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Date	November 2020	Marks available	3	Reference code	20N.1.SL.TZ0.T_12
Level	Standard Level	Paper	Paper 1 (with calculator from previous syllabus)	Time zone	Time zone 0
Command term	Find	Question number	T_12	Adapted from	N/A

Question

Jean-Pierre jumps out of an airplane that is flying at constant altitude. Before opening his parachute, he goes through a period of freefall.

Jean-Pierre’s vertical speed during the time of freefall, $S$ $S$ , in ${m s}^{- 1}$ ${m s}^{- 1}$ , is modelled by the following function.

$S (t) = K - 60 (1 . 2^{- t}), t \geq 0$ $S (t) = K - 60 (1 . 2^{- t}), t \geq 0$

where $t$ $t$ , is the number of seconds after he jumps out of the airplane, and $K$ $K$ is a constant. A sketch of Jean-Pierre’s vertical speed against time is shown below.

Jean-Pierre’s initial vertical speed is $0 {m s}^{- 1}$ $0 {m s}^{- 1}$ .

Find the value of $K$ $K$ .

[2]

a.

In the context of the model, state what the horizontal asymptote represents.

[1]

b.

Find Jean-Pierre’s vertical speed after $10$ $10$ seconds. Give your answer in $km h^{- 1}$ $km h^{- 1}$ .

[3]

c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure. It appeared in a paper that permitted the use of a calculator, and so might not be suitable for all forms of practice.

$0 = K - 60 (1 . 2^{0})$ $0 = K - 60 (1 . 2^{0})$ (M1)

Note: Award (M1) for correctly substituted function equated to zero.

$(K =) 60$ $(K =) 60$ (A1) (C2)

[2 marks]

a.

the (vertical) speed that Jean-Pierre is approaching (as $t$ $t$ increases) (A1) (C1)
OR
the limit of the (vertical) speed of Jean-Pierre (A1) (C1)

Note: Accept “maximum speed” or “terminal speed”.

[1 mark]

b.

$(S =) 60 - 60 (1 . 2^{- 10})$ $(S =) 60 - 60 (1 . 2^{- 10})$ (M1)

Note: Award (M1) for correctly substituted function.

$(S =) 50.3096 \dots ({m s}^{- 1})$ $(S =) 50.3096 \dots ({m s}^{- 1})$ (A1)(ft)

Note: Follow through from part (a).

$181 ({km h}^{- 1}) (181.114 \dots ({km h}^{- 1}))$ $181 ({km h}^{- 1}) (181.114 \dots ({km h}^{- 1}))$ (A1)(ft) (C3)

Note: Award the final (A1)(ft) for correct conversion of their speed to $km h^{- 1}$ $km h^{- 1}$ .

[3 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

Syllabus sections

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

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

On the following grid, sketch the graph of $f$ $f$ .
EXN.2.SL.TZ0.9a:
Show that $T_{0} = 100$ $T_{0} = 100$ .
EXN.2.SL.TZ0.9c:
Find the temperature of the water when $t = 15$ $t = 15$ .
21M.1.SL.TZ1.8d:
Solve $f (x) > 0$ $f (x) > 0$ for $x > 0$ $x > 0$ .
21M.2.SL.TZ2.6a:
Show that $A_{0} = 100$ $A_{0} = 100$ .
21N.2.SL.TZ0.2b:
Sketch the graph of $f$ $f$ on the following grid.
21M.1.SL.TZ2.5a:
Find the exact value of $a$ $a$ .
21M.2.SL.TZ2.6c:
Find, correct to the nearest $10$ $10$ years, the time taken after the plant’s death for $25 %$ $25 %$ of the carbon-14 to decay.
20N.1.SL.TZ0.S_4a:
Find the value of $a$ $a$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ $f^{- 1} (x)$ .
21N.2.SL.TZ0.2a:
Find the values of $x$ $x$ for which $f (x) = 0$ $f (x) = 0$ .
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ $f^{- 1} (\sqrt{32})$ .
EXN.2.SL.TZ0.9b:
Show that $k = \frac{1}{10} \ln \frac{10}{7}$ $k = \frac{1}{10} \ln \frac{10}{7}$ .
21M.2.SL.TZ2.6b:
Show that $k = \frac{\ln 2}{5730}$ $k = \frac{\ln 2}{5730}$ .
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 $K$ $K$ .
21N.1.AHL.TZ0.3:
Solve the equation $\log_{3} \sqrt{x} = \frac{1}{2 \log_{2} 3} + \log_{3} (4 x^{3})$ $\log_{3} \sqrt{x} = \frac{1}{2 \log_{2} 3} + \log_{3} (4 x^{3})$ , where $x > 0$ $x > 0$ .
21N.1.SL.TZ0.8a:
Show that $a = 8$ $a = 8$ .
21N.1.SL.TZ0.8d.ii:
Find the value of $p$ $p$ and the value of $q$ $q$ .
21N.1.SL.TZ0.8d.i:
Show that $27, p, q$ $27, p, q$ and $125$ $125$ are four consecutive terms in a geometric sequence.

Topic 2—Functions