DP Mathematics: Applications and Interpretation Questionbank

Show that $b\approx 0.00939$.

Find the angle through which Mars rotates on its axis each hour.

Show that the maximum value of $\omega =1.98$, correct to three significant figures.

Hence show that $a=1.6$, correct to two significant figures.

Find the value of $c$.

Find the value of $d$.

Find the minimum value of $\omega$.

the maximum value of $R\left(t\right)$.

the minimum value of $R\left(t\right)$.

Find the height of point $\text{A}$ above the ground.

Calculate the number of seconds it takes for the water wheel to complete one rotation.

Hence find the number of rotations the water wheel makes in one hour.

Find $k$.

Find $\text{A}\hat{\text{O}}\text{B}$.

Determine the rate of change of $h$ when the top of the bucket is at $\text{B}$.

Write down a sequence of transformations that will transform the graph of $y=\cos x$ onto the graph of $y=f\left(x\right)$.

Assuming a carrying capacity of $5000$ use the given values of $N\left(0\right)$ and $N\left(1\right)$ to calculate the parameters $C$ and $k$.

Use these parameters to calculate the value of $N\left(8\right)$ predicted by this model.

Comment on the likelihood of the fish population reaching $5000$.

A function $f$ is of the form $f\left(t\right)=p{\text{e}}^{q \cos \left(rt\right)}, p, q, r\in {\mathrm{ℝ}}^{+}$. Part of the graph of $f$ is shown.

The points $\text{A}$ and $\text{B}$ have coordinates $\text{A}(0, 6.5)$ and $\text{B}(5.2, 0.2)$, and lie on $f$.

The point $\text{A}$ is a local maximum and the point $\text{B}$ is a local minimum.

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

Find the value of $a$, $b$, $c$ and $d$.

the distance she runs in 20 minutes.

Calculate her distance after 8 minutes. Give your answer in km, correct to 3 decimal places.

her maximum speed, in ms^–1.

Give two reasons why the prediction in part (b)(ii) might be lower than 14.

the number of new people infected on day 6.

the day when the total number of people infected will be greater than 1000.

Use an exponential regression to find the value of $a$ and of $b$, correct to 4 decimal places.

Hence predict the total number of people infected by this disease after several months.

$k$.

Use the logistic model to find the day when the rate of increase of people infected is greatest.

Explain why the number of degrees of freedom is 2.

Perform a ${\chi }^2$ goodness of fit test at the 5% significance level. You should clearly state your hypotheses, the p-value, and your conclusion.

Use your answer to part (a) to show that the model predicts 16.7 people will be infected on the first day.

$L$.

$c$.

The solution to the differential equation is given by

$Q\left(t\right)=\frac{L}{1+C{\text{e}}^{-kt}}$

where $C$ is a constant.

Using your answer to part (f)(i), estimate the percentage of computers in city X that are expected to have been infected by the virus over a long period of time.

Find the value of $q$;

Find the difference in height between low tide and high tide.

There are two high tides on 12 December 2017. At what time does the second high tide occur?

How much time is there between the first low tide and the next high tide?

Find when the seat is 30 m above the ground for the third time.

Find the value of $p$;

Find the value of $r$.

Find the first time when the ball’s speed is changing at a rate of 2 cm s⁻².

Find the maximum speed of the ball.

For the graph of $f$, write down the period.

For the graph of $f$, write down the amplitude.

Find the coordinates of A.

Hence, write $f\left(x\right)$ in the form $p\text{cos}\left(x+r\right)$.