4 Find the first three terms in the binomial expansion of \(\sqrt { 4 + x }\) in ascending powers of \(x\).
State the set of values of \(x\) for which the expansion is valid.
show that \(\frac { y - 2 } { y + 1 } = A \mathrm { e } ^ { x ^ { 3 } }\), where \(A\) is a constant.
(ii) Hence, given that \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } ( y - 2 ) ( y + 1 ) ,$$ 6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)
7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

# Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sqrt{4+x} = 2\left(1+\frac{x}{4}\right)^{\frac{1}{2}}$ | M1 | dealing with $\sqrt{4}$ (or terms in $4^{\frac{1}{2}}, 4^{-\frac{1}{2}}$, etc) |
| $= 2\left(1 + \frac{1}{2}\cdot\frac{x}{4} + \frac{\frac{1}{2}\cdot\frac{-1}{2}}{2}\left(\frac{x}{4}\right)^2 + \ldots\right)$ | M1, A1 | correct binomial coefficients; correct unsimplified expression for $(1+x/4)^{\frac{1}{2}}$ or $(4+x)^{\frac{1}{2}}$ |
| $= 2\left(1 + \frac{1}{8}x - \frac{1}{128}x^2 + \ldots\right)$ | | |
| $= 2 + \frac{1}{4}x - \frac{1}{64}x^2 + \ldots$ | A1 | cao |
| Valid for $-1 < x/4 < 1 \Rightarrow -4 < x < 4$ | B1 [5] | |

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4 Find the first three terms in the binomial expansion of $\sqrt { 4 + x }$ in ascending powers of $x$.\\
State the set of values of $x$ for which the expansion is valid.\\
show that $\frac { y - 2 } { y + 1 } = A \mathrm { e } ^ { x ^ { 3 } }$, where $A$ is a constant.\\
(ii) Hence, given that $x$ and $y$ satisfy the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } ( y - 2 ) ( y + 1 ) ,$$

6 Solve the equation $\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }$.

Section B (36 marks)\\
7 A straight pipeline AB passes through a mountain. With respect to axes $\mathrm { O } x y z$, with $\mathrm { O } x$ due East, $\mathrm { O } y$ due North and $\mathrm { O } z$ vertically upwards, A has coordinates $( - 200,100,0 )$ and B has coordinates $( 100,200,100 )$, where units are metres.\\

\hfill \mbox{\textit{OCR MEI C4 2010 Q4 [5]}}