19M.2.AHL.TZ2.H_4

pestleMathematicsAAHLPaper 219M· sl-2-3-graphing, ahl-2-15-solutions-of-inequalitiessource ↗

Sketch the graphs $y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x$ and $y = 1 + {\text{cos}}\,x$ on the following axes for 0 < $x$ ≤ 9.

[2]

Hence solve ${\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0$ in the range 0 < $x$ ≤ 9.

[4]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

A1A1

Note: Award A1 for each correct curve, showing all local max & mins.

Note: Award A0A0 for the curves drawn in degrees.

[2 marks]

$x$ = 1.35, 4.35, 6.64 (M1)

Note: Award M1 for attempt to find points of intersections between two curves.

0 < $x$ < 1.35 A1

Note: Accept $x$ < 1.35.

4.35 < $x$ < 6.64 A1A1

Note: Award A1 for correct endpoints, A1 for correct inequalities.

Note: Award M1FTA1FTA0FTA0FT for 0 < $x$ < 7.31.

Note: Accept $x$ < 7.31.

[4 marks]

Examiners’ report

[N/A]