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Date	November 2019	Marks available	5	Reference code	19N.3.AHL.TZ0.Hdm_2
Level	Additional Higher Level	Paper	Paper 3	Time zone	Time zone 0
Command term	Find	Question number	Hdm_2	Adapted from	N/A

Question

In parts (b) and (c), ${\left( {abc \ldots } \right)_n}$ denotes the number ${abc \ldots }$ written in base $n$ , where $n \in {\mathbb{Z}^ + }$ . For example, ${\left( {359} \right)_n} = 3{n^2} + 5n + 9$ .

State Fermat’s little theorem.

[2]

a.i.

Find the remainder when ${15^{1207}}$ is divided by $13$ .

[5]

a.ii.

Convert ${\left( {7A2} \right)_{16}}$ to base $5$ , where ${\left( A \right)_{16}} = {\left( {10} \right)_{10}}$ .

[4]

Consider the equation ${\left( {1251} \right)_n} + {\left( {30} \right)_n} = {\left( {504} \right)_n} + {\left( {504} \right)_n}$ .

Find the value of $n$ .

[4]

Markscheme

EITHER

${a^p} \equiv a\left( {{\text{mod}}\,p} \right)$ A1

where $p$ is prime A1

${a^{p - 1}} \equiv 1\left( {{\text{mod}}\,p} \right)$ A1

where $p$ is prime and $p$ does not divide $a$ (or equivalent statement) A1

[2 marks]

a.i.

${15^{1207}} \equiv {2^{1207}}\left( {{\text{mod}}\,13} \right)$

${2^{12}} \equiv 1\left( {{\text{mod}}\,13} \right)$ (M1)(A1)

${2^{1207}} = {\left( {{2^{12}}} \right)^{100}}{2^7}$ (M1)

${2^{1207}}\left( { \equiv {2^7}} \right) \equiv 11\left( {{\text{mod}}\,13} \right)$ (M1)A1

the remainder is $11$

Note: Award as above for using $15$ instead of $2$ .

[5 marks]

a.ii.

${\left( {7A2} \right)_{16}} = 7 \times {16^2} + 10 \times 16 + 2$ M1

$= 1954$ A1

EITHER

$5\left| \!{\underline {\, {1954} \,}} \right.$

$390\,r\,4$

$78\,r\,0$

$15\,r\,3$ M1

$3\,r\,0$

$0\,r\,3$

$1954 = 3 \times {5^4} + 0 \times {5^3} + 3 \times {5^2} + 0 \times {5^1} + 4$ M1

THEN

${\left( {7A2} \right)_{16}} = {\left( {30304} \right)_5}$ A1

[4 marks]

the equation can be written as

${n^3} + 2{n^2} + 5n + 1 + 3n = 2\left( {5{n^2} + 4} \right)$ M1A1

$\Rightarrow {n^3} - 8{n^2} + 8n - 7 = 0$ (M1)

Note: The (M1) is for an attempt to solve the original equation.

$n = 7$ A1

[4 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.7—Laws of exponents and logs

Show 49 related questions

22M.1.SL.TZ1.8b.i:
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22M.1.AHL.TZ1.10b.ii:
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22M.1.SL.TZ2.8b:
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22M.2.AHL.TZ2.12e:
By solving the logistic differential equation, show that its solution can be expressed in the form

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21N.2.AHL.TZ0.10d:
Sketch the graph of $f$ for $- 30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.
SPM.1.SL.TZ0.6b:
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19N.3.AHL.TZ0.Hdm_2c:
Consider the equation ${\left( {1251} \right)_n} + {\left( {30} \right)_n} = {\left( {504} \right)_n} + {\left( {504} \right)_n}$ .

Find the value of $n$ .
SPM.1.SL.TZ0.6a:
Show that ${\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2}$ .
22M.1.SL.TZ1.8a.ii:
Given that $p > 0$ and $S_{\infty} = 3 + \sqrt{3}$ , find the value of $x$ .
22M.1.SL.TZ1.8b.ii:
Write down $d$ in the form $k \ln x$ , where $k \in ℚ$ .
22M.1.AHL.TZ1.10a.ii:
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22M.1.AHL.TZ1.10a.iii:
Given that $p > 0$ and $S_{\infty} = 3 + \sqrt{3}$ , find the value of $x$ .
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 $n$ .
18M.1.AHL.TZ1.H_5:
Solve ${\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}$ .
17N.1.AHL.TZ0.H_1:
Solve the equation ${\log _2}(x + 3) + {\log _2}(x - 3) = 4$ .
20N.3.AHL.TZ0.Hdm_3a.i:
Write down the remainder when $14^{2022}$ is divided by $7$ .
19N.3.AHL.TZ0.Hdm_2b:
Convert ${\left( {7A2} \right)_{16}}$ to base $5$ , where ${\left( A \right)_{16}} = {\left( {10} \right)_{10}}$ .
18M.2.SL.TZ1.S_8b:
Use the regression equation to estimate the value of y when x = 3.57.
19M.1.AHL.TZ2.H_7:
Solve the simultaneous equations

