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1.2 Algebra and Functions

1.2 Algebra and Functions

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Question 201

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−14​z>3
a.

Given 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−3Q​z>3

find the value of the constant PPP and show that Q=5Q = 5Q=5.

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b.

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.

[4]
c.

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+bln⁡2a + b \ln 2a+bln2, where aaa and bbb are constants to be found.

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Markscheme

1.2 Algebra and Functions Questions

  1. A Level
  2. /Maths
  3. /1.2 Algebra and Functions

566 exam-style questions on OCR A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Indices, 1.2.2 Surds, 1.2.3 Simultaneous equations, 1.2.4 Quadratic functions, 1.2.5 Completing the square, 1.2.6 Solving quadratic equations, 1.2.7 Inequalities, 1.2.8 Expressing solutions of inequalities, 1.2.9 Representing inequalities graphically, 1.2.10 Manipulating polynomials, 1.2.11 Simplifying rational expressions, 1.2.12 The modulus function (A-level only), 1.2.13 Graphs of functions, 1.2.14 Sketching curves from equations, 1.2.15 Sketching reciprocal curves, 1.2.16 Interpreting solutions graphically, 1.2.17 Intersection points of graphs, 1.2.18 Proportional relationships, 1.2.19 Graph of the modulus of a linear function (A-level only), 1.2.20 Solving modulus equations graphically (A-level only), 1.2.21 Definition of a function (A-level only), 1.2.22 Inverse and composite functions (A-level only), 1.2.23 Simple graph transformations, 1.2.24 Combinations of graph transformations (A-level only), 1.2.25 Partial fractions (A-level only), and 1.2.26 Models in context. Each one has a worked solution and a mark scheme showing where the marks go.

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