(a) Write \(7\sqrt{3}\left(5\sqrt{3} - 4\right) + \sqrt{27}\) in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [3]
(b) In this question all lengths are in centimetres.
Diagram not drawn to scale
This shape is made from two rectangles. The area of this shape is \(33\sqrt{2} - 18\).
Showing all your working, find the value of \(x\). Give your answer in the form \(c + d\sqrt{2}\), where \(c\) and \(d\) are integers. [5]
Mark scheme (a)
Scheme
Marks
Comments
\(35\times3 - 28\sqrt{3} + 3\sqrt{3}\) or better
M2
with at most one sign or arithmetic slip
M1 for any one of 105, \(-28\sqrt{3}\), \(3\sqrt{3}\) si
\(105 - 25\sqrt{3}\)
A1
Allow e.g. \(-25\sqrt{3} + 105\);
Mark scheme (b)
Scheme
Marks
Comments
\(2\left(6 - \sqrt{2}\right) + x\times5\sqrt{2}\) \((= 33\sqrt{2} - 18)\) si
B1
or \(2\left(6 - \sqrt{2} - x\right) + x(5\sqrt{2} + 2)\) \((= 33\sqrt{2} - 18)\) si omitted brackets may be recovered in later work
Forms a correct equation for the area and either isolates \(x\) term: \(5x\sqrt{2} = 33\sqrt{2} - 18 - 12 + 2\sqrt{2}\) or collects terms: \(30 + 5x\sqrt{2} = 35\sqrt{2}\)
M1
FT ‘their \(2\left(6 - \sqrt{2}\right) + x\times5\sqrt{2}\)’ \(= 33\sqrt{2} - 18\) oe providing expression is dimensionally correct for an area and at most one error in the expression;
allow one further error in forming and rearranging the equation e.g. sign, bracketing or arithmetic slip
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