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Date	May 2021	Marks available	3	Reference code	21M.1.SL.TZ2.5
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 2
Command term	Find	Question number	5	Adapted from	N/A

Question

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

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

Syllabus sections

Topic 2—Functions » SL 2.9—Exponential and logarithmic functions

20N.1.SL.TZ0.S_4b:
The $x$ -intercept of the graph of $f$ is $(5, 0)$ .

On the following grid, sketch the graph of $f$ .
EXN.2.SL.TZ0.9a:
Show that $T_{0} = 100$ .
EXN.2.SL.TZ0.9c:
Find the temperature of the water when $t = 15$ .
21M.1.SL.TZ1.8d:
Solve $f (x) > 0$ for $x > 0$ .
21M.2.SL.TZ2.6a:
Show that $A_{0} = 100$ .
21N.2.SL.TZ0.2b:
Sketch the graph of $f$ on the following grid.
21M.2.SL.TZ2.6c:
Find, correct to the nearest $10$ years, the time taken after the plant’s death for $25 %$ of the carbon-14 to decay.
20N.1.SL.TZ0.S_4a:
Find the value of $a$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ .
21N.2.SL.TZ0.2a:
Find the values of $x$ for which $f (x) = 0$ .
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ .
EXN.2.SL.TZ0.9b:
Show that $k = \frac{1}{10} \ln \frac{10}{7}$ .
21M.2.SL.TZ2.6b:
Show that $k = \frac{\ln 2}{5730}$ .
20N.1.SL.TZ0.T_12b:
In the context of the model, state what the horizontal asymptote represents.
20N.1.SL.TZ0.T_12a:
Find the value of $K$ .
20N.1.SL.TZ0.T_12c:
Find Jean-Pierre’s vertical speed after $10$ seconds. Give your answer in $km h^{- 1}$ .
21N.1.AHL.TZ0.3:
Solve the equation $\log_{3} \sqrt{x} = \frac{1}{2 \log_{2} 3} + \log_{3} (4 x^{3})$ , where $x > 0$ .
21N.1.SL.TZ0.8a:
Show that $a = 8$ .
21N.1.SL.TZ0.8d.ii:
Find the value of $p$ and the value of $q$ .
21N.1.SL.TZ0.8d.i:
Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

Topic 2—Functions

Question

Markscheme

Examiners report

Syllabus sections

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