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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2017

Paper 1

Put $L=|x_n|$ and choose perpendicular unit vectors $e,f$ such that $x_n=Le$ and $x_{n+1}=\lambda Lf$. Direction vectors for the two lines are
$$
x_{n+2}-x_{n+1}=-\lambda L(\lambda e+f)
$$
and
$$
x_{n+3}-x_n=-L(e+\lambda^3f).
$$
Their scalar product and norms give
$$
\cos\theta
=\frac{\lambda^2L^2(1+\lambda^2)}
{\lambda L^2\sqrt{1+\lambda^2}\sqrt{1+\lambda^6}}.
$$
Therefore the acute angle between the straight lines satisfies
$$
\boxed{\cos\theta
=\frac{\lambda\sqrt{1+\lambda^2}}{\sqrt{1+\lambda^6}}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 5A / b / iv / Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 5A / b / iv / Solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 5A / b / iv / Solution

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