A train is travelling at a constant speed of km/h along a straight level track from towards

The train will travel along a section of track

Section passes along a bridge over a valley.

Section passes through a tunnel in a mountain.

Section is 6.2 km long.

From to , the curve of the valley and the mountain, directly below and above the train track, is modelled by the graph of
 

where and are real numbers.
 

All measurements are in kilometres.

1. The curve defined from to passes through . The gradient of the curve at is – 0.06 and the curve has a turning point at  .
2.  i. From this information write down three simultaneous equations in , and .   (3 marks)
   

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3. ii. Hence show that  , and .   (2 marks)
   

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2. Find, giving exact values
3.   i. the coordinates of .   (2 marks)
   

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4.  ii. the length of the tunnel.   (1 mark)
   

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5. iii. the maximum depth of the valley below the train track.   (1 mark)
   

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The driver sees a large rock on the track at a point , 6.2 km from . The driver puts on the brakes at the instant that the front of the train comes out of the tunnel at .

From its initial speed of km/h, the train slows down from point so that its speed km/h is given by

,

where km is the distance of the front of the train from and is a real constant.

3. Find the value of in terms of .   (1 mark)
   

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4. Find the exact distance from the front of the train to the large rock when the train finally stops.   (2 marks)
   

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