The gradient of the line that is perpendicular to the graph of a relation at any point    is half the gradient of the line joining and the point  .

The relation satisfies the differential equation

Find    at the point  for the curve defined by the relation  .

Give your answer in the form  , where  , .  (5 marks)

A horizontal beam is supported at its endpoints, which are 2 m apart. The deflection metres of the beam measured downwards at a distance metres from the support at the origin is given by the differential equation  .
 

1. Given that both the inclination, , and the deflection, , of the beam from the horizontal at    are zero, use the differential equation above to show that  .   (2 marks)
   

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2. Find the angle of inclination of the beam to the horizontal at the origin . Give your answer as a positive acute angle in degrees, correct to one decimal place.   (2 marks)
   

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3. Find the value of , in metres, where the maximum deflection occurs, and find the maximum deflection, in metres.   (3 marks)
   

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4. Find the maximum angle of inclination of the beam to the horizontal in the part of the beam where  . Give your answer as a positive acute angle in degrees, correct to one decimal place.   (2 marks)
   

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The gradient of the tangent to a curve at any point    is half the gradient of the line segment joining and the point  .

The coordinates of points on the curve satisfy the differential equation

A.   

B.   

C.   

D.   

E.