# Miscellaneous (IB SL)

$\def\iixi#1#2{\left(\begin{array}{c}#1\\#2\end{array}\right)}\let\frac=\dfrac$
1 (IB/sl/2019/November/Paper2/q2)
[Maximum mark: 5]
Consider the lines $L_{1}$ and $L_{2}$ with respective equations
$$L_{1}: y=-\frac{2}{3} x+9 \text { and } L_{2}: y=\frac{2}{5} x-\frac{19}{5} .$$
 (a) Find the point of intersection of $L_{1}$ and $L_{2}$. 

A third line, $L_{3}$, has gradient $-\frac{3}{4}$
 (b) Write down a direction vector for $L_{3}$. 

$L_{3}$ passes through the intersection of $L_{1}$ and $L_{2}$.
(c) Write down a vector equation for $L_{3}$. 

 2 (IB/sl/2016/November/Paper1/q7) [Maximum mark: 7] Let $f(x)=m-\frac{1}{x}$, for $x \neq 0 .$ The line $y=x-m$ intersects the graph of $f$ in two distinct points. Find the possible values of $m$.

 3 (IB/sl/2015/May/paper2tz2/q7) [Maximum mark: 8] Let $f(x)=k x^{2}+k x$ and $g(x)=x-0.8$. The graphs of $f$ and $g$ intersect at two distinct points. Find the possible values of $k$.

 4 (IB/sl/2019/May/paper1tz2/q6) [Maximum mark: 7] Solve $\log _{4}(2-x)=\log _{16}(13-4 x)$.

5 (IB/sl/2018/November/Paper1/q4)
[Maximum mark: 6]
Let $b=\log _{2} a$, where $a>0$. Write down each of the following expressions in terms of $b$.
 (a) $\log _{2} a^{3}$
 (b) $\log _{2} 8 a$
 (c) $\log _{k} a$

6 (IB/sl/2018/May/paper1tz2/q7)
[Maximum mark: 8]
An arithmetic sequence has $u_{1}=\log _{c}(p)$ and $u_{2}=\log _{c}(p q)$, where $c>1$ and $p, q>0$.
 (a) Show that $d=\log _{\mathrm{c}}(g)$.
 (b) Let $p=c^{2}$ and $q=c^{3}$. Find the value of $\sum_{n=1}^{20} n_{*}$.

 7 (IB/sl/2017/November/Paper1/q7) [Maximum mark: 7] Consider $f(x)=\log _{x}\left(6 x-3 x^{2}\right)$, for $0 < x < 2$, where $k>0$. The equation $f(x)=2$ has exactly one solution. Find the value of $k$.

 8 (IB/sl/2017/May/paper1tz2/q7) [Maximum mark: 7] Solve $\log _{2}(2 \sin x)+\log _{2}(\cos x)=-1$, for $2 \pi < x < \frac{5 \pi}{2}$.

9 (IB/sl/2016/May/paper1tz2/q3)
[Maximum mark: 6]
Let $x=\ln 3$ and $y=\ln 5$. Write the following expressions in terms of $x$ and $y$.
 (a) $\ln \left(\frac{5}{3}\right)$. 
 (b) $\ln 45$. 

10 (IB/sl/2015/May/paper1tz1/q3)
[Maximum mark: 6]
 (a) Given that $2^{\prime \prime}=8$ and $2^{n}=16$, write down the value of $m$ and of $n$. 
 (b) Hence or otherwise solve $8^{2 x+1}=16^{2 x-3}$. 

11 (IB/sl/2016/May/paper2tz1/q7)
[Maximum mark: 8]
Note: One decade is 10 years
A population of rare birds, $P$, can be modelled by the equation $P_{0}=P_{3} \mathrm{e}^{\text {th }}$, where $P_{0}$ is the initial population, and $t$ is measured in decades. After one decade, it is estimated that $\frac{P_{1}}{P_{0}}=0.9$.
 (a) (i) Find the value of $k$. (ii) Interpret the meaning of the value of $k$. 
 (b) Find the least number of whole years for which $\frac{P_{t}}{P_{0}} < 0.75$. 

12 (IB/sl/2019/November/Paper1/q2)
[Maximum mark: 6]
In a class of 30 students, 18 are fluent in Spanish, 10 are fluent in French, and 5 are not fluent in either of these languages. The following venn diagram shows the events 'fluent in Spanish" and "fluent in French".
The values $m, n, p$ and $q$ represent numbers of students.
 (a) Write down the value of $q$. 
 (b) Find the value of $n$. 
 (c) Write down the value of $m$ and of $p$. 

13 (IB/sl/2018/November/Paper2/q1)
[Maximum mark: 6]
In a group of 35 students, some take art class $(A)$ and some take music class $(M) .5$ of these students do not take either class. This information is shown in the following Venn diagram.
 (a) Write down the number of students in the group who take art class. 
 (b) One student from the group is chosen at random. Find the probability that (i) the student does not take art class: (ii) the student takes either art class or music class, but not both. 

14 (IB/sl/2017/May/paper1tz1/q1)
[Maximum mark: 6]
In a group of 20 girls, 13 take history and 8 take economics. Three girls take both history and economics, as shown in the following venn diagram. The values $p$ and $q$ represent numbers of girls.
 (a) Find the value of (i) $p$; (ii) $q$. 
 (b) A girl is selected at random. Find the probability that she takes economics but not history. 

1](a) (12,1) (b) $\iixi{-4}{3}$ (c) $r=\iixi{12}{1} +t \iixi{-4}{3}$

2] $\quad m<-1$ or $m>1$

3] $k<0.2, k>5$

4] $x=-3$

5](a) $3 b$ (b) $3+b$ (c) $\frac{b}{3}$

6](a) show (b) 610

7] $k=\sqrt{3}$

8] $x=\frac{25 \pi}{12}, \frac{29 \pi}{12}$

9] (a) $y-x$ (b) $2 x+y$

10](a) $\quad m=3, n=4$ (b) $x=7.5$

11](a)(i) $\quad k=\ln 0.9$ (ii) population is decreasing (b) 28

12](a) 5 (b) 3 (c) 15,7

13](a) $17$ (b) (i) $\frac{18}{35}$ (ii) $\frac{24}{35}$

14](a)(i) $\quad p=10$ (ii) $q=2$ (b) $\frac{1}{4}$