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Question 692

A deep-sea research probe is descending through a thermal layer. Its altitude h h\,h in metres, relative to the layer's boundary, is modeled by the function h(t)h(t)h(t) for t>0t > 0t>0, where t t\,t is the time in seconds since the descent began. It is known that:

  • at time t=4t = 4t=4, the probe is at point P(4,−10)P(4, -10)P(4,−10)
  • the rate of change of altitude is given by h′(t)=5t2+at+b3th'(t) = \dfrac{5t^2 + at + b}{3\sqrt{t}}h′(t)=3t​5t2+at+b​, where a a\,a and b b\,b are constants
  • at the instant t=4t = 4t=4, the rate of change of altitude is 12 m s-1
a.

Show that 4a+b=−84a + b = -84a+b=−8.

[3]
b.

Given also that a+2b=−23a + 2b = -23a+2b=−23

Find, in simplest form, the expression for h(t)h(t)h(t).

[6]
c.

The probe's mission data is later adjusted by a time-shift, such that the altitude is modeled by H(t)=h(t−3)H(t) = h(t - 3)H(t)=h(t−3). Given that point P P\,P on the original model is transformed to point Q Q\,Q on the new model,

State the coordinates of QQQ.

[1]
Markscheme

Integration Questions

  1. A Level
  2. /Maths
  3. /Integration

864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.

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