OCR MEI C4 (Core Mathematics 4)

Using partial fractions, find \(\int \frac{x}{(x+1)(2x+1)} dx\). [7]

1. Express \(\cos \theta + \sqrt{3} \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(\alpha\) is acute, expressing \(\alpha\) in terms of \(\pi\). [4]
2. Write down the derivative of \(\tan \theta\). Hence show that \(\int_0^{\frac{\pi}{6}} \frac{1}{(\cos \theta + \sqrt{3} \sin \theta)^2} d\theta = \frac{\sqrt{3}}{4}\). [4]

In a chemical process, the mass \(M\) grams of a chemical at time \(t\) minutes is modelled by the differential equation $$\frac{dM}{dt} = -\frac{M}{2(1 + \frac{t}{2})}$$

1. Find \(\int \frac{1}{1 + \frac{t}{2}} dt\) [3]
2. Find constants \(A\), \(B\) and \(C\) such that $$\frac{1}{t(1 + \frac{t}{2})} = \frac{A}{t} + \frac{Bt + C}{1 + \frac{t}{2}}$$ [5]
3. Use integration, together with your results in parts (i) and (ii), to show that $$M \sim \frac{K}{.1 + \frac{t}{2}}$$ where \(K\) is a constant. [6]
4. When \(t = 1\), \(M = 25\). Calculate \(K\) What is the mass of the chemical in the long term? [4]

The growth of a tree is modelled by the differential equation $$10\frac{dh}{dt} = 20 - h$$ where \(h\) is its height in metres and the time \(t\) is in years. It is assumed that the tree is grown from seed, so that \(h = 0\) when \(t = 0\).

1. Write down the value of \(h\) for which \(\frac{dh}{dt} = 0\), and interpret this in terms of the growth of the tree. [1]
2. Verify that \(h = 20(1 - e^{-0.1t})\) satisfies this differential equation and its initial condition. [5]

The alternative differential equation $$200\frac{dh}{dt} = 400 - h^2$$ is proposed to model the growth of the tree. As before, \(h = 0\) when \(t = 0\).

1. Using partial fractions, show by integration that the solution to the alternative differential equation is $$h = \frac{20(1 - e^{-0.2t})}{1 + e^{-0.2t}}$$ [9]
2. What does this solution indicate about the long-term height of the tree? [1]
3. After a year, the tree has grown to a height of 2m. Which model fits this information better? [3]

1. Find the first three non-zero terms of the binomial expansion of \(\frac{1}{\sqrt{4-x^2}}\) for \(|x| < 2\). [4]
2. Use this result to find an approximation for \(\int_0^1 \frac{1}{\sqrt{4-x^2}} dx\), rounding your answer to 4 significant figures. [2]
3. Given that \(\int \frac{1}{\sqrt{4-x^2}} dx = \arcsin\left(\frac{1}{2}x\right) + c\), evaluate \(\int_0^1 \frac{1}{\sqrt{4-x^2}} dx\), rounding your answer to 4 significant figures. [1]