OCR MEI C3 (Core Mathematics 3) 2014 June

Evaluate \(\int_0^{\frac{\pi}{4}} (1 - \sin 3x) \, dx\), giving your answer in exact form. [3]

Find the exact gradient of the curve \(y = \ln(1 - \cos 2x)\) at the point with \(x\)-coordinate \(\frac{1}{4}\pi\). [5]

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

Fig. 4 shows the curve \(y = f(x)\), where $$f(x) = a + \cos bx, \quad 0 \leq x \leq 2\pi,$$ and \(a\) and \(b\) are positive constants. The curve has stationary points at \((0, 3)\) and \((2\pi, 1)\). \includegraphics{figure_4}

1. Find \(a\) and \(b\). [2]
2. Find \(f^{-1}(x)\), and state its domain and range. [5]

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

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

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]
2. 'If \(m\) and \(n\) are consecutive even numbers, then \(mn\) is divisible by 8.' [2]

Fig. 8 shows the curve \(y = f(x)\), where \(f(x) = \frac{x}{\sqrt{2 + x^2}}\). \includegraphics{figure_8}

1. Show algebraically that \(f(x)\) is an odd function. Interpret this result geometrically. [3]
2. Show that \(f'(x) = \frac{2}{(2 + x^2)^{\frac{3}{2}}}\). Hence find the exact gradient of the curve at the origin. [5]
3. Find the exact area of the region bounded by the curve, the \(x\)-axis and the line \(x = 1\). [4]
4. 1. Show that if \(y = \frac{x}{\sqrt{2 + x^2}}\), then \(\frac{1}{y^2} = \frac{2}{x^2} + 1\). [2]
   2. Differentiate \(\frac{1}{y^2} = \frac{2}{x^2} + 1\) implicitly to show that \(\frac{dy}{dx} = \frac{2y^3}{x^3}\). Explain why this expression cannot be used to find the gradient of the curve at the origin. [4]

Fig. 9 shows the curve \(y = xe^{-2x}\) together with the straight line \(y = mx\), 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. \includegraphics{figure_9}

1. Show that the \(x\)-coordinate of P is \(-\frac{1}{2}\ln m\). [3]
2. Find, in terms of \(m\), the gradient of the tangent to the curve at P. [4]

You are given that OP and this tangent are equally inclined to the \(x\)-axis.

1. Show that \(m = e^{-2}\), and find the exact coordinates of P. [4]
2. Find the exact area of the shaded region between the line OP and the curve. [7]