OCR MEI C3 (Core Mathematics 3) 2011 January

Given that \(y = \sqrt[3]{1 + x^2}\), find \(\frac{dy}{dx}\). [4]

Solve the inequality \(|2x + 1| \geqslant 4\). [4]

The area of a circular stain is growing at a rate of \(1 \text{ mm}^2\) per second. Find the rate of increase of its radius at an instant when its radius is \(2\) mm. [5]

Use the triangle in Fig. 4 to prove that \(\sin^2 \theta + \cos^2 \theta = 1\). For what values of \(\theta\) is this proof valid? [3] \includegraphics{figure_4}

1. On a single set of axes, sketch the curves \(y = e^x - 1\) and \(y = 2e^{-x}\). [3]
2. Find the exact coordinates of the point of intersection of these curves. [5]

A curve is defined by the equation \((x + y)^2 = 4x\). The point \((1, 1)\) lies on this curve. By differentiating implicitly, show that \(\frac{dy}{dx} = \frac{2}{x + y} - 1\). Hence verify that the curve has a stationary point at \((1, 1)\). [4]

Fig. 7 shows the curve \(y = f(x)\), where \(f(x) = 1 + 2 \arctan x\), \(x \in \mathbb{R}\). The scales on the \(x\)- and \(y\)-axes are the same. \includegraphics{figure_7}

1. Find the range of f, giving your answer in terms of \(\pi\). [3]
2. Find \(f^{-1}(x)\), and add a sketch of the curve \(y = f^{-1}(x)\) to the copy of Fig. 7. [5]

1. Use the substitution \(u = 1 + x\) to show that $$\int_0^1 \frac{x^3}{1 + x} dx = \int_a^b \left( u^2 - 3u + 3 - \frac{1}{u} \right) du,$$ where \(a\) and \(b\) are to be found. Hence evaluate \(\int_0^1 \frac{x^3}{1 + x} dx\), giving your answer in exact form. [7] Fig. 8 shows the curve \(y = x^2 \ln(1 + x)\). \includegraphics{figure_8}
2. Find \(\frac{dy}{dx}\). Verify that the origin is a stationary point of the curve. [5]
3. Using integration by parts, and the result of part (i), find the exact area enclosed by the curve \(y = x^2 \ln(1 + x)\), the \(x\)-axis and the line \(x = 1\). [6]

Fig. 9 shows the curve \(y = f(x)\), where \(f(x) = \frac{1}{\cos^2 x}\), \(-\frac{1}{2}\pi < x < \frac{1}{2}\pi\), together with its asymptotes \(x = \frac{1}{2}\pi\) and \(x = -\frac{1}{2}\pi\). \includegraphics{figure_9}

1. Use the quotient rule to show that the derivative of \(\frac{\sin x}{\cos x}\) is \(\frac{1}{\cos^2 x}\). [3]
2. Find the area bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac{1}{4}\pi\). [3]

The function \(g(x)\) is defined by \(g(x) = \frac{1}{2}f(x + \frac{1}{4}\pi)\).

1. Verify that the curves \(y = f(x)\) and \(y = g(x)\) cross at \((0, 1)\). [3]
2. State a sequence of two transformations such that the curve \(y = f(x)\) is mapped to the curve \(y = g(x)\). On the copy of Fig. 9, sketch the curve \(y = g(x)\), indicating clearly the coordinates of the minimum point and the equations of the asymptotes to the curve. [8]
3. Use your result from part (ii) to write down the area bounded by the curve \(y = g(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = -\frac{1}{4}\pi\). [1]