1.7 Laws of exponents and logarithms questions for IB Maths AA

Show that $\mathrm{P}(X\le 1)={\displaystyle \frac{n+1}{2^n}}$.

Find the least value of $n$ such that the probability of obtaining at least two heads is greater than 0.99.

A blade is tested once and the result is positive. Find the probability that the blade has a crack.

The blade in part (a) is tested a second time and the result is again positive. Find the probability that the blade has a crack.

A blade is tested $n$ times and every result is positive. Find the least value of $n$ for which the probability that the blade has a crack is greater than 0.999.

Consider the case $\mathrm{P}(F)=0.1$ and $a=0.8$. A component is tested once and fails. Find the probability that it is faulty.

Show that $\mathrm{O}(F\mid \text{fail})={\displaystyle \frac{a}{1-a}}\times \mathrm{O}(F)$.

Find a similar expression for $\mathrm{O}(F\mid \text{pass})$ in terms of $a$ and $\mathrm{O}(F)$.

Write down $\mathrm{P}(R_k\mid F)$ and $\mathrm{P}(R_k\mid F^{\mathrm{\prime }})$.

Hence show that $\mathrm{O}(F\mid R_k)={\left({\displaystyle \frac{a}{1-a}}\right)}^k\times \mathrm{O}(F)$.

A component is tested $m+n$ times. It fails $m$ of the tests and passes the other $n$, in some order. Let $S$ be the event that these results are obtained, in that order. Show that $\mathrm{O}(F\mid S)={\left({\displaystyle \frac{a}{1-a}}\right)}^{m-n}\times \mathrm{O}(F)$.

Explain briefly the significance of the result in part (e) for a component that fails exactly as many tests as it passes.

A component is tested 10 times. It fails 6 tests and passes 4. Find the probability that it is faulty.

A component is tested repeatedly and fails every test. Find the least number of tests after which the probability that it is faulty is greater than 0.999.

A faulty component is tested 10 times. Find the probability that, from its results, the probability that it is faulty is calculated to be greater than 0.5.

Find $f^{\mathrm{\prime }}(x)$.

Find the least value of $a$ such that $f$ is increasing for all $x\ge a$.

Let $g$ be the function $f$ restricted to the domain $x\ge \ln 3$.

Explain why the inverse function $g^{-1}$ exists.

Find the domain of $g^{-1}$.

The regression line of $y$ on $t$ has equation $y=at+b$. Find the value of $a$ and the value of $b$.

Write down the value of $r$, Pearson's product-moment correlation coefficient, for these data.

A scatter diagram suggests that the relationship between $t$ and $y$ is not linear. Let $z=\ln y$.

The regression line of $z$ on $t$ has equation $z=ct+d$. Find the value of $c$ and the value of $d$.

Find the value of Pearson's product-moment correlation coefficient for $t$ and $z$. Hence state which of the two regression lines gives the better model for the data.

Show that the equation $z=ct+d$ can be written in the form $y=A{e}^{kt}$, and find the value of $A$ and the value of $k$.

Use the model $y=A{e}^{kt}$ to estimate the number of downloads in week 9.

Comment on the reliability of your estimate in part (f).

According to the model $y=A{e}^{kt}$, find the first week in which the number of downloads is more than 100 thousand.

Use your GDC to solve the inequality $2^x>x^2$, for $x>0$.

Use your GDC to solve the inequality $3^x>x^3$, for $x>0$.

Show that, for $x>0$, the inequality $a^x>x^a$ is equivalent to ${\displaystyle \frac{\ln x}{x}}<{\displaystyle \frac{\ln a}{a}}$.

Let $g(x)={\displaystyle \frac{\ln x}{x}}$, for $x>0$.

Find the coordinates of the stationary point on the graph of $y=g(x)$ and show that it is a maximum.

Hence show that ${\mathrm{e}}^x\ge x^{\mathrm{e}}$ for all $x>0$, with equality only when $x=\mathrm{e}$.

Hence determine which is larger, ${\mathrm{e}}^{\pi }$ or ${\pi }^{\mathrm{e}}$, justifying your answer.

You are given that $g(x)<0$ for $0<x<1$, $g(1)=0$ and $g(x)\to 0$ as $x\to \mathrm{\infty }$.

Now let $1<a<\mathrm{e}$. Explain why the equation $g(x)=g(a)$ has exactly two solutions for $x>0$, one of which is $x=a$, and the other of which, $x=b$, satisfies $b>\mathrm{e}$.

Hence write down, in terms of $a$ and $b$, the solution of $a^x>x^a$ for $x>0$, where $1<a<\mathrm{e}$, and state the value of $b$ when $a=2$.

Suggest how the solution of $a^x>x^a$ changes when $a>\mathrm{e}$. Use part (b) to support your answer.

Find the set of values of $a>1$ for which $a^x>x^a$ is true for all $x>a$.