OCR C3 (Core Mathematics 3) 2015 June

1 Find the equation of the tangent to the curve \(y = \frac { 5 x + 4 } { 3 x - 8 }\) at the point \(( 2 , - 7 )\).

2 It is given that \(\theta\) is the acute angle such that \(\cot \theta = 4\). Without using a calculator, find the exact value of

1. \(\tan \left( \theta + 45 ^ { \circ } \right)\),
2. \(\operatorname { cosec } \theta\).

3 The volume, \(V\) cubic metres, of water in a reservoir is given by $$V = 3 ( 2 + \sqrt { h } ) ^ { 6 } - 192 ,$$ where \(h\) metres is the depth of the water. Water is flowing into the reservoir at a constant rate of 150 cubic metres per hour. Find the rate at which the depth of water is increasing at the instant when the depth is 1.4 metres.

4 It is given that \(| x + 3 a | = 5 a\), where \(a\) is a positive constant. Find, in terms of \(a\), the possible values of $$| x + 7 a | - | x - 7 a |$$

5
\includegraphics[max width=\textwidth, alt={}, center]{00a4be37-c095-4d9c-a1cd-d03b8ab1d411-2_455_643_1327_694} The diagram shows the curve \(y = \mathrm { e } ^ { 3 x } - 6 \mathrm { e } ^ { 2 x } + 32\).

1. Find the exact \(x\)-coordinate of the minimum point and verify that the \(y\)-coordinate of the minimum point is 0 .
2. Find the exact area of the region (shaded in the diagram) enclosed by the curve and the axes.

6
\includegraphics[max width=\textwidth, alt={}, center]{00a4be37-c095-4d9c-a1cd-d03b8ab1d411-3_553_579_274_726} The diagram shows the curve \(y = 8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right)\). The end-points \(A\) and \(B\) of the curve have coordinates ( \(a , - 4 \pi\) ) and ( \(b , 4 \pi\) ) respectively.

1. State the values of \(a\) and \(b\).
2. It is required to find the root of the equation \(8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right) = x\).
   (a) Show by calculation that the root lies between 1.7 and 1.8.
   (b) In order to find the root, the iterative formula $$x _ { n + 1 } = p + \sin \left( q x _ { n } \right) ,$$ with a suitable starting value, is to be used. Determine the values of the constants \(p\) and \(q\) and hence find the root correct to 4 significant figures. Show the result of each step of the iteration process.

7

1. Find the exact value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\).
2. Use Simpson's rule with two strips to show that an approximate value of \(\int _ { 1 } ^ { 9 } ( 7 x + 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\) can be expressed in the form \(m + n \sqrt [ 3 ] { 36 }\), where the values of the constants \(m\) and \(n\) are to be stated.
3. Use the results from parts (i) and (ii) to find an approximate value of \(\sqrt [ 3 ] { 36 }\), giving your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers. \section*{Question 8 begins on page 4.}

8 The functions \(f\) and \(g\) are defined as follows: $$\begin{gathered} \mathrm { f } ( x ) = 2 + \ln ( x + 3 ) \text { for } x \geqslant 0
\mathrm {~g} ( x ) = a x ^ { 2 } \text { for all real values of } x , \text { where } a \text { is a positive constant. } \end{gathered}$$

1. Given that \(\operatorname { gf } \left( \mathrm { e } ^ { 4 } - 3 \right) = 9\), find the value of \(a\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
3. Given that \(\mathrm { ff } \left( \mathrm { e } ^ { N } - 3 \right) = \ln \left( 53 \mathrm { e } ^ { 2 } \right)\), find the value of \(N\).

9 It is given that \(\mathrm { f } ( \theta ) = \sin \left( \theta + 30 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right)\).

1. Show that \(\mathrm { f } ( \theta ) = \cos \theta\). Hence show that $$f ( 4 \theta ) + 4 f ( 2 \theta ) \equiv 8 \cos ^ { 4 } \theta - 3 .$$
2. Hence
   (a) determine the greatest and least values of \(\frac { 1 } { \mathrm { f } ( 4 \theta ) + 4 \mathrm { f } ( 2 \theta ) + 7 }\) as \(\theta\) varies,
   (b) solve the equation $$\sin \left( 12 \alpha + 30 ^ { \circ } \right) + \cos \left( 12 \alpha + 60 ^ { \circ } \right) + 4 \sin \left( 6 \alpha + 30 ^ { \circ } \right) + 4 \cos \left( 6 \alpha + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < \alpha < 60 ^ { \circ }\). \section*{END OF QUESTION PAPER}