\) years, the list price is \(\pounds V\). When \(t = 5 , V = 5000\). Aloke, Ben and Charlie all run outlets for used cars. Each of them has a different model for the depreciation.

1. Aloke claims that the rate of depreciation is constant. Write this claim as a differential equation.
   Solve the differential equation and hence find the value of a car that is 7 years old according to this model.
   Explain why this model breaks down for large \(t\).
2. Ben believes that the rate of depreciation is inversely proportional to the square root of the age of the car. Express this claim as a differential equation and hence find the value of a car that is 7 years old according to this model.
   Does this model ever break down?
3. Charlie believes that a better model is given by the differential equation $$\frac { \mathrm { d } V } { \mathrm {~d} t } = k V$$ Solve this differential equation and find the value of the car after 7 years according to this model.
   Does this model ever break down?
4. Further investigation reveals that the average value of this particular type of car when 8 years old is \(\pounds 3000\). Find the value of \(V\) when \(t = 8\) for the three models above. Which of the three models best predicts the value of \(V\) at this time?

9 Beside a major route into a county town the authorities decide to build a large pyramid. Fig. 9.1 shows this pyramid, ABCDE O is the centre point of the horizontal base BCDE . A coordinate system is defined with O as the origin. The \(x\)-axis and \(y\)-axis are horizontal and the \(z\)-axis is vertical, as shown in Fig. 9.1 The vertices of the pyramid are $$A ( 0,0,6 ) , B ( - 4 , - 4,0 ) , C ( 4 , - 4,0 ) , D ( 4,4,0 ) \text { and } E ( - 4,4,0 ) .$$ \begin{figure}[h] \end{figure} The pyramid is supported by a vertical pole OA and there are also support rods from O to points on the triangular faces \(\mathrm { ABC } , \mathrm { ACD } , \mathrm { ADE }\) and AEB . One of the rods, ON , is shown in fig.9.2 which shows one quarter of the pyramid. \begin{figure}[h] \end{figure} M is the mid-point of the line BC .

1. Write down the coordinates of M.
2. Write down the vector \(\overrightarrow { \mathrm { AM } }\) and hence the coordinates of the point N which divides \(\overrightarrow { \mathrm { AM } }\) so that the ratio \(\mathrm { AN } : \mathrm { NM } = 2 : 1\).
3. Show that ON is perpendicular to both AM and BC .
4. Hence write down the equation of the plane ABC in its simplest form.
5. Find the angle between the face ABC and the ground.