OCR C3 (Core Mathematics 3) 2014 June

1 Given that \(y = 4 x ^ { 2 } \ln x\), find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = \mathrm { e } ^ { 2 }\).

2 By first using appropriate identities, solve the equation $$5 \cos 2 \theta \operatorname { cosec } \theta = 2$$ for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

3

1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sqrt { x } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
2. Deduce an approximation to \(\int _ { 0 } ^ { 2 } \left( 1 + 10 \mathrm { e } ^ { \sqrt { x } } \right) \mathrm { d } x\).

4 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 3 } + 4 \quad \text { and } \quad \mathrm { g } ( x ) = \sqrt [ 3 ] { x - 10 }$$

1. Evaluate \(\mathrm { f } ^ { - 1 } ( - 50 )\).
2. Show that \(\operatorname { fg } ( x ) = 2 x - 16\).
3. Differentiate \(\operatorname { gf } ( x )\) with respect to \(x\).

5

1. The mass, \(M\) grams, of a substance at time \(t\) years is given by $$M = 58 \mathrm { e } ^ { - 0.33 t }$$ Find the rate at which the mass is decreasing at the instant when \(t = 4\). Give your answer correct to 2 significant figures.
2. The mass of a second substance is increasing exponentially. The initial mass is 42.0 grams and, 6 years later, the mass is 51.8 grams. Find the mass at a time 24 years after the initial value.

6
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_524_720_246_676} The diagram shows the curve \(y = x ^ { 4 } - 8 x\).

1. By sketching a second curve on the copy of the diagram, show that the equation $$x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0$$ has two real roots. State the equation of the second curve.
2. The larger root of the equation \(x ^ { 4 } + x ^ { 2 } - 8 x - 9 = 0\) is denoted by \(\alpha\).
   (a) Show by calculation that \(2.1 < \alpha < 2.2\).
   (b) Use an iterative process based on the equation $$x = \sqrt [ 4 ] { 9 + 8 x - x ^ { 2 } } ,$$ with a suitable starting value, to find \(\alpha\) correct to 3 decimal places. Give the result of each step of the iterative process.

7
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-3_547_851_1749_605} The diagram shows the curve \(y = \sqrt { \frac { 3 } { 4 x + 1 } }\) for \(0 \leqslant x \leqslant 20\). The point \(P\) on the curve has coordinates \(\left( 20 , \frac { 1 } { 9 } \sqrt { 3 } \right)\). The shaded region \(R\) is enclosed by the curve and the lines \(x = 0\) and \(y = \frac { 1 } { 9 } \sqrt { 3 }\).

1. Find the exact area of \(R\).
2. Find the exact volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis.

8
\includegraphics[max width=\textwidth, alt={}, center]{33a2b09d-0df9-48d6-9ee9-e0a1ec345f41-4_616_1024_296_516} The diagram shows the curve \(y = \frac { 2 x + 4 } { x ^ { 2 } + 5 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of the two stationary points.
2. The function g is defined for all real values of \(x\) by $$\mathrm { g } ( x ) = \left| \frac { 2 x + 4 } { x ^ { 2 } + 5 } \right| .$$ (a) Sketch the curve \(y = \mathrm { g } ( x )\) and state the range of g .
   (b) It is given that the equation \(\mathrm { g } ( x ) = k\), where \(k\) is a constant, has exactly two distinct real roots. Write down the set of possible values of \(k\).

9

1. Express \(5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence
   (a) give details of the transformations needed to transform the curve \(y = 5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) to the curve \(y = \sin \theta\),
   (b) find the smallest positive value of \(\beta\) satisfying the equation $$5 \cos \left( \frac { 1 } { 3 } \beta - 40 ^ { \circ } \right) + 3 \cos \left( \frac { 1 } { 3 } \beta + 20 ^ { \circ } \right) = 3 .$$ \section*{END OF QUESTION PAPER}