A research engineer determines that the stability of a feedback loop in a control system depends on the roots of the cubic polynomial P(y)=8y3+18y2+y−6P(y) = 8y^3 + 18y^2 + y - 6P(y)=8y3+18y2+y−6.
Use the factor theorem to prove that (y+2)(y + 2)(y+2) is a factor of P(y)P(y)P(y).
Hence, using algebraic methods, express P(y)P(y)P(y) as a product of three linear factors.
A specific operating mode of the loop requires the condition P(tanϕ)=0P(\tan \phi) = 0P(tanϕ)=0. For the range π2<ϕ<π\frac{\pi}{2} < \phi < \pi2π<ϕ<π, determine all possible values of ϕ\phiϕ, giving your answers to three significant figures.
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.