A-LevelMathematicsAlgebra9709/21 · May/June 2023 · Q48 Marks

The polynomial p(x) is defined by p(x) = 2x³ + 3x² + kx − 30, where k is a constant. It is given that (x – 3) is a factor of p(x).

(a)Find the value of k.[2]
(b)Hence find the quotient when p(x) is divided by (x – 3) and factorise p(x) completely.[3]
(c)It is given that a is one of the roots of the equation p(x) = 0. Given also that the equation |4y – 5| = a is satisfied by two real values of y, find these two values of y.[3]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[M1] Substitute x = 3, equate to zero and attempt solution Condone 51/3
[A1] Obtain k = -17
[M1] Divide by x-3 at least as far as 2x² + mx
[A1] Obtain 2x² + 9x +10
[A1] Obtain (x-3)(2x+5)(x + 2) SC-no attempt at division (or equivalent) and only correct fully factorised form shown: award B1 only.
[B1 FT] Solve linear equation 4y-5=3 to obtain y = 2 following any positive root from their factorised p(x).
[M1] Attempt solution of linear equation 4y-5=-3 or equivalent with RHS of equation being – (their positive root).
[A1] Obtain ½ and 2 and no other solutions

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