A-LevelMathematicsAlgebra9709/22 · Oct/Nov 2024 · Q510 Marks

The polynomial p(x) is defined by p(x) = ax³ + bx² - ax + 8, where a and b are constants. It is given that (x + 2) is a factor of p(x), and that the remainder is 24 when p(x) is divided by (x-2).

(a)Find the values of a and b.[4]
(b)Factorise p(x) and hence show that the equation p(x) = 0 has exactly one real root.[3]
(c)Solve the equation p(½ cosec θ) = 0 for -90° < θ < 90°.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.

[M1] (a) Substitute x = -2 and equate to zero
[M1] (a) Substitute x = 2 and equate to 24
[A1] (a) Obtain -6a+4b+8=0 and 6a+4b-16=0
[A1] (a) Obtain a = 2 and b=1
[M1] (b) Divide by x + 2 at least as far as the x term
[A1] (b) Obtain (x + 2)(2x² - 3x + 4)
[A1] (b) Conclude with reference to root -2, discriminant is –23 and no further root
[B1] (c) State cosec θ = -4
[M1] (c) Attempt to find at least one value of θ from sin θ = ± (1/4)
[A1] (c) Obtain -14.5 only and no others in the range

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Topic: Algebra, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Oct/Nov 2024 Paper 2 Variant 2, question 5.

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