The concentration of a bioactive compound, σ(z)\sigma(z)σ(z), in mg/L, at a depth zzz meters below the surface of a lake is modeled by the function:
σ(z)=z4−z3−2z2+z−14z2−z−6z>3 \sigma(z) = \frac{z^4 - z^3 - 2z^2 + z - 14}{z^2 - z - 6} \quad z > 3 σ(z)=z2−z−6z4−z3−2z2+z−14z>3Given that
σ(z)≡z2+P+Qz−3z>3 \sigma(z) \equiv z^2 + P + \frac{Q}{z - 3} \quad z > 3 σ(z)≡z2+P+z−3Qz>3find the value of the constant PPP and show that Q=5Q = 5Q=5.
Find the equation of the tangent to the concentration curve at the point where z=4z = 4z=4. Give your answer in the form σ=mz+c\sigma = mz + cσ=mz+c, where mmm and ccc are constants to be found.
A researcher calculates the total mass potential between depths z=4z = 4z=4 and z=5z = 5z=5, which is represented by the area RRR bounded by the curve σ(z)\sigma(z)σ(z), the zzz-axis, and the vertical lines z=4z = 4z=4 and z=5z = 5z=5. Calculate the exact value of this area, writing your answer in the form a+bln2a + b \ln 2a+bln2, where aaa and bbb are constants to be found.
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.