Easy -2.5 This is a 1-mark question requiring only basic understanding of sampling concepts—identifying that qualitative observations or existence claims are less affected by sample size than quantitative estimates. It requires minimal statistical knowledge and no calculation, making it significantly easier than typical A-level maths questions.
The survey described in the article was based on a small sample.
State one conclusion which is unlikely to be influenced by the size of the sample. [1]
\(\Rightarrow \cos\theta = 0,\) and \(\tan\theta = \frac{1}{2}\)
\(\theta = 26.6°, -153.4°, -90°, 90°\)
Answer
Marks
M1
correct trig identity used
M1
factorising oe
M1
both needed and \(\cot\theta = 1/\tan\theta\) soi
B3,2,1,0
accept 27°, -153° as above
(omission of \(\cot\theta=0\) could gain M1, M1, M0, B1) ...........................................
M1
correct trig equivalents and a one line equation (or common denominator) formed
M1
use of Pythagoras and factorising
M1
accept 27°, -153° as above
B3,2,1,0
in both cases, -1 if extra solutions in the range are given ( dependent on at least B1 being scored)-not their incorrect solutions eg 26.6°,-153.4°, 0°, 180°, -180° would obtain B1 to lose both of these, at least B2 would need to be scored.
Total: [6]
**Answer:** $\operatorname{cosec}^2\theta = 1 + \cot^2\theta$
$\Rightarrow 1 + \cot^2\theta = 1 + 2\cot\theta$
$\Rightarrow \cot^2\theta - 2\cot\theta = 0$
$\Rightarrow \cot\theta(\cot\theta - 2) = 0$
$\Rightarrow \cot\theta = 0,$ **and** $\cot\theta = 2, \tan\theta = \frac{1}{2}$
$\theta = 26.6°, -153.4°, -90°, 90°$
...................................................................
**OR** $\frac{1}{\sin^2\theta} = 1 + \frac{2\cos\theta}{\sin\theta} + \frac{2\cos\theta}{\sin\theta}$
$\Rightarrow \sin^2\theta + 2\sin\theta\cos\theta - 1 = 0$
$\Rightarrow 2\sin\theta\cos\theta - \cos^2\theta = 0$
$\Rightarrow \cos\theta(\sin\theta - \cos\theta) = 0$
$\Rightarrow \cos\theta = 0,$ **and** $\tan\theta = \frac{1}{2}$
$\theta = 26.6°, -153.4°, -90°, 90°$
| M1 | correct trig identity used |
| M1 | factorising oe |
| M1 | both needed and $\cot\theta = 1/\tan\theta$ soi |
| B3,2,1,0 | accept 27°, -153° as above |
| | (omission of $\cot\theta=0$ could gain M1, M1, M0, B1) ........................................... |
| M1 | correct trig equivalents and a one line equation (or common denominator) formed |
| M1 | use of Pythagoras and factorising |
| M1 | accept 27°, -153° as above |
| B3,2,1,0 | |
| | **in both cases, -1 if extra solutions in the range are given ( dependent on at least B1 being scored)-not their incorrect solutions eg 26.6°,-153.4°, 0°, 180°, -180° would obtain B1 to lose both of these, at least B2 would need to be scored.** |
**Total: [6]**
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The survey described in the article was based on a small sample.
State one conclusion which is unlikely to be influenced by the size of the sample. [1]
\hfill \mbox{\textit{OCR MEI C4 2011 Q5 [1]}}