DP Mathematics: Applications and Interpretation Questionbank

Find $\left(f\circ g\right)\left(x\right)$.

Solve the equation $\left(f\circ g\right)\left(x\right)=x$.

Hence or otherwise, given that $g\left(2a\right)=f^{-1}\left(2a\right)$, find the value of $a$.

Write down the other solution of $f\left(x\right)=k$.

Complete the table below placing a tick (✔) to show whether the unknown parameters $a$ and $b$ are positive, zero or negative. The row for $c$ has been completed as an example.

State the values of $x$ for which $f\left(x\right)$ is decreasing.

Find the value of $K$.

In the context of the model, state what the horizontal asymptote represents.

Find Jean-Pierre’s vertical speed after $10$ seconds. Give your answer in $\text{km} {\text{h}}^{-1}$ .

Find an equation which links $C$ and $r$.

Find, to the nearest USD, the cost of disk that has a radius of $1.1$cm.

Find the value of $a$.

Find the value of $b$.

Find the value of $c$.

Find the value of $t$ at which the population first reaches $2200$.

Hence find an expression for $h\left(x\right)$.

Find the equation for Jorge’s model.

Use this model to predict the number of fish in the lake when $t=8$.

Assuming the proportion of marked fish in the second sample is equal to the proportion of marked fish in the lake, show that the estate manager will estimate there are now $2000$ fish in the lake.

Find the total cost of buying $40$ litres of gas at Leon’s gas station.

Find the value of $p$.

Use this model to find the percentage of information retained by his students $36$ hours after Professor Wei’s lecture.

State a mathematical reason why Professor Wei might believe this.

Write down one possible limitation of the domain of the model.

Find $L^{-1}\left(70\right)$.

Show that $I=\frac{9}{d^2}$.

Sketch the curve of $I$ on the axes below showing clearly the point $(1.5, 4)$.

Calculate the amount of the drug remaining in the body $10$ hours after the injection.

Find the value of

(i)     $a$.

(ii)    $b$.

(iii)   $c$.

Write down the equation of the axis of symmetry of the graph.

the initial dose of the drug.

Find the value of $p$.

the percentage of the drug that leaves the body each hour.

Find the value of $h$ when $\theta =160°$.

Write down the amplitude of $g\left(\theta \right)$.

Find $\frac{dy}{dx}$.

Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.

Hence find the cross-sectional area of the tunnel.

Hence find the maximum height of the tunnel.

Write down the integral which can be used to find the cross-sectional area of the tunnel.

Find $\frac{dy}{dx}$.

Hence find the maximum height of the tunnel.

$x=4$.

$x=6$.

Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.

Write down the integral which can be used to find the cross-sectional area of the tunnel.

Hence find the cross-sectional area of the tunnel.

Find the equation of the least squares regression quadratic curve for these four points.

By considering the gradient of this curve when $x=4$, explain why it may not be a good model.

Find the equation of the new model.

Write down the value of $A$.

Find the value of $k$.

Write down the value of $c$.

Hence form two equations in terms of $a$ and $b$.

Hence find the equation of the quadratic curve.

Find the area of the shaded region in Irina’s design.

maximum value of $h$.

Find the period of the function.

Find the height of $\text{C}$ above the ground when $t=2$.

At any given instant, find the probability that point $\text{C}$ is visible from Tim’s window.

minimum value of $h$.

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

The wind speed increases. The blades rotate at twice the speed, but still at a constant rate.

At any given instant, find the probability that Tim can see point $\text{C}$ from his window. Justify your answer.

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

Write down the amplitude of the function.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

Calculate the angle, in degrees, that the blade $\text{[BC]}$ turns through in one second.

The wind speed increases and the blades rotate faster, but still at a constant rate.

Given that point $\text{C}$ is now higher than $110 \text{m}$ for $1$ second during each complete rotation, find the time for one complete rotation.

Find the period of the function.

maximum value of $h$.

minimum value of $h$.

Find the height of $\text{C}$ above the ground when $t=2$.

Find the time, in seconds, it takes for the blade $\text{[BC]}$ to make one complete rotation under these conditions.

Write down the amplitude of the function.

Sketch the function $h\left(t\right)$ for $0\le t\le 5$, clearly labelling the coordinates of the maximum and minimum points.

Find the time, in seconds, that point $\text{C}$ is above a height of $100 \text{m}$, during each complete rotation.

Find the value of $t$ when the ball hits the ground.

Find the value of $b$.

Write down the height of the ball above the ground at the instant it is hit by the bat.

Find the value of $a$.

Given $0<M<8$, find the range for $N$.

