$($a$)$ Describe two contrasting situations where eccentric corrections might prove useful to a surveyor doing fieldwork.

$($b$)$ Using the notation in the above sketch and the Sine rule, derive an expression for $a_1$ in termsof $d,$ Join A and ${}_1.$

$($c$)$ If Join A is $275{46}^{\text{'}},$ Join B is $83{13}^{\text{'}},{}_1$ is $1{22}^{\text{'}}$ and ${}_2$ is $0{56}^{\text{'}},$ then calculate the oriented bearings from the Eccentric setup to Trig A and to Trig B$.$

$($d$)$ Using the notation in the above sketch and the Cosine rule, derive an expression for the distance Eccentric $-$ Trig B in terms of Join B$,$ d$,{}_2$ and ${}_2.$

$($e$)$ Draw a sketch of an ideal traverse and list five important features of such a traverse.

$($f$)$ Using sketches, give three cases of errors going unnoticed in a circular traverse.

$($a$)$ Explain what limitations and disadvantages are associated with a laser scanner?

$($b$)$ Describe two alternative methods for taking into account refraction in trigonometrical heighting over long distances. Also explain what checks you would carry out.

$($c$)$ What is a coordinate transformation? Explain fully how you would carry out a simple transformation using a hand calculator in a situation where there are three survey marks, two of which have coordinates in both an old and a new survey system and the third of which has coordinates in the old system only. Detail how you would obtain coordinates for the third mark in the new system and also how you would check these coordinates.

$($d$)$ For a minimum constraint resection $(3$ fixed points$)$ draw a diagram of the three main scenarios possible. For each case, draw the tangents of the fixing lines and the circles they are tangent to$,$ and highlight any potential geometrical weakness.

$($e$)$ Draw a labelled sketch to illustrate how a combination of GPS and traditional methods $($using distances and bearings from a Total Station$)$ may save time and cost in a survey. What check could be done, for example, to confirm that the EDM does not have a scale factor and

that there have been no errors in the polar calculations?