| Date | May 2012 | Marks available | 4 | Reference code | 12M.2.hl.TZ2.6 |
| Level | HL only | Paper | 2 | Time zone | TZ2 |
| Command term | Sketch | Question number | 6 | Adapted from | N/A |
Question
Sketch the curve \(y = \frac{{\cos x}}{{\sqrt {{x^2} + 1} }},{\text{ }} - 4 \leqslant x \leqslant 4\) showing clearly the coordinates of the x-intercepts, any maximum points and any minimum points.
Write down the gradient of the curve at x = 1 .
Find the equation of the normal to the curve at x = 1 .
Markscheme
A1A1A1A1
Note: Award A1 for correct shape. Do not penalise if too large a domain is used,
A1 for correct x-intercepts,
A1 for correct coordinates of two minimum points,
A1 for correct coordinates of maximum point.
Accept answers which correctly indicate the position of the intercepts, maximum point and minimum points.
[4 marks]
gradient at x = 1 is –0.786 A1
[1 mark]
gradient of normal is \(\frac{{ - 1}}{{ - 0.786}}( = 1.272...)\) (A1)
when x = 1, y = 0.3820... (A1)
Equation of normal is y – 0.382 = 1.27(x – 1) A1
\(( \Rightarrow y = 1.27x - 0.890)\)
[3 marks]
Examiners report
Most candidates were able to make a meaningful start to this question, but many made errors along the way and hence only a relatively small number of candidates gained full marks for the question. Common errors included trying to use degrees, rather than radians, trying to use algebraic methods to find the gradient in part (b) and trying to find the equation of the tangent rather than the equation of the normal in part (c).
Most candidates were able to make a meaningful start to this question, but many made errors along the way and hence only a relatively small number of candidates gained full marks for the question. Common errors included trying to use degrees, rather than radians, trying to use algebraic methods to find the gradient in part (b) and trying to find the equation of the tangent rather than the equation of the normal in part (c).
Most candidates were able to make a meaningful start to this question, but many made errors along the way and hence only a relatively small number of candidates gained full marks for the question. Common errors included trying to use degrees, rather than radians, trying to use algebraic methods to find the gradient in part (b) and trying to find the equation of the tangent rather than the equation of the normal in part (c).
