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1.5 Coordinate Geometry

1.5 Coordinate Geometry

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Question 112

The profile of a suspended track for a high-speed transport system is modelled by the curve CCC. The path of the track starts at point P P\,P and ends at point QQQ, as defined by the parametric equations

x=12t2+2t+10 x = \frac{1}{2}t^2 + \frac{2}{t} + 10 x=21​t2+t2​+10 h=3t+12t h = 3t + \frac{12}{t} h=3t+t12​

where 1≤t≤61 \le t \le 61≤t≤6.

The horizontal distance from a sensor at the origin is x x\,x metres, and h h\,h is the height of the track above the ground in metres.

P P\,P is the point on the track where t=1t = 1t=1 and Q Q\,Q is the point where t=6t = 6t=6.

a.

Safety regulations require that the difference in height between the start point P P\,P and the end point Q Q\,Q must be less than 6 metres. Show that the track meets this requirement.

[2]
bi.

Find an expression for dhdx\displaystyle \frac{dh}{dx}dxdh​ in terms of ttt.

[3]
bii.

A vertical reinforcement pillar is placed between the ground and the lowest point R R\,R on the track. Find the height of this pillar.

[3]
biii.

Calculate the acute angle the track makes with the horizontal at the end point QQQ. Give your answer to the nearest degree.

[2]
Markscheme

1.5 Coordinate Geometry Questions

  1. A Level
  2. /Maths
  3. /1.5 Coordinate Geometry

178 exam-style questions on OCR (MEI) A Level Maths 1.5 Coordinate Geometry, covering 1.5.1 Equation of a straight line y = mx + c, 1.5.2 Gradients of parallel and perpendicular lines, 1.5.3 Distance between two points, 1.5.4 Midpoint of a line segment, 1.5.5 Form the equation of a straight line, 1.5.6 Draw a line given its equation, 1.5.7 Point of intersection of two lines, 1.5.8 Straight line models, 1.5.9 Intersection of a line and a curve, 1.5.10 Intersection of a line and a circle, 1.5.11 Equation of a circle, 1.5.12 Circle properties, 1.5.13 Parameter and parametric equations (A-level only), 1.5.14 Convert between cartesian and parametric forms (A-level only), 1.5.15 Equation of a circle in parametric form (A-level only), 1.5.16 Gradient of a parametric curve (A-level only), 1.5.17 Parametric equations in modelling (A-level only), and 1.5 Coordinate Geometry. Each one has a worked solution and a mark scheme showing where the marks go.

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