21M.1.SL.TZ2.5

pestleMathematicsAASLPaper 121M· sl-2-9-exponential-and-logarithmic-functions, sl-5-4-tangents-and-normalsource ↗

Consider the function $f$ defined by $f (x) = \ln (x^{2} - 16)$ for $x > 4$ .

The following diagram shows part of the graph of $f$ which crosses the $x$ -axis at point $A$ , with coordinates $(a, 0)$ . The line $L$ is the tangent to the graph of $f$ at the point $B$ .

Find the exact value of $a$ .

[3]

Given that the gradient of $L$ is $\frac{1}{3}$ , find the $x$ -coordinate of $B$ .

[6]

Markscheme / solution

$\ln (x^{2} - 16) = 0$ (M1)

$e^{0} = x^{2} - 16 (= 1)$

$x^{2} = 17$ OR $x = \pm \sqrt{17}$ (A1)

$a = \sqrt{17}$ A1

[3 marks]

attempt to differentiate (must include $2 x$ and/or $\frac{1}{x^{2} - 16}$ ) (M1)

$f' (x) = \frac{2 x}{x^{2} - 16}$ A1

setting their derivative $= \frac{1}{3}$ M1

$\frac{2 x}{x^{2} - 16} = \frac{1}{3}$

$x^{2} - 16 = 6 x$ OR $x^{2} - 6 x - 16 = 0$ (or equivalent) A1

valid attempt to solve their quadratic (M1)

$x = 8$ A1

Note: Award A0 if the candidate’s final answer includes additional solutions (such as $x = - 2, 8$ ).

[6 marks]

Examiners’ report

[N/A]