19N.1.SL.TZ0.S_5

pestleMathematicsAASLPaper 119N· sl-2-7-solutions-of-quadratic-equations-and-inequalities-discriminant-and-nature-of-rootssource ↗

Consider the function $f$ , with derivative $f'\left( x \right) = 2{x^2} + 5kx + 3{k^2} + 2$ where $x{\text{, }}k \in \mathbb{R}$ .

Show that the discriminant of $f'\left( x \right)$ is ${k^2} - 16$ .

[2]

Given that $f$ is an increasing function, find all possible values of $k$ .

[4]

Markscheme / solution

correct substitution into ${b^2} - 4ac$ (A1)

eg ${\left( {5k} \right)^2} - 4\left( 2 \right)\left( {3{k^2} + 2} \right)$ , ${\left( {5k} \right)^2} - 8\left( {3{k^2} + 2} \right)$

correct expansion of each term A1

eg $25{k^2} - 24{k^2} - 16$ , $25{k^2} - \left( {24{k^2} + 16} \right)$

${k^2} - 16$ AG N0

[2 marks]

valid approach M1

eg $f'\left( x \right) > 0$ , $f'\left( x \right) \geqslant 0$

recognizing discriminant $< 0$ or $\leqslant 0$ M1

eg $D < 0$ , ${k^2} - 16 \leqslant 0$ , ${k^2} < 16$

two correct values for $k$ /endpoints (even if inequalities are incorrect) (A1)

eg $k = \pm 4$ , $k < - 4$ and $k > 4$ , $\left| k \right| < 4$

correct interval A1 N2

eg $- 4 < k < 4$ , $- 4 \leqslant k \leqslant 4$

Note: Candidates may work with an equation, then write the intervals with inequalities at the end. If inequalities are not seen until the candidate’s final correct answer, M0M0A1A1 may be awarded.
If candidate is working with incorrect inequalitie(s) at the beginning, then gets the correct final answer, award M0M0A1A0 or M1M0A1A0 or M0M1A1A0 in line with the markscheme.

[4 marks]

Examiners’ report

[N/A]