19M.1.SL.TZ2.S_7

pestleMathematicsAASLPaper 119M· sl-3-7-circular-functions-graphs-composites-transformationssource ↗

Consider the graph of the function $f\left( x \right) = 2\,{\text{sin}}\,x$ , 0 ≤ $x$ < $2\pi$ . The graph of $f$ intersects the line $y = - 1$ exactly twice, at point A and point B. This is shown in the following diagram.

Consider the graph of $g\left( x \right) = 2\,{\text{sin}}\,px$ , 0 ≤ $x$ < $2\pi$ , where $p$ > 0.

Find the greatest value of $p$ such that the graph of $g$ does not intersect the line $y = - 1$ .

Markscheme / solution

recognizing period of $g$ is larger than the period of $f$ (M1)

eg sketch of $g$ with larger period (may be seen on diagram), A at $x = 2\pi$ ,

image of A when $x > 2\pi$ , $\frac{{7\pi }}{6} \to 2\pi$ , $2\,{\text{sin}}\,\left( {2\pi p} \right) = - 1$ , $\frac{{7\pi }}{6} \times k = 2\pi$

correct working (A1)

eg $\frac{{7\pi }}{6} \cdot \frac{1}{p} = 2\pi$ , $2\pi p = \frac{{7\pi }}{6}$ , $\frac{{12}}{7}$

$p = \frac{7}{12}$ $\left( {{\text{accept}}\,\,p < \frac{7}{12}\,\,{\text{or}}\,\,p \leqslant \frac{7}{12}} \right)$ A1 N2

[3 marks]

Examiners’ report

[N/A]