19M.1.SL.TZ2.S_9

pestleMathematicsAISLPaper 119M· sl-4-1-concepts-reliability-and-sampling-techniquessource ↗

Let $\theta$ be an obtuse angle such that ${\text{sin}}\,\theta = \frac{3}{5}$ .

Let $f\left( x \right) = {{\text{e}}^x}\,{\text{sin}}\,x - \frac{{3x}}{4}$ .

Find the value of ${\text{tan}}\,\theta$ .

[4]

Line $L$ passes through the origin and has a gradient of ${\text{tan}}\,\theta$ . Find the equation of $L$ .

[2]

Find the derivative of $f$ .

[5]

The following diagram shows the graph of $f$ for 0 ≤ $x$ ≤ 3. Line $M$ is a tangent to the graph of $f$ at point P.

Given that $M$ is parallel to $L$ , find the $x$ -coordinate of P.

[4]

Markscheme / solution

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

evidence of valid approach (M1)

eg sketch of triangle with sides 3 and 5, ${\text{co}}{{\text{s}}^2}\,\theta = 1 - {\text{si}}{{\text{n}}^2}\,\theta$

correct working (A1)

eg missing side is 4 (may be seen in sketch), ${\text{cos}}\,\theta = \frac{4}{5}$ , ${\text{cos}}\,\theta = - \frac{4}{5}$

${\text{tan}}\,\theta = - \frac{3}{4}$ A2 N4

[4 marks]

correct substitution of either gradient or origin into equation of line (A1)

(do not accept $y = mx + b$ )

eg $y = x\,{\text{tan}}\,\theta$ , $y - 0 = m\left( {x - 0} \right)$ , $y = mx$

$y = - \frac{3}{4}x$ A2 N4

Note: Award A1A0 for $L = - \frac{3}{4}x$ .

[2 marks]

$\frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{ - 3x}}{4}} \right) = - \frac{3}{4}$ (seen anywhere, including answer) A1

choosing product rule (M1)

eg $uv' + vu'$

correct derivatives (must be seen in a correct product rule) A1A1

eg ${\text{cos}}\,x$ , ${{\text{e}}^x}$

$f'\left( x \right) = {{\text{e}}^x}\,{\text{cos}}\,x + {{\text{e}}^x}\,{\text{sin}}\,x - \frac{3}{4}$ $\left( { = {{\text{e}}^x}\,\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right) - \frac{3}{4}} \right)$ A1 N5

[5 marks]

valid approach to equate their gradients (M1)

eg $f' = {\text{tan}}\,\theta$ , $f' = - \frac{3}{4}$ , ${{\text{e}}^x}\,{\text{cos}}\,x + {{\text{e}}^x}\,{\text{sin}}\,x - \frac{3}{4} = - \frac{3}{4}$ , ${{\text{e}}^x}\,\left( {{\text{cos}}\,x + {\text{sin}}\,x} \right) - \frac{3}{4} = - \frac{3}{4}$

correct equation without ${{\text{e}}^x}$ (A1)

eg ${\text{sin}}\,x = - {\text{cos}}\,x$ , ${\text{cos}}\,x + {\text{sin}}\,x = 0$ , $\frac{{ - {\text{sin}}\,x}}{{{\text{cos}}\,x}} = 1$

correct working (A1)

eg ${\text{tan}}\,\theta = - 1$ , $x = 135^\circ$

$x = \frac{{3\pi }}{4}$ (do not accept $135^\circ$ ) A1 N1

Note: Do not award the final A1 if additional answers are given.

[4 marks]

Examiners’ report

[N/A]