OCR C3 (Core Mathematics 3) 2009 June

\includegraphics{figure_1} Each diagram above shows part of a curve, the equation of which is one of the following: $$y = \sin^{-1} x, \quad y = \cos^{-1} x, \quad y = \tan^{-1} x, \quad y = \sec x, \quad y = \cosec x, \quad y = \cot x.$$ State which equation corresponds to

1. Fig. 1, [1]
2. Fig. 2, [1]
3. Fig. 3. [1]

\includegraphics{figure_2} The diagram shows the curve with equation \(y = (2x - 3)^2\). The shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\). Find the exact volume obtained when the shaded region is rotated completely about the \(x\)-axis. [5]

The angles \(\alpha\) and \(\beta\) are such that $$\tan \alpha = m + 2 \quad \text{and} \quad \tan \beta = m,$$ where \(m\) is a constant.

1. Given that \(\sec^2 \alpha - \sec^2 \beta = 16\), find the value of \(m\). [3]
2. Hence find the exact value of \(\tan(\alpha + \beta)\). [3]

It is given that \(\int_a^{3a} (e^{5x} + e^x) dx = 100\), where \(a\) is a positive constant.

1. Show that \(a = \frac{1}{5}\ln(300 + 3e^a - 2e^{3a})\). [5]
2. Use an iterative process, based on the equation in part (i), to find the value of \(a\) correct to 4 decimal places. Use a starting value of 0.6 and show the result of each step of the process. [4]

The functions f and g are defined for all real values of \(x\) by $$f(x) = 3x - 2 \quad \text{and} \quad g(x) = 3x + 7.$$ Find the exact coordinates of the point at which

1. the graph of \(y = fg(x)\) meets the \(x\)-axis, [3]
2. the graph of \(y = g(x)\) meets the graph of \(y = g^{-1}(x)\), [3]
3. the graph of \(y = |f(x)|\) meets the graph of \(y = |g(x)|\). [4]

\includegraphics{figure_3} The diagram shows the curve with equation \(x = (37 + 10y - 2y^2)^{\frac{1}{2}}\).

1. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [2]
2. Hence find the equation of the tangent to the curve at the point \((7, 3)\), giving your answer in the form \(y = mx + c\). [5]

1. Express \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin(\theta - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
2. Hence
   1. solve, for \(0° < \theta < 360°\), the equation \(8 \sin \theta - 6 \cos \theta = 9\), [4]
   2. find the greatest possible value of $$32 \sin x - 24 \cos x - (16 \sin y - 12 \cos y)$$ as the angles \(x\) and \(y\) vary. [3]

\includegraphics{figure_4} The diagram shows the curves \(y = \ln x\) and \(y = 2 \ln(x - 6)\). The curves meet at the point \(P\) which has \(x\)-coordinate \(a\). The shaded region is bounded by the curve \(y = 2 \ln(x - 6)\) and the lines \(x = a\) and \(y = 0\).

1. Give details of the pair of transformations which transforms the curve \(y = \ln x\) to the curve \(y = 2 \ln(x - 6)\). [3]
2. Solve an equation to find the value of \(a\). [4]
3. Use Simpson's rule with two strips to find an approximation to the area of the shaded region. [3]

1. Show that, for all non-zero values of the constant \(k\), the curve $$y = \frac{kx^2 - 1}{kx^2 + 1}$$ has exactly one stationary point. [5]
2. Show that, for all non-zero values of the constant \(m\), the curve $$y = e^{mx}(x^2 + mx)$$ has exactly two stationary points. [7]