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Nov 2021 p31 q10
1936
The complex number 1 + 2i is denoted by u. The polynomial 2x^3 + ax^2 + 4x + b, where a and b are real constants, is denoted by p(x). It is given that u is a root of the equation p(x) = 0.
(a) Find the values of a and b.
(b) State a second complex root of this equation.
(c) Find the real factors of p(x).
(d) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities |z - u| ≤ √5 and arg z ≤ 1/4 π.
(ii) Find the least value of Im z for points in the shaded region. Give your answer in an exact form.
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