16. The following diagram shows a sketch of \(y = \sin x\) for values of \(x\) from 0° to 360°.
(a) Given that \(\sin 62^\circ = 0{\cdot}8829\), correct to 4 decimal places, write down all the solutions of the equation\[\sin x = -0{\cdot}8829\]for values of \(x\) from 0° to 360°. [2]
(b)
(i) Use the following axes to sketch the graph of \(y = -\sin x\) for values of \(x\) from 0° to 360°. You must indicate any important values on the \(y\)-axis. [2]
(ii) Use the following axes to sketch the graph of \(y = \sin x + 1\) for values of \(x\) from 0° to 360°. You must indicate any important values on the \(y\)-axis. [2]
Mark scheme (a)
Scheme
Marks
Comments
242° and 298° with no other values
B2
B1 for either angle. Check diagram. Penalise –1 for each extra value within range (beyond 2 attempts). Ignore extra (correct or incorrect) values outside the required range. If only two angles offered and no marks gained, award SC1 for sight of both 180° + 62° and 360° – 62°.
Mark scheme (b)
Scheme
Marks
Comments
(i) Reflection in \(x\)-axis with maximum at (270°, +1), minimum at (90°, –1)
B2
Mark clear intention. Must be the correct shape, i.e. a single cycle of a negative sine curve, with \(x\)-intercepts at \(x\) = 0, \(x\) = 180° and \(x\) = 360°, minimum at \(x\) = 90°, maximum at \(x\) = 270°. Accept any clear indication of \(y\)-coordinates.
If not B2, award B1 for one of the following:
Fully correct shape and position (both for 0 ⩽ \(x\) ⩽ 180° and for 180°⩽ \(x\) ⩽ 360°) without correct coordinates indicated
OR
Correct shape and position either for 0 ⩽ \(x\) ⩽ 180° or for 180°⩽ \(x\) ⩽ 360° (a sine curve entirely between \(y\) = –1 and \(y\) = 1) AND indication of \(y\) = –1 and \(y\) = 1.
SC1 for a graph which is fully correct (including labelling) other than having pointed minimum and maximum (formed from straight lines).
(ii) Vertical translation +1 with maximum at (90°, +2), minimum at (270°, 0), with \(y\)-intercept at +1.
B2
Mark clear intention. Must be the correct shape, i.e. a single cycle of a sine curve, with consistent \(y\) values at \(x\) = 0, \(x\) = 180° and \(x\) = 360°, maximum at \(x\) = 90°, minimum at \(x\) = 270°.
Accept any clear indication of \(y\)-coordinates. Must have correct points for \(x\) = 180° and \(x\) = 360°. 1 and 2 indicated on the \(y\)-axis.
If not B2, award B1 for one of the following:
Fully correct shape and position (both for 0 ⩽ \(x\) ⩽ 180° and for 180°⩽ \(x\) ⩽ 360°) without correct coordinates indicated
OR
Correct shape and position either for 0 ⩽ \(x\) ⩽ 180° or for 180°⩽ \(x\) ⩽ 360° (a sine curve entirely between \(y\) = 0 and \(y\) = 2) AND indication of \(y\) = 1 and \(y\) = 2.
SC1 for a graph which is fully correct (including labelling) other than having pointed minimum and maximum (formed from straight lines).
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