OCR MEI C3 (Core Mathematics 3) 2011 June

Solve the equation \(|2x - 1| = |x|\). [4]

Given that \(f(x) = 2\ln x\) and \(g(x) = e^x\), find the composite function \(gf(x)\), expressing your answer as simply as possible. [3]

1. Differentiate \(\frac{\ln x}{x^2}\), simplifying your answer. [4]
2. Using integration by parts, show that \(\int \frac{\ln x}{x^2} \, dx = -\frac{1}{x}(1 + \ln x) + c\). [4]

The height \(h\) metres of a tree after \(t\) years is modelled by the equation $$h = a - be^{-kt},$$ where \(a\), \(b\) and \(k\) are positive constants.

1. Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres, find the values of \(a\) and \(b\). [3]
2. Given also that the tree grows to a height of 6 metres in 8 years, find the value of \(k\), giving your answer correct to 2 decimal places. [3]

Given that \(y = x^2\sqrt{1 + 4x}\), show that \(\frac{dy}{dx} = \frac{2x(5x + 1)}{\sqrt{1 + 4x}}\). [5]

A curve is defined by the equation \(\sin 2x + \cos y = \sqrt{3}\).

1. Verify that the point P \((\frac{\pi}{6}, \frac{\pi}{6})\) lies on the curve. [1]
2. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at the point P. [5]

1. Multiply out \((3^n + 1)(3^n - 1)\). [1]
2. Hence prove that if \(n\) is a positive integer then \(3^{2n} - 1\) is divisible by 8. [3]

\includegraphics{figure_8} Fig. 8 shows the curve \(y = f(x)\), where \(f(x) = \frac{1}{e^x + e^{-x} + 2}\).

1. Show algebraically that \(f(x)\) is an even function, and state how this property relates to the curve \(y = f(x)\). [3]
2. Find \(f'(x)\). [3]
3. Show that \(f(x) = \frac{e^x}{(e^x + 1)^2}\). [2]
4. Hence, using the substitution \(u = e^x + 1\), or otherwise, find the exact area enclosed by the curve \(y = f(x)\), the \(x\)-axis, and the lines \(x = 0\) and \(x = 1\). [5]
5. Show that there is only one point of intersection of the curves \(y = f(x)\) and \(y = \frac{1}{4}e^x\), and find its coordinates. [5]

Fig. 9 shows the curve \(y = f(x)\). The endpoints of the curve are P \((-\pi, 1)\) and Q \((\pi, 3)\), and \(f(x) = a + \sin bx\), where \(a\) and \(b\) are constants. \includegraphics{figure_9}

1. Using Fig. 9, show that \(a = 2\) and \(b = \frac{1}{2}\). [3]
2. Find the gradient of the curve \(y = f(x)\) at the point \((0, 2)\). Show that there is no point on the curve at which the gradient is greater than this. [5]
3. Find \(f^{-1}(x)\), and state its domain and range. Write down the gradient of \(y = f^{-1}(x)\) at the point \((2, 0)\). [6]
4. Find the area enclosed by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \pi\). [4]