2025 Nov TZ3 P1 Q7

Consider a triangle \(\mathrm{PQR}\) where \(\mathrm{PR} = 3, \mathrm{QR} = 2\) and \(\mathrm{P\hat{R}Q} = 60^{\circ}\).

1. Find \(\mathrm{PQ}\).

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Now consider a general triangle \(\mathrm{ABC}\) where \(\mathrm{AC} = x + 2, \mathrm{BC} = 3x - 1\) and \(\mathrm{A\hat{C}B} = 60^{\circ}\).

2. Show that \(\mathrm{AB} = \sqrt{7x^2 - 7x + 7}\).

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3. Given that \(\mathrm{AB} = 7\),

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   1. find the value of \(x\);

   2. find the area of triangle \(\mathrm{ABC}\).