Find the average number of earthquakes in a year with a magnitude of at least $7.2$.

Find the value of $a$.

Find the value of $b$.

Write down the value of $d$.

Write down three simultaneous equations for $a, b$ and $c$.

Hence, or otherwise, find the values of $a, b$ and $c$.

Determine whether the crate will fit through the archway. Justify your answer.

Use the model to determine the total length of time, in years, for which a graduate is expected to be in debt after graduating from university.

Determine an equation for the axis of symmetry of the parabola that models the archway.

the equation of the principal axis.

Write down one reason why a quadratic function would not be a good model for the number of hours of daylight per day, across a number of years.

Hence or otherwise find the equation of this model in the form:

$f\left(t\right)=a \cos \left(bt\right)+d$

the amplitude.

the period.

Find the population of the Bulbul birds at the start of the migration season.

According to this model, find the smallest possible population of Bulbul birds during the migration season.

Calculate the time taken for the population to decrease below 1400.

Find the population of the Bulbul birds after 5 days.

Using the values in the table and your answer to part (b), sketch the graph of $y=d\left(s\right)$ for 0 ≤ $s$ ≤ 10 and −10 ≤ $d$ ≤ 60, clearly showing the vertex.

Hence, identify why Model A may not be appropriate at lower speeds.

Use Model B to calculate an estimate for the braking distance at a speed of $20\text{m}{\text{s}}^{-1}$.

Calculate the percentage error in the estimate in part (e).

Write down a second equation to represent Model A, when the speed is $10\text{m}{\text{s}}^{-1}$.

Find the values of $p$ and $q$.

Find the coordinates of the vertex of the graph of $y=d\left(s\right)$.

It is found that once a driver realizes the need to stop their vehicle, 1.6 seconds will elapse, on average, before the brakes are engaged. During this reaction time, the vehicle will continue to travel at its original speed.

A truck approaches an intersection with speed $s\text{m}{\text{s}}^{-1}$. The driver notices the intersection’s traffic lights are red and they must stop the vehicle within a distance of $330\text{m}$.

Using model B and taking reaction time into account, calculate the maximum possible speed of the truck if it is to stop before the intersection.

$a$.

$b$.

$d$.

Calculate the number of revolutions of the Ferris wheel per ride.

Find in which city, X or Y, the computer virus is spreading more easily. Justify your answer using your results from part (b).

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.

$L$.

$c$.

Assuming that the ground is horizontal and the ball is not hit by the arrow, find the $x$ coordinate of the point where the ball lands.

Using the data in the table write down the equation for an appropriate non-linear regression model.

By considering large values of $t$ write down one criticism of the model found in (b)(ii).

Use your answer from part (b)(ii) to estimate the time taken for the number of infected computers to double.

Write down the maximum and minimum value of $v$.

Write down two transformations that will transform the graph of $y=v\left(t\right)$ onto the graph of $y=i\left(t\right)$.

Sketch the graph of $y=p\left(t\right)$ for 0 ≤ $t$ ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.

Find the total time in the interval 0 ≤ $t$ ≤ 0.02 for which $p\left(t\right)$ ≥ 3.

Find $p_{av}$(0.007).

With reference to your graph of $y=p\left(t\right)$ explain why $p_{av}\left(T\right)$ > 0 for all $T$ > 0.

Given that $p\left(t\right)$ can be written as $p\left(t\right)=a\text{sin}\left(b\left(t-c\right)\right)+d$ where $a$, $b$, $c$, $d$ > 0, use your graph to find the values of $a$, $b$, $c$ and $d$.

The equation of the axis of symmetry is x = p. Find p.

Hence, show that a = 2.

Let $f\left(x\right)=p{x}^2+qx-4p$, where p ≠ 0. Find Find the number of roots for the equation $f\left(x\right)=0$.

Justify your answer.

Find the equation of the axis of symmetry of the graph of $f$.

Write down the value of $h$.

Find the value of $k$.

The graph of a second function, $g$, is obtained by a reflection of the graph of $f$ in the $y$-axis, followed by a translation of $\left(\begin{array}{c}-3\\ 6\end{array}\right)$.

Find the coordinates of the vertex of the graph of $g$.

Use the model to find the volume of the barrel.

The empty barrel is being filled with water. The volume $V{\text{m}}^3$ of water in the barrel after $t$ minutes is given by $V=0.8(1-{\text{e}}^{-0.1t})$. How long will it take for the barrel to be half-full?

Find the equation of the axis of symmetry of the graph of $f$.

(i)     Write down the value of $h$.

(ii)     Find the value of $k$.

Show that $f(2\pi )=2\pi$.

