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1.5 B: Algebra and functions

1.5 B: Algebra and functions

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Question 257

The rate of production of a biofuel in a research facility, R(t)R(t)R(t) kilolitres per hour, is modelled by the formula

R(t)=4t3+5t2+8t+15t2+3,1≤t≤3 R(t) = \frac{4t^3 + 5t^2 + 8t + 15}{t^2 + 3}, \quad 1 \le t \le 3 R(t)=t2+34t3+5t2+8t+15​,1≤t≤3

where ttt is the time in hours from the start of a production cycle. Given that

R(t)≡At+B+Ct+Dt2+3 R(t) \equiv At + B + \frac{Ct + D}{t^2 + 3} R(t)≡At+B+t2+3Ct+D​
a.

(i) find the values of the constants AAA, BBB and CCC.

(ii) show that D=0D = 0D=0.

[4]
b.

Hence, using algebraic integration, calculate the total volume of biofuel produced between t=1t = 1t=1 and t=3t = 3t=3, giving your answer in the form p+qln⁡kp + q \ln kp+qlnk, where ppp, qqq and kkk are integers and kkk is prime.

[5]
Markscheme

1.5 B: Algebra and functions Questions

  1. A Level
  2. /Maths
  3. /1.5 B: Algebra and functions

532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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