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Date May 2019 Marks available 6 Reference code 19M.2.AHL.TZ2.H_9
Level Additional Higher Level Paper Paper 2 Time zone Time zone 2
Command term Sketch Question number H_9 Adapted from N/A

Question

Consider the polynomial P ( z ) z 4 6 z 3 2 z 2 + 58 z 51 , z C .

Sketch the graph of y = x 4 6 x 3 2 x 2 + 58 x 51 , stating clearly the coordinates of any maximum and minimum points and intersections with axes.

[6]
b.

Hence, or otherwise, state the condition on k R such that all roots of the equation P ( z ) = k are real.

[2]
c.

Markscheme

shape       A1

x -axis intercepts at (−3, 0), (1, 0) and y -axis intercept at (0, −51)       A1A1

minimum points at (−1.62, −118) and (3.72, 19.7)       A1A1

maximum point at (2.40, 26.9)       A1

Note: Coordinates may be seen on the graph or elsewhere.

Note: Accept −3, 1 and −51 marked on the axes.

[6 marks]

b.

from graph, 19.7 ≤ k  ≤ 26.9       A1A1

Note: Award A1 for correct endpoints and A1 for correct inequalities.

[2 marks]

c.

Examiners report

[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion
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Topic 1—Number and algebra » AHL 1.14—Complex roots of polynomials, conjugate roots, De Moivre’s, powers & roots of complex numbers
Topic 1—Number and algebra
Topic 5 —Calculus

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