1.9 Laws of logarithms questions for IB Maths AI

Find an expression for $f^{-1}(x)$. The domain is not required.

Solve the equation $f(x)=f^{-1}(x)$.

Find the intensity of the sound at a distance of 10 m from the speaker. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Hence find the sound level at a distance of 10 m from the speaker.

Show that $(L\circ I)(d)=a-20{\log }_{10}d$, and find the value of $a$.

Find an expression for $L^{-1}(x)$.

Use your answer to part (d) to find the intensity of a sound of 95 dB.

The city council requires the sound level at the nearest house to be no more than 85 dB. Find the minimum distance between the speaker and the nearest house.

Write down a differential equation for $T$, using a positive constant $k$.

By solving your differential equation, show that $T=22+A{\mathrm{e}}^{-kt}$, where $A$ is a constant.

Write down the value of $A$.

After 20 minutes the core temperature of a loaf is 61 °C.

Show that $k=0.0320$, correct to three significant figures.

A loaf can be sliced once its core temperature has fallen to 35 °C.

Use the model to find how long after leaving the oven a loaf can be sliced. Give your answer to the nearest minute.

Find the rate at which the core temperature is changing at the moment the loaf leaves the oven.

The head baker, Jonas, thinks that $k$ should be estimated from more than one reading. He records the core temperature of a loaf every 10 minutes.

Complete the table, giving each value of $y$ correct to three decimal places.

Find the equation of the regression line of $y$ on $t$.

Hence find Jonas's estimates for $k$ and for $A$.

Write down the value of the coefficient of determination, $R^2$, for the regression in part (f)(ii), and interpret it in context.

On busy days the ovens warm the cooling room. Jonas models the room temperature as $22+0.15t$ °C and uses the differential equation

Use Euler's method with a step length of 5 minutes to estimate the core temperature of a loaf after 40 minutes on a busy day.

On a busy day, the measured core temperature of a loaf after 40 minutes is 45.2 °C. Suggest how Jonas could use Euler's method to obtain a more accurate estimate.

Write down the value of $f(0)$.

On the same axes, sketch the graph of $y=f^{-1}(x)$.

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

Find the height of the balloon 10 minutes after it is released.

Find the atmospheric pressure at this height.

Find an expression for $(p\circ h)(t)$.

The expression for $(p\circ h)(t)$ can be written in the form $101.3k^t$. Find the value of $k$, giving your answer to six significant figures.

Find the time, in minutes, after which the pressure around the balloon has fallen to 70 kPa.

Find an expression for $p^{-1}(x)$.

Valentina is hiking in the Andes. Her barometer reads 62 kPa. Use your answer to part (f) to estimate her altitude.

State one reason why the estimate in part (g) may not be accurate.

The matrix $\mathit{A}$ can be written as $k\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$, where $k>0$ and $0\mathrm{°}<\theta <90\mathrm{°}$.

(i) Find the value of $k$ and the value of $\theta$.

(ii) Hence describe fully the transformation represented by $\mathit{A}$.

Find the coordinates of the images under $T$ of the points $(8,0)$ and $(8,8)$.

Find $\det \mathit{A}$, and hence find the area of $S_1$.

Amara imagines continuing the pattern indefinitely. Show that the total area of all the squares $S_0,S_1,S_2,\dots$ is finite, and find this total area.

The point $C$ is the invariant point of $T$, so that $T$ maps $C$ to itself.

Find the coordinates of $C$.

Let $\mathit{c}$ be the position vector of $C$. Show that, for every position vector $\mathit{x}$, $T(\mathit{x})-\mathit{c}=\mathit{A}(\mathit{x}-\mathit{c})$.

Hence show that the distance $C{P}_n={0.75}^n\times C{P}_0$.

Amara draws the squares $S_0,S_1,S_2,\dots$ only while the vertex $P_n$ is at least 0.2 cm from $C$.

Find the number of squares in her logo.

Amara joins $P_0$ to $P_1$, $P_1$ to $P_2$, and so on, with straight line segments.

(i) Show that $P_{n+1}-P_n$, as a vector, is ${\mathit{A}}^n\left(\begin{array}{c}8\\ 0\end{array}\right)$.

(ii) Hence find the total length of this path if it were continued indefinitely.

Amara's colleague, Femi, suggests replacing $\mathit{A}$ with $\mathit{B}=\left(\begin{array}{cc}0.45& 0.6\\ 0.6& -0.45\end{array}\right)$.

The matrix $\mathit{B}$ represents a reflection in the line $y=(\tan \alpha )x$ combined with an enlargement, centred at the origin, with scale factor 0.75.

Find the equation of the line of reflection.

Find ${\mathit{B}}^2$. Hence explain why Femi's design would not produce a spiral of squares.