A-LevelMathematicsAlgebra9709/33 · May/June 2024 · Q76 Marks
Let f(x) = 8x³ + 54x² - 17x-21.
- (a)Show that x + 7 is a factor of f(x).[1]
- (b)Find the quotient when f(x) is divided by x+7.[2]
- (c)Hence solve the equation 8 cos³θ + 54 cos²θ - 17 cosθ - 21 = 0, for 0° ≤ θ ≤ 360°.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.
- [B1] Show 8 × (-7)³ + 54 × (-7)² – 17 × (-7) – 21 = 0 This is sufficient if no errors seen.
- [M1] Commence division and reach partial quotient of the form 8x² + 2x or 8x³ + 54x² – 17x – 21 = (x + 7)(Ax² + Bx + C) and reach A = 8 and B = ± 2 or C = -3
- [A1] Obtain quotient 8x² – 2x – 3 with no errors seen
- [M1] Solve quadratic from (b) to obtain a value for θ = cos⁻¹(-1/2) or cos⁻¹(3/4)
- [A1] Obtain one answer, e.g. θ = 120°
- [A1] Obtain three further answers, e.g. θ = 240°, 41.4° and 318.6° (condone 319°) and no others in the interval
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