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A-LevelMathematicsAlgebra9709/21 · Oct/Nov 2021 · Q610 Marks

The polynomials f(x) and g(x) are defined by f(x) = 4x³ + ax² + 8x + 15 and g(x) = x² + bx + 18, where a and b are constants.

  1. (a)Given that (x + 3) is a factor of f(x), find the value of a.[2]
  2. (b)Given that the remainder is 40 when g(x) is divided by (x – 2), find the value of b.[2]
  3. (c)When a and b have these values, factorise f(x) – g(x) completely.[3]
  4. (d)Hence solve the equation f(cosec θ) – g(cosec θ) = 0 for 0 < θ < 2π.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

  1. [M1] Substitute x=-3, equate to zero and attempt solution for a
  2. [A1] Obtain a=13
  3. [M1] Substitute x=2, equate to 40 and attempt solution for b
  4. [A1] Obtain b=9
  5. [B1] Identify x+3 as factor of f(x) - g(x)
  6. [M1] Attempt, by division or equivalent, to find quadratic factor
  7. [A1] Obtain (x+3)(2x-1)(2x+1)
  8. [M1] Attempt correct process to find at least 1 value from cosec θ = k where k < -1
  9. [A1] Obtain 3.48 or 5.94
  10. [A1] Obtain a second correct solution

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About this question

Topic
Algebra, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source
A-Level Mathematics, Oct/Nov 2021 Paper 2 Variant 1, question 6.

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