${\text{lo}}{{\text{g}}_2}6x = 1 + 2\,{\text{lo}}{{\text{g}}_2}y$

$1 + {\text{lo}}{{\text{g}}_6}x = {\text{lo}}{{\text{g}}_6}\left( {15y - 25} \right)$ .
20N.3.AHL.TZ0.Hdm_3b.ii:
The base $13$ number $1 y 93 y 25$ is divisible by $6$ . Find the possible values of the digit $y$ .
18M.1.AHL.TZ2.H_11a:
Show that ${\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x$ where $r,\,x \in {\mathbb{R}^ + }$ .
20N.3.AHL.TZ0.Hdm_3a.ii:
Use Fermat’s little theorem to find the remainder when $14^{2022}$ is divided by $17$ .
17M.2.AHL.TZ2.H_6:
Given that ${\log _{10}}\left( {\frac{1}{{2\sqrt 2 }}\left( {p + 2q} \right)} \right) = \frac{1}{2}\left( {{{\log }_{10}}p + {{\log }_{10}}q} \right),{\text{ }}p > 0,{\text{ }}q > 0$ , find $p$ in terms of $q$ .
20N.3.AHL.TZ0.Hdm_3b.i:
Prove that a number in base $13$ is divisible by $6$ if, and only if, the sum of its digits is divisible by $6$ .
18M.2.SL.TZ1.S_8a:
Find the value of a and of b.
EXN.1.SL.TZ0.2:
Solve the equation $2 \ln x = \ln 9 + 4$ . Give your answer in the form $x = p e^{q}$ where $p, q \in ℤ^{+}$ .
22M.1.AHL.TZ1.1:
Find the value of $\int_{1}^{9} (\frac{3 \sqrt{x} - 5}{\sqrt{x}}) d x$ .
21N.2.AHL.TZ0.10a.ii:
$y$ -axis.
22M.1.AHL.TZ1.10a.i:
Show that $p = \pm \frac{1}{\sqrt{3}}$ .
22M.1.SL.TZ1.2a:
The expression $\frac{3 \sqrt{x} - 5}{\sqrt{x}}$ can be written as $3 - 5 x^{p}$ . Write down the value of $p$ .
22M.1.SL.TZ1.5:
Consider the curve with equation $y = (2 x - 1) e^{k x}$ , where $x \in ℝ$ and $k \in ℚ$ .

The tangent to the curve at the point where $x = 1$ is parallel to the line $y = 5 e^{k} x$ .

Find the value of $k$ .
22M.1.SL.TZ1.8a.i:
Show that $p = \pm \frac{1}{\sqrt{3}}$ .
21N.1.SL.TZ0.8b:
Write down an expression for $f^{- 1} (x)$ .
21N.2.AHL.TZ0.10c:
The oblique asymptote of the graph of $f$ can be written as $y = a x + b$ where $a, b \in ℚ$ .

Find the value of $a$ and the value of $b$ .
21N.1.SL.TZ0.8c:
Find the value of $f^{- 1} (\sqrt{32})$ .
17M.1.AHL.TZ1.H_1:
Find the solution of ${\log _2}x - {\log _2}5 = 2 + {\log _2}3$ .
16N.1.AHL.TZ0.H_7:
Solve the equation ${4^x} + {2^{x + 2}} = 3$ .
18M.2.SL.TZ1.S_8c:
The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.
19N.3.AHL.TZ0.Hdm_2a.i:
State Fermat’s little theorem.
19N.1.AHL.TZ0.H_10d:
The area enclosed by the graph of $y = f\left( x \right)$ and the line $y = 4$ can be expressed as ${\text{ln}}\,v$ . Find the value of $v$ .
21N.2.AHL.TZ0.10e.i:
Express $\frac{1}{f (x)}$ in partial fractions.
21N.2.AHL.TZ0.10e.ii:
Hence find the exact value of $\int_{0}^{3} \frac{1}{f (x)} d x$ , expressing your answer as a single logarithm.
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.
21N.2.AHL.TZ0.10a.i:
$x$ -axis.
21N.2.AHL.TZ0.10b:
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