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Date	November 2021	Marks available	5	Reference code	21N.1.SL.TZ0.8
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	8	Adapted from	N/A

Question

Consider the function $f (x) = a^{x}$ where $x, a \in ℝ$ and $x > 0, a > 1$ .

The graph of $f$ contains the point $(\frac{2}{3}, 4)$ .

Consider the arithmetic sequence $\log_{8} 27, \log_{8} p, \log_{8} q, \log_{8} 125,$ where $p > 1$ and $q > 1$ .

Show that $a = 8$ .

[2]

a.

Write down an expression for $f^{- 1} (x)$ .

[1]

b.

Find the value of $f^{- 1} (\sqrt{32})$ .

[3]

c.

Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

[4]

d.i.

Find the value of $p$ and the value of $q$ .

[5]

d.ii.

Markscheme

$f (\frac{2}{3}) = 4$ OR $a^{\frac{2}{3}} = 4$ (M1)

$a = 4^{\frac{3}{2}}$ OR $a = {(2^{2})}^{\frac{3}{2}}$ OR $a^{2} = 64$ OR $\sqrt[3]{a} = 2$ A1

$a = 8$ AG

[2 marks]

a.

$f^{- 1} (x) = \log_{8} x$ A1

Note: Accept $f^{- 1} (x) = \log_{a} x$ .
Accept any equivalent expression for $f^{- 1}$ e.g. $f^{- 1} (x) = \frac{\ln x}{\ln 8}$ .

[1 mark]

b.

correct substitution (A1)

$\log_{8} \sqrt{32}$ OR $8^{x} = 32^{\frac{1}{2}}$

correct working involving log/index law (A1)

$\frac{1}{2} \log_{8} 32$ OR $\frac{5}{2} \log_{8} 2$ OR $\log_{8} 2 = \frac{1}{3}$ OR $\log_{2} 2^{\frac{5}{2}}$ OR $\log_{2} 8 = 3$ OR $\frac{\ln 2^{\frac{5}{2}}}{\ln 2^{3}}$ OR $2^{3 x} = 2^{\frac{5}{2}}$

$f^{- 1} (\sqrt{32}) = \frac{5}{6}$ A1

[3 marks]

c.

METHOD 1

equating a pair of differences (M1)

$u_{2} - u_{1} = u_{4} - u_{3} (= u_{3} - u_{2})$

$\log_{8} p - \log_{8} 27 = \log_{8} 125 - \log_{8} q$

$\log_{8} 125 - \log_{8} q = \log_{8} q - \log_{8} p$

$\log_{8} (\frac{p}{27}) = \log_{8} (\frac{125}{q}), \log_{8} (\frac{125}{q}) = \log_{8} (\frac{q}{p})$ A1A1

$\frac{p}{27} = \frac{125}{q}$ and $\frac{125}{q} = \frac{q}{p}$ A1

$27, p, q$ and $125$ are in geometric sequence AG

Note: If candidate assumes the sequence is geometric, award no marks for part (i). If $r = \frac{5}{3}$ has been found, this will be awarded marks in part (ii).

METHOD 2

expressing a pair of consecutive terms, in terms of $d$ (M1)

$p = 8^{d} \times 27$ and $q = 8^{2 d} \times 27$ OR $q = 8^{2 d} \times 27$ and $125 = 8^{3 d} \times 27$

two correct pairs of consecutive terms, in terms of $d$ A1

$\frac{8^{d} \times 27}{27} = \frac{8^{2 d} \times 27}{8^{d} \times 27} = \frac{8^{3 d} \times 27}{8^{2 d} \times 27}$ (must include 3 ratios) A1

all simplify to $8^{d}$ A1

$27, p, q$ and $125$ are in geometric sequence AG

[4 marks]

d.i.

METHOD 1 (geometric, finding $r$ )

$u_{4} = u_{1} r^{3}$ OR $125 = 27 {(r)}^{3}$ (M1)

$r = \frac{5}{3}$ (seen anywhere) A1

$p = 27 r$ OR $\frac{125}{q} = \frac{5}{3}$ (M1)

$p = 45, q = 75$ A1A1

METHOD 2 (arithmetic)

$u_{4} = u_{1} + 3 d$ OR $\log_{8} 125 = \log_{8} 27 + 3 d$ (M1)

$d = \log_{8} (\frac{5}{3})$ (seen anywhere) A1

$\log_{8} p = \log_{8} 27 + \log_{8} (\frac{5}{3})$ OR $\log_{8} q = \log_{8} 27 + 2 \log_{8} (\frac{5}{3})$ (M1)

$p = 45, q = 75$ A1A1

METHOD 3 (geometric using proportion)

recognizing proportion (M1)

$p q = 125 \times 27$ OR $q^{2} = 125 p$ OR $p^{2} = 27 q$

two correct proportion equations A1

attempt to eliminate either $p$ or $q$ (M1)

$q^{2} = 125 \times \frac{125 \times 27}{q}$ OR $p^{2} = 27 \times \frac{125 \times 27}{p}$

$p = 45, q = 75$ A1A1

[5 marks]

d.ii.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.i.

