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1 Unit 1: Pure Mathematics A

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Question 28

The path of a high-precision laser cutter across a titanium plate is defined by the parametric equations

x=t2+8t,y=12t(8−t),a≤t≤b x = t^2 + 8t, \quad y = \frac{12}{t(8-t)}, \quad a \le t \le b x=t2+8t,y=t(8−t)12​,a≤t≤b

where ttt is the time in seconds, and aaa and bbb are constants. The cutter's path intersects the reference line y=1y = 1y=1 at two points representing the start and end of a specific cooling phase.

a.

Find the value of aaa and the value of bbb, where b>ab > ab>a.

[2]
b.

The region RRR is bounded by the path and the line y=1y = 1y=1. Show that the area of region RRR is given by

M−k∫abt+4t(8−t) dt M - k \int_{a}^{b} \frac{t+4}{t(8-t)} \, dt M−k∫ab​t(8−t)t+4​dt

where MMM and kkk are constants to be found.

[4]
ci.

Express t+4t(8−t)\frac{t+4}{t(8-t)}t(8−t)t+4​ in partial fractions.

[3]
cii.

Hence, use algebraic integration to find the exact area of RRR, giving your answer in the form P−Qln⁡3P - Q \ln 3P−Qln3, where PPP and QQQ are integers.

[4]
Markscheme

1 Unit 1: Pure Mathematics A Questions

  1. A Level
  2. /Maths
  3. /1 Unit 1: Pure Mathematics A

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