OCR MEI C3 (Core Mathematics 3) 2012 January

Differentiate \(x^2 \tan 2x\). [3]

The functions \(\text{f}(x)\) and \(\text{g}(x)\) are defined as follows. $$\text{f}(x) = \ln x, \quad x > 0$$ $$\text{g}(x) = 1 + x^2, \quad x \in \mathbb{R}$$ Write down the functions \(\text{fg}(x)\) and \(\text{gf}(x)\), and state whether these functions are odd, even or neither. [4]

Show that \(\int_0^{\frac{\pi}{2}} x \cos \frac{1}{2} x \, dx = \frac{\sqrt{2}}{2} \pi + 2\sqrt{2} - 4\). [5]

Prove or disprove the following statement: 'No cube of an integer has 2 as its units digit.' [2]

Each of the graphs of \(y = \text{f}(x)\) and \(y = \text{g}(x)\) below is obtained using a sequence of two transformations applied to the corresponding dashed graph. In each case, state suitable transformations, and hence find expressions for \(\text{f}(x)\) and \(\text{g}(x)\).

1. \includegraphics{figure_5i} [3]
2. \includegraphics{figure_5ii} [3]

Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius \(r\) metres of the oil slick \(t\) hours after the start of the leak is modelled by the equation $$r = 20(1 - e^{-0.2t}).$$

1. Find the radius of the slick when \(t = 2\), and the rate at which the radius is increasing at this time. [4]
2. Find the rate at which the area of the slick is increasing when \(t = 2\). [4]

Fig. 7 shows the curve \(x^3 + y^3 = 3xy\). The point P is a turning point of the curve. \includegraphics{figure_7}

1. Show that \(\frac{dy}{dx} = \frac{y - x^2}{y^2 - x}\). [4]
2. Hence find the exact \(x\)-coordinate of P. [4]

Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x-2}}\), together with the lines \(y = x\) and \(x = 11\). The curve meets these lines at P and Q respectively. R is the point \((11, 11)\). \includegraphics{figure_8}

1. Verify that the \(x\)-coordinate of P is 3. [2]
2. Show that, for the curve, \(\frac{dy}{dx} = \frac{x-4}{2(x-2)^{\frac{3}{2}}}\). Hence find the gradient of the curve at P. Use the result to show that the curve is not symmetrical about \(y = x\). [7]
3. Using the substitution \(u = x - 2\), show that \(\int_3^{11} \frac{x}{\sqrt{x-2}} \, dx = 25\frac{1}{3}\). Hence find the area of the region PQR bounded by the curve and the lines \(y = x\) and \(x = 11\). [9]

Fig. 9 shows the curves \(y = \text{f}(x)\) and \(y = \text{g}(x)\). The function \(y = \text{f}(x)\) is given by $$\text{f}(x) = \ln \left( \frac{2x}{1+x} \right), \quad x > 0.$$ The curve \(y = \text{f}(x)\) crosses the \(x\)-axis at P, and the line \(x = 2\) at Q. \includegraphics{figure_9}

1. Verify that the \(x\)-coordinate of P is 1. Find the exact \(y\)-coordinate of Q. [2]
2. Find the gradient of the curve at P. [Hint: use \(\frac{a}{b} = \ln a - \ln b\).] [4]

The function \(\text{g}(x)\) is given by $$\text{g}(x) = \frac{e^x}{2-e^x}, \quad x < \ln 2.$$ The curve \(y = \text{g}(x)\) crosses the \(y\)-axis at the point R.

1. Show that \(\text{g}(x)\) is the inverse function of \(\text{f}(x)\). Write down the gradient of \(y = \text{g}(x)\) at R. [5]
2. Show, using the substitution \(u = 2 - e^x\) or otherwise, that \(\int_0^{\ln \frac{4}{3}} \text{g}(x) dx = \ln \frac{3}{2}\). Using this result, show that the exact area of the shaded region shown in Fig. 9 is \(\ln \frac{32}{27}\). [Hint: consider its reflection in \(y = x\).] [7]