[N/A]

d.ii.

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

Show 82 related questions

22M.1.SL.TZ1.8b.i:
Show that $p = \frac{2}{3}$ .
22M.1.AHL.TZ1.10b.ii:
Write down $d$ in the form $k \ln x$ , where $k \in ℚ$ .
19M.2.SL.TZ1.T_5a:
Calculate, in CAD, the total amount John pays for the bicycle.
18M.2.AHL.TZ1.H_1b:
Calculate the number of positive terms in the sequence.
19M.1.SL.TZ1.S_10c:
At point P, the graphs of $f$ and $g$ intersect for the 21st time. Find the coordinates of P.
18N.1.SL.TZ0.S_3a:
Find u₈.
22M.3.AHL.TZ1.1e:
By using a suitable table of values or otherwise, determine the smallest positive integer, greater than $1$ , that is both a triangular number and a pentagonal number.
18M.2.SL.TZ1.S_7b:
Hence find the value of n such that $\sum\limits_{k = 1}^n {{x_k} = 861}$ .
18M.2.SL.TZ1.S_7a:
Given that x_k_{+ 1} = x_k + a, find a.
17M.1.AHL.TZ2.H_3b:
the value of $r$ ;
17N.1.SL.TZ0.S_2a:
Find the common difference.
19N.1.SL.TZ0.S_1a:
Find the common difference.
22M.3.AHL.TZ1.1d:
Hence show that $P_{5} (n) = \frac{n (3 n - 1)}{2}$ for $n \in ℤ^{+}$ .
22M.1.SL.TZ1.8b.ii:
Write down $d$ in the form $k \ln x$ , where $k \in ℚ$ .
22M.1.AHL.TZ1.10b.i:
Show that $p = \frac{2}{3}$ .
22M.1.AHL.TZ1.10b.iii:
The sum of the first $n$ terms of the series is $\ln (\frac{1}{x^{3}})$ .

Find the value of $n$ .
22M.1.SL.TZ2.2b:
Given that the $n$ ^th term of this sequence is $- 33$ , find the value of $n$ .
22M.1.SL.TZ2.2c:
Find the common difference, $d$ .
18M.2.AHL.TZ1.H_7b:
Find the approximate number of fish in the lake at the start of 2042.
17N.1.SL.TZ0.S_2b:
Find the tenth term.
19N.1.SL.TZ0.S_1c:
Find the sum of the first $20$ terms.
19M.2.SL.TZ1.S_7b.ii:
Find the value of $m$ .
EXN.2.SL.TZ0.7c.i:
Find the value of $N$ .
EXN.2.SL.TZ0.7b:
Given that $J_{n}$ follows a geometric sequence, state the value of the common ratio, $r$ .
EXN.2.SL.TZ0.7d:
Find Jane’s total earnings at the start of her $10$ th year of employment. Give your answer correct to the nearest dollar.
20N.2.SL.TZ0.T_5g:
Show that the tank will never be completely filled using this pump.
17M.1.AHL.TZ1.H_7a:
find the value of $d$ .
21M.1.SL.TZ1.3:
Consider an arithmetic sequence where $u_{8} = S_{8} = 8$ . Find the value of the first term, $u_{1}$ , and the value of the common difference, $d$ .
17M.1.SL.TZ2.S_1a:
Find the common difference.
17M.1.SL.TZ2.S_1b:
Find the tenth term.
20N.2.SL.TZ0.T_5b:
Show that the volume of the tank is $624 000 m^{3}$ , correct to three significant figures.
19M.1.SL.TZ1.S_10d:
The following diagram shows part of the graph of $g$ reflected in the $x$ -axis. It also shows part of the graph of $f$ and the point P.