Find the coordinates of ${\text{P}}_0$ and of ${\text{P}}_1$.

Find the equation of $L$.

Show that the distance between the $x$-coordinates of ${\text{P}}_k$ and ${\text{P}}_{k+1}$ is $2\pi$.

A saw has a toothed edge which is 300 mm long. Find the number of complete teeth on this saw.

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

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

Find the value of $p$;

Find the value of $q$;

Find the value of $r$.

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

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

Find the value of $p$.

Find the value of $q$.

Use the model to find the depth of the water 10 hours after high tide.

The range of $f$ is $k$ ≤ $f\left(x\right)$ ≤ $m$. Find $k$ and $m$.

Find the range of $g$.

Find the value of $b$ and of $c$.

Find the period of $g$.

The equation $g\left(x\right)=12$ has two solutions where $\pi$ ≤ $x$ ≤ $\frac{4\pi }{3}$. Find both solutions.

(i)     Find the value of $c$.

(ii)     Show that $b=\frac{\pi }{6}$.

(iii)     Find the value of $a$.

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

(i)     Find $w$.

(ii)     Hence or otherwise, find the maximum positive rate of change of $g$.

Use this information to write down an equation involving $x$ and $y$.

Use this ratio to write down $y$ in terms of $x$.

Find the value of $x$ and of $y$.

On graph paper, draw a scatter diagram for these data. Use a scale of 2 cm to represent 5 hours on the $x$-axis and 2 cm to represent 10 points on the $y$-axis.

(i)     $\overline{x}$, the mean number of hours spent on social media;

(ii)     $\overline{y}$, the mean number of IB Diploma points.

Plot the point $(\overline{x},\overline{y})$ on your scatter diagram and label this point M.

Write down the equation of the regression line $y$ on $x$ for these eight male students.

Draw the regression line, from part (e), on your scatter diagram.

Use the given equation of the regression line to estimate the number of IB Diploma points that this girl obtained.

Write down a reason why this estimate is not reliable.

Find the value of a.

Find the coordinates of R.

Line L₃ is parallel to line L₂ and passes through the point (2, 3).

Find the equation of line L₃. Give your answer in the form y = mx + c.

Calculate $f(1)$.

Sketch the graph of $y=f(x)$ for $-7⩽x⩽4$ and $-30⩽y⩽30$.

Write down the equation of the vertical asymptote.

Write down the coordinates of the $x$-intercept.

Write down the possible values of $x$ for which and $f^{\prime }(x)>0$.

Find the solution of $f(x)=g(x)$.

Write down the amount of money Jashanti saves per month.

Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car.

Calculate how much extra money Jashanti needs.

Using only this information, write down an equation in terms of a and b.

Using this information, write down a second equation in terms of a and b.

Hence find the value of a and of b.

The graph intersects the x-axis at a second point, P.

Find the x-coordinate of P.

Find the value of $a$ and of $b$.

Find $g^{\prime }(x)$.

Show that $k=6$.

Find the equation of the tangent to the graph of $y=g(x)$ at $x=2$. Give your answer in the form $y=mx+c$.

Use your answer to part (a) and the value of $k$, to find the $x$-coordinates of the stationary points of the graph of $y=g(x)$.

Find $g^{\prime }(-1)$.

Hence justify that $g$ is decreasing at $x=-1$.

Find the $y$-coordinate of the local minimum.

Write down the value of $c$.

Find the value of $a$ and of $b$.

Write down the second $x$-intercept of the function.

Consider the following graphs of quadratic functions.

The equation of each of the quadratic functions can be written in the form $y=a{x}^2+bx+c$, where $a\ne 0$.

Each of the sets of conditions for the constants $a$, $b$ and $c$, in the table below, corresponds to one of the graphs above.

Write down the number of the corresponding graph next to each set of conditions.

Find the cost of producing 70 shirts.

Find the value of s.

Find the number of shirts produced when the cost of production is lowest.

Find f'(x)

Find the gradient of the graph of f at $x=-\frac{1}{2}$.

Find the x-coordinate of the point at which the normal to the graph of f has gradient $-\frac{1}{8}$.

Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.

Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.

Write down an expression for $W$ in terms of $p$.

Find the price, $p$, that will give Maria the highest weekly profit.

Find the number of fruit flies which were placed in the container.

Find the number of fruit flies that are in the container after 6 days.

The maximum capacity of the container is 8000 fruit flies.

Find the number of days until the container reaches its maximum capacity.

Calculate the volume of this pan.

Find the radius of the sphere in cm, correct to one decimal place.

Find the value of $a$.

Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.

Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.

In the context of this model, state what the value of 19 represents.

Find the value of A.

Interpret what A represents in this context.

Find the time since the experiment began for the bacteria population to be equal to 40A.

Write down what the value of 150 represents in the context of the question.

Find the value of $a$.

Find the total time that the baking tin is in the oven.

State the domain of $P$.

Find the distance from the centre of Orangeton to the point at which the road meets the highway.

This straight road crosses the highway and then carries on due north.

State whether the straight road will ever cross the river. Justify your answer.

Find the value of $a$.

Find the time, in years, for the population to decrease to 20 turtles.

There is a number $m$ beyond which the turtle population will not decrease.

Find the value of $m$. Justify your answer.

Write down the maximum amount of yeast in this solution.

Write down, and simplify, an expression for the car’s value when Gabriella purchased it.

Find the value of $k$.

Find the value of $n$.

Find the exact value of each of the zeros of $f$.

Expand the expression for $f(x)$.

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

Use your answer to part (b)(ii) to find the values of $x$ for which $f$ is increasing.

Draw the graph of $f$ for $-3⩽x⩽3$ and $-40⩽y⩽20$. Use a scale of 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 5 units on the $y$-axis.

Write down the coordinates of the point of intersection.

Find the value of k.

Using your value of k , find f ′(x).

Use your answer to part (b) to show that the minimum value of f(x) is −22 .

Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.

Write down the equation of the vertical asymptote.

Write down the equation of the horizontal asymptote.

Calculate the value of x for which f(x) = 0 .

Sketch the curve for −1 < x < 3 and −2 < y < 12.

A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.

Find $\frac{\text{dy}}{\text{dx}}$.

Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.

Write down the $y$-intercept of the graph.

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

Show that $a=8$.

Find $f(2)$.

Write down the $x$-coordinates of these two points;

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Write down the range of $f(x)$.

Write down the number of possible solutions to the equation $f(x)=5$.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.

Write down the derivative of $f$.

Find the point on the graph of $f$ at which the gradient of the tangent is equal to 6.

Write down a formula for $A$, the surface area to be coated.

Express this volume in $\text{c}{\text{m}}^3$.

Write down, in terms of $r$ and $h$, an equation for the volume of this water container.

Show that $A=\pi r^2+\frac{1000000}{r}$.

Find $\frac{\text{d}A}{\text{d}r}$.

Using your answer to part (e), find the value of $r$ which minimizes $A$.

Find the value of this minimum area.

Find the least number of cans of water-resistant material that will coat the area in part (g).

Find $\frac{\text{d}y}{\text{d}x}$.

Find the gradient of this tangent at point P.

Find the equation of this tangent. Give your answer in the form y = mx + c.

Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.

Use your graphic display calculator to find the zero of f (x).

Use your graphic display calculator to find the coordinates of the local minimum point.

Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.

Find the value of $P$ if no vases are sold.

Differentiate $P\left(x\right)$.

Find $f^{\prime }\left(x\right)$.

Find the gradient of the graph of $y=f\left(x\right)$ at $x=2$.

Find the equation of the tangent line to the graph of $y=f\left(x\right)$ at $x=2$. Give the equation in the form $ax+by+d=0$ where, $a$, $b$, and $d\in \mathbb{Z}$.

Write down the number of Elvis impersonators in $1977$.

Calculate the time taken for the number of Elvis impersonators to reach $130 000$.

Calculate the number of Elvis impersonators when $t=70$.

The world population in $2047$ is projected to be $9 500 000 000$ people.

Use this information to explain why the model for the number of Elvis impersonators is unrealistic.

Write down the initial design fee charged for each order.

Find the total amount charged for an order of $94$ T-shirts.

Write down the number of T-shirts in an order for which EnYear charged $63$ euros.

An order of $p$ T-shirts will be charged the same price by both M-Line and EnYear.

Find the value of $p$.

Find the value of $c$.

Write down the equation for the axis of symmetry of the graph.

Use the symmetry of the graph to show that the second solution is $x=4$.

Write down the $x$-intercepts of the graph.

On graph paper, draw the graph of $y=f\left(x\right)$ for  $-10\le x\le 4$  and  $-14\le y\le 14$. Use a scale of $1 \text{cm}$ to represent $1$ unit on the $x$-axis and $1 \text{cm}$ to represent $2$ units on the $y$-axis.

Write down the equation of $T$.

Draw the tangent $T$ on your graph.

Given $f\left(a\right)=5.5$ and $f\text{'}\left(a\right)=-6$, state whether the function, $f$, is increasing or decreasing at $x=a$. Give a reason for your answer.