(i) Sketch a diagram to illustrate this information. Label the points \({\text{A, B, C}}\). Show the angles \({90^ \circ }\), \({60^ \circ }\) and the length \(7.3{\text{ cm}}\) on your diagram.

(ii) Find the length of \({\text{BC}}\).

Point \({\text{D}}\) is on the straight line \({\text{AC}}\) extended and is such that angle \({\text{CDB}}\) is \({20^ \circ }\).

(i) Show the point \({\text{D}}\) and the angle \({20^ \circ }\) on your diagram.

(ii) Find the size of angle \({\text{CBD}}\).

Unit penalty (UP) is applicable where indicated in the left hand column.

(i)

     (A1)

For \({\text{A}}\), \({\text{B}}\), \({\text{C}}\), \(7.3\), \({60^ \circ }\), \({90^ \circ }\), shown in correct places     (A1)

Note: The \({90^ \circ }\) should look like \({90^ \circ }\) (allow \( \pm {10^ \circ }\))

(ii) Using \(\tan 60\) or \(\tan 30\)     (M1)

(UP)     \(4.21{\text{ cm}}\)     (A1)(ft)

Note: (ft) on their diagram

Or

Using sine rule with their correct values     (M1)

(UP)     \( = 4.21{\text{ cm}}\)     (A1)(ft)

Or

Using special triangle \(\frac{{7.3}}{{\sqrt 3 }}\)     (M1)

(UP)     \(4.21{\text{ cm}}\)     (A1)(ft)

Or

Any other valid solution

Note: If A and B are swapped then \({\text{BC}} = 8.43{\text{ cm}}\)     (C3)

[3 marks]

(i) For \({\text{ACD}}\) in a straight line and all joined up to \({\text{B}}\), for \({20^ \circ }\) shown in correct place and \({\text{D}}\) labelled. \({\text{D}}\) must be on \({\text{AC}}\) extended.     (A1)

(ii) \({\text{B}}\hat {\text{C}}{\text{D}} = {120^ \circ }\)     (A1)

\({\text{C}}\hat {\text{B}}{\text{D}} = {40^ \circ }\)     (A1)     (C3)

[3 marks]

The initial diagram was well drawn by most candidates but few could extend \({\text{AC}}\) to find \({\text{D}}\). The point \({\text{D}}\) was either drawn between \({\text{A}}\) and \({\text{C}}\) or on \({\text{CA}}\) extended. When on \({\text{CA}}\) extended the candidates could be awarded A1 follow through for the angle. A surprising number of candidates could not find the correct answer for the length of \({\text{BC}}\).

The initial diagram was well drawn by most candidates but few could extend \({\text{AC}}\) to find \({\text{D}}\). The point \({\text{D}}\) was either drawn between \({\text{A}}\) and \({\text{C}}\) or on \({\text{CA}}\) extended. When on \({\text{CA}}\) extended the candidates could be awarded A1 follow through for the angle. A surprising number of candidates could not find the correct answer for the length of \({\text{BC}}\).