Find an expression for the area of the shaded region. Do not calculate the value of the expression.
19M.1.SL.TZ1.S_10a:
Find the two smallest non-zero values of $x$ for which $f(x) = g(x)$ .
19M.2.SL.TZ2.S_10b:
Find the exact value of ${S_k}$ .
19M.2.SL.TZ2.S_10c:
Show that $F = 3240$ .
EXN.1.SL.TZ0.4b:
Prove that the sum of the first $n$ terms of this arithmetic sequence is a square number.
EXN.2.SL.TZ0.7a:
Show that $H_{n} = 2400 n + 67 600$ .
21M.2.SL.TZ2.3a:
Given that the $k$ th term of the sequence is zero, find the value of $k$ .
18M.1.SL.TZ1.T_7b:
On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.
18M.2.AHL.TZ1.H_7a:
Show that there will be approximately 2645 fish in the lake at the start of 2020.
EXN.2.SL.TZ0.7c.ii:
For the value of $N$ found in part (c) (i), state Helen’s annual salary and Jane’s annual salary, correct to the nearest dollar.
19N.1.SL.TZ0.S_1b:
Find the first term.
21M.2.SL.TZ2.3b:
Let $S_{n}$ denote the sum of the first $n$ terms of the sequence.

Find the maximum value of $S_{n}$ .
EXN.1.SL.TZ0.4a:
Show that $u_{1} = 4$ .
18N.2.AHL.TZ0.H_1a:
Find the common ratio of this sequence.
22M.1.SL.TZ2.2a:
State the value of the first term, $u_{1}$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ .
17N.2.SL.TZ0.T_2b:
Find the value of $k$ .
18M.1.SL.TZ1.T_7a.ii:
For that day find the weight of Sergei’s first lift.
20N.2.SL.TZ0.T_5c:
Write down the common difference, $d$ .
18M.2.SL.TZ2.T_4b:
Calculate the café’s total profit for the first 12 weeks.
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
19M.2.SL.TZ1.S_7a:
Write down the first three non-zero terms of ${w_n}$ .
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ .
17M.1.AHL.TZ1.H_7b:
determine the value of $\sum\limits_{r = 1}^N {{u_r}}$ .
19M.2.SL.TZ2.S_10d:
An infinite geometric series is given as ${S_\infty } = a + \frac{a}{{\sqrt 2 }} + \frac{a}{2} + \ldots$ , $a \in {\mathbb{Z}^ + }$ .

Find the largest value of $a$ such that ${S_\infty } < F$ .
16N.1.SL.TZ0.T_13a:
Find the year in which the comet was seen from Earth for the fifth time.
17M.1.SL.TZ2.S_1c:
Find the sum of the first ten terms of the sequence.
17M.1.SL.TZ1.T_5a:
Find the distance from the base of this ladder to the top rung.
17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
18M.1.SL.TZ1.T_7a.i:
For that day find how much weight was added after each lift.
16N.1.SL.TZ0.T_13b:
Determine how many times the comet has been seen from Earth up to the year 2014.
17M.1.SL.TZ2.T_5a:
Diagram $n$ is formed with 52 sticks. Find the value of $n$ .
17N.2.SL.TZ0.T_2e:
Calculate the total distance Carlos runs in the first year.
20N.2.SL.TZ0.T_5e.ii:
Write down the number of hours that the pump was pumping fuel into the tank.
17M.1.SL.TZ1.T_5b:
The company also makes a ladder that is 1050 cm long.

Find the maximum number of rungs in this 1050 cm long ladder.
20N.2.SL.TZ0.T_5d:
Find the amount of fuel pumped into the tank in the $13th$ hour.
20N.2.SL.TZ0.T_5f:
Find the total amount of fuel pumped into the tank in the first $8$ hours.
19M.2.SL.TZ1.T_5e:
John purchased the bicycle in 2008.

Justify why John should not insure his bicycle in 2019.
17M.1.SL.TZ2.T_5b:
Find the total number of sticks used by Tomás for all 24 diagrams.
20N.2.SL.TZ0.T_5e.i:
Find the value of $n$ such that $u_{n} = 0$ .
20N.2.SL.TZ0.T_5a:
Find $h$ , the height of the tank.
21N.1.SL.TZ0.8a:
Show that $a = 8$ .
21N.1.SL.TZ0.8d.i:
Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.
19M.2.SL.TZ1.T_5b:
Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
19M.2.SL.TZ1.T_5c:
Calculate, in years, when the bicycle value will be less than 50 USD.
19M.2.SL.TZ1.T_5d:
Find the total amount John has paid to insure his bicycle for the first 5 years.
17N.2.SL.TZ0.T_2a.i:
Write down the distance Rosa runs in the third training session;
17N.2.SL.TZ0.T_2a.ii:
Write down the distance Rosa runs in the $n$ th training session.
17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
17N.2.SL.TZ0.T_2d:
Find the distance Carlos runs in the fifth month of training.
18M.2.SL.TZ2.T_4d:
Calculate the tea-shop’s total profit for the first 12 weeks.

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