OCR MEI C3 (Core Mathematics 3) 2014 June

1 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } ( 1 - \sin 3 x ) \mathrm { d } x\), giving your answer in exact form.

3 Solve the equation \(| 3 - 2 x | = 4 | x |\).

4 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = a + \cos b x , 0 \leqslant x \leqslant 2 \pi ,$$ and \(a\) and \(b\) are positive constants. The curve has stationary points at \(( 0,3 )\) and \(( 2 \pi , 1 )\). \begin{figure}[h] \end{figure}

1. Find \(a\) and \(b\).
2. Find \(\mathrm { f } ^ { - 1 } ( x )\), and state its domain and range.

5 A spherical balloon of radius \(r \mathrm {~cm}\) has volume \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\). The balloon is inflated at a constant rate of \(10 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of \(r\) when \(r = 8\).

6 The value \(\pounds V\) of a car \(t\) years after it is new is modelled by the equation \(V = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants which depend on the make and model of the car.

1. Brian buys a new sports car. Its value is modelled by the equation $$V = 20000 \mathrm { e } ^ { - 0.2 t } .$$ Calculate how much value, to the nearest \(\pounds 100\), this car has lost after 1 year.
2. At the same time as Brian buys his car, Kate buys a new hatchback for \(\pounds 15000\). Her car loses \(\pounds 2000\) of its value in the first year. Show that, for Kate's car, \(k = 0.143\) correct to 3 significant figures.
3. Find how long it is before Brian's and Kate's cars have the same value.

7 Either prove or disprove each of the following statements.

1. 'If \(m\) and \(n\) are consecutive odd numbers, then at least one of \(m\) and \(n\) is a prime number.'
2. 'If \(m\) and \(n\) are consecutive even numbers, then \(m n\) is divisible by 8 .'

8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\). \begin{figure}[h] \end{figure}

1. Show algebraically that \(\mathrm { f } ( x )\) is an odd function. Interpret this result geometrically.
2. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \left( 2 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }\). Hence find the exact gradient of the curve at the origin.
3. Find the exact area of the region bounded by the curve, the \(x\)-axis and the line \(x = 1\).
4. (A) Show that if \(y = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\), then \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\).
   (B) Differentiate \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\) implicitly to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y ^ { 3 } } { x ^ { 3 } }\). Explain why this expression cannot be used to find the gradient of the curve at the origin.

9 Fig. 9 shows the curve \(y = x \mathrm { e } ^ { - 2 x }\) together with the straight line \(y = m x\), where \(m\) is a constant, with \(0 < m < 1\). The curve and the line meet at O and P . The dashed line is the tangent at P . \begin{figure}[h] \end{figure}

1. Show that the \(x\)-coordinate of P is \(- \frac { 1 } { 2 } \ln m\).
2. Find, in terms of \(m\), the gradient of the tangent to the curve at P . You are given that OP and this tangent are equally inclined to the \(x\)-axis.
3. Show that \(m = \mathrm { e } ^ { - 2 }\), and find the exact coordinates of P .
4. Find the exact area of the shaded region between the line OP and the curve. \section*{END OF QUESTION PAPER} \section*{OCR \(^ { \text {N } }\)}