OCR MEI C4 (Core Mathematics 4)

1 Using partial fractions, find \(\int \frac { x } { ( x + 1 ) ( 2 x + 1 ) } \mathrm { d } x\).

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 \(\pi\).
2. Write down the derivative of \(\tan \theta\). Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { ( \cos \theta + \sqrt { 3 } \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { \sqrt { 3 } } { 4 }\).

3 In a chemical process, the mass \(\boldsymbol { M }\) grams of a chemical at time \(\boldsymbol { t }\) minutes is modelled by the differential equation $$\underset { d t } { d \underline { \underline { M } } } - \underset { \mathrm { z } \left( \left. \right| ^ { \prime } + \mathrm { z } ^ { 2 } \right) ^ { \prime } } { M _ { - } }$$

1. Find \({ } _ { \mathrm { f } } \overline { 1 } ; \overline { 12 } d t\)
   (li) Find constants \(\boldsymbol { A } , \boldsymbol { B }\) and \(\boldsymbol { C }\) such lhat $$\begin{array} { c c } 1 & B t + C
   \hdashline t \left( I + t ^ { 2 } \right) & I \end{array} . + \begin{array} { c c } I & I + 1 ^ { 2 } . \end{array}$$
2. Use integration, together with your results in parts (i) and (ii), to show that $$M = \frac { K t } { J \sqrt { + , 2 } } ,$$ where \(\boldsymbol { K }\) is a constant.
3. When \(\boldsymbol { t } = \mathrm { I } , \boldsymbol { M } = 25\). Calculate \(\boldsymbol { K }\) What is the mass of the chemical in the long term?

4 The growth of a tree is modelled by the differential equation $$10 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 20 - h$$ 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 \(\frac { \mathrm { d } h } { \mathrm {~d} t } = 0\), and interpret this in terms of the growth of the tree.
2. Verify that \(h = 20 \left( 1 - \mathrm { e } ^ { - 0.1 t } \right)\) satisfies this differential equation and its initial condition. The alternative differential equation $$200 \frac { \mathrm {~d} h } { \mathrm {~d} t } = 400 - h ^ { 2 }$$ is proposed to model the growth of the tree. As before, \(h = 0\) when \(t = 0\).
3. Using partial fractions, show by integration that the solution to the alternative differential equation is $$h = \frac { 20 \left( 1 - \mathrm { e } ^ { - 0.2 t } \right) } { 1 + \mathrm { e } ^ { - 0.2 t } }$$
4. What does this solution indicate about the long-term height of the tree?
5. After a year, the tree has grown to a height of 2 m . Which model fits this information better?

5

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 \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to
   4 significant figures.
3. Given that \(\int \frac { 1 } { \sqrt { 4 x ^ { 2 } } } \mathrm {~d} x = \arcsin \left( \frac { 1 } { 2 } x \right) + c\), evaluate \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 x ^ { 2 } } } \mathrm {~d} x\), rounding your answer to 4 significant figures.