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Date November 2012 Marks available 2 Reference code 12N.2.sl.TZ0.3
Level SL only Paper 2 Time zone TZ0
Command term Write down Question number 3 Adapted from N/A

Question

Let \(f(x) = {x^3} - 2x - 4\) . The following diagram shows part of the curve of f .


The curve crosses the x-axis at the point P.

Write down the x-coordinate of P.

[1]
a.

Write down the gradient of the curve at P.

[2]
b.

Find the equation of the normal to the curve at P, giving your equation in the form \(y = ax + b\) .

[3]
c.

Markscheme

\(x = 2\) (accept \((2{\text{, }}0)\))    A1     N1

[1 mark]

a.

evidence of finding gradient of f at \(x = 2\)     (M1) 

e.g. \(f'(2)\)

the gradient is 10     A1     N2

[2 marks]

b.

evidence of negative reciprocal of gradient     (M1)

e.g. \(\frac{{ - 1}}{{f'(x)}}\) , \( - \frac{1}{{10}}\) 

evidence of correct substitution into equation of a line     (A1)

e.g. \(y - 0 = \frac{{ - 1}}{{10}}(x - 2)\) , \(0 = - 0.1(2) + b\)

\(y = - \frac{1}{{10}}x + \frac{2}{{10}}\) (accept \(a = - 0.1\) , \(b = 0.2\) )     A1     N2

[3 marks]

c.

Examiners report

This question was generally done well.

a.

Most candidates did not use their GDC in part (b), resulting in a variety of careless errors occasionally arising either in differentiating or substituting.

b.

There were some candidates who did not know the relationship between gradients of perpendicular lines while others found the equation of the tangent rather than the normal in part (c).

c.

Syllabus sections

Topic 2 - Functions and equations » 2.2 » Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range.
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