2025 SST PRELIMS P2 Q9

2025 SST PRELIMS P2 Q9

Secondary 4
8 marks
2025 School of Science and Technology Prelims A Math Paper 2

The diagram shows a garden plot designed in the shape of a trapezium \(OPQR\) inscribed in a quadrant \(OPQS\) with a fixed radius \(x\) metres and angle \(QOR = \theta\) radians.

  1. Show that the area, \(A \text{ m}^2\), of trapezium \(OPQR\) is given by[3]
    \[A = \frac{1}{4}x^2(\sin 2\theta + 2\cos \theta).\]
  2. Given that \(\theta\) can vary, find
    1. the value of \(\theta\) for which \(A\) has a stationary value.[3]
    2. the stationary value of \(A\) at \(x = 10\) and determine whether it is a maximum or a minimum.[2]

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Answer:(i) \(A=\frac14x^2(\sin2\theta+2\cos\theta)\). (ii)(a) \(\theta=\frac\pi6\text{ rad}\). (ii)(b) \(A=\frac{75\sqrt3}{2}=65.0\text{ m}^2\) to 3 significant figures, maximum.

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