DP Mathematics: Applications and Interpretation Questionbank

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

Find $k$.

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

Hence write down the domain of $h$.

The price of gas at Erica’s gas station is $\$1.30$ per litre. A customer must buy a minimum of $10$ litres of gas. The total cost at Erica’s gas station is cheaper than Leon’s gas station when $x>k$.

Find the minimum value of $k$.

On day $n$ of the fitness programmes Daniella runs more than Charlie for the first time.

Find the value of $n$.

Show that $I=\frac{9}{d^2}$.

Whilst swimming, Scarlett can hear the siren only if the sound intensity at her location is greater than $1.5\times {10}^{-6}$ ${\text{W\hspace{0.05em}m}}^{-2}$.

Find the values of $d$ where Scarlett cannot hear the siren.

Find the value of

(i)     $a$.

(ii)    $b$.

(iii)   $c$.

Hence find the cross-sectional area of the tunnel.

Hence find the maximum height of the tunnel.

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

Find the range of $h$.

Use the data in the second table to find the value of $m$ and the value of $b$ for the regression line, $\ln x=m(\ln d)+b$.

Find $g\left(0\right)$.

maximum value of $h$.

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

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

Find the number of each type of ticket sold.

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

Find the value of $b$.

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.

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).

Determine the maximum amount of coffee the cafe can make that will not result in a loss of money for the morning.

Solve this system of three equations to find the value of $a$, $b$ and $c$.

Show that $4a+2b+c=34$.

Given that $\underset{T\to \infty }{\lim }k=A$ and $\underset{T\to 0}{\lim }k=0$, sketch the graph of $k$ against $T$.

Show that $\text{ln}y=qx+\text{ln}p$.

By finding the equation of a suitable regression line, show that $p=1.83$ and $q=0.986$.

$c$.

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

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

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

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 value of $x$ and of $y$.

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

Write down the coordinates of the point of intersection.

Find $f^{\prime }\left(x\right)$.

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 $g$.

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

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

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

Find the value of $p$.

Find the value of $b$.

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

Interpret what A represents in this context.

Find the $y$-intercept of the graph of $f\left(x\right)+3$.

Calculate the volume of this pan.

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

Write down the derivative of $f$.

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

Describe the transformation $f\left(x+1\right)$.

Find the remainder when $p\left(x\right)$ is divided by $\left(x-2\right)$.

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

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

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

Find the inverse function $f^{-1}$, stating its domain.

Sketch the graph of $y=g\left(x\right)$. State the equations of any asymptotes and the coordinates of any intercepts with the axes.

Sketch the graph of $y=f\left(x\right)$ for $-\frac{5\pi }{8}⩽x⩽\frac{\pi }{8}$.

With reference to your graph, explain why $f$ is a function on the given domain.

Hence, show that there are no solutions to $(g^{-1}{)}^{\prime }(x)=0$.

Show that $(f\circ g)(x)=x^4-4x^2+3$.

The equation $(f\circ g)(x)=k$ has exactly two solutions, for $0⩽x⩽2.25$. Find the possible values of $k$.

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

Show that $k=6$.

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 the equation of $L_2$. Give your answer in the form $ax+by+d=0$, where $a, b, d\in \mathrm{\mathbb{Z}}$.

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

Find the value of $q$;

Find the period of $g$.

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

Find the value of $n$.

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

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

Kaelani has $450 \text{PGK}$.

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.

Find the value of $c$.

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

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.

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 $k$.

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

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

Find an expression for the inverse function$f^{-1}\left(x\right)$. The domain is not required.

The endpoints of the minute hand and hour hand meet when $\theta =k$.

Find the smallest possible value of $k$.

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

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

In the context of the question, interpret your answer to part (b)(i).

Write down the value of $c$.

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

Assuming that the model found in part (a) remains valid, estimate the percentage of trees in stock when $d=25$.

On the same set of axes draw the graph of $y=g\left(x\right)$, showing any intercepts and asymptotes.

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

Find the value of $b$.

Find the value of $a$.

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

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

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

the equation of the principal axis.

For the first year of the model, find the length of time when there are more than $10$ hours and $30$ minutes of daylight per day.

The outer dome of a large cathedral has the shape of a hemisphere of diameter 32 m, supported by vertical walls of height 17 m. It is also supported by an inner dome which can be modelled by rotating the curve $y=33-0.08x^3$ through 360° about the $y$-axis between $y$ = 0 and $y$ = 33, as indicated in the diagram.

Find the volume of the space between the two domes.

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

Find the value of this area.

Hence explain how a straight line graph could be drawn using the coordinates in the table.

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

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

Sketch the graph of $x$ against $t$, labelling the maximum point of the graph with its coordinates.

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

(i)     Write down the value of $h$.

(ii)     Find the value of $k$.

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

Expand the expression for $f(x)$.

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

Find the gradient of this tangent at point P.

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

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

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

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

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).

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

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 $d$.

Write down the amount of money Jashanti saves per month.

Find the inverse of $f$.

Find $f(8)$.

Find the range of values of $k$ for which $f(x)=k$ has no solution.

Write down the domain and range of $f^{-1}$.

find the value of $f\left(4\right)$.

Sketch the curve $y=f\left(x\right)$, 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

Sketch the graph of $y=f\left(\left|x\right|\right)$.

Determine the area of the region enclosed between the graph of $y=f\left(\left|x\right|\right)$, the $x$-axis and the lines with equations $x=-1$ and $x=1$.

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

Explain why $f$ has no inverse on the given domain.

Find the inverse function $g^{-1}$ and state its domain.

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

On the following grid, sketch the graph of $(f\circ g)(x)$, for $0⩽x⩽2.25$.

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

Find the value of $c$.

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

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

For the value of $a$ found in part (b), find an expression for $g^{-1}\left(x\right)$.

Find the value of $r$.

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

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

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.

Find the value of $b$.

Use your answer to part (b) to write down the value of $n$ to the nearest integer.

Find the equation of the least squares regression line of ${\log }_{10} P$ against ${\log }_{10} d$.

Explain why this graph indicates that $P$ is inversely proportional to $d^n$.

Find the value of $t$ at which the ball hits the ground.

Find an expression for the height $h$ of the ball at time $t$.

The graph of the function $f\left(x\right)=\ln x$ is translated by $\left(\begin{array}{c}a\\ b\end{array}\right)$ so that it then passes through the points $(0, 1)$ and $({\text{e}}^3, 1+\ln 2)$ .

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

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

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

Hence find the cross-sectional area of the tunnel.

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

Find $h^{-1}\left(10\right)$.

Write down the range of $h^{-1}$.

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

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

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 $b$.

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

Find an expression for $f^{-1}\left(x\right)$. You are not required to state a domain.

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.

The rate, $A$, of a chemical reaction at a fixed temperature is related to the concentration of two compounds, $B$ and $C$, by the equation

$A=k{B}^x{C}^y$, where $x$, $y$, $k\in \mathbb{R}$.

A scientist measures the three variables three times during the reaction and obtains the following values.

Find $x$, $y$ and $k$.

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.

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

Find the new value of the parameter identified in part (e)(i).

when the height of the object is zero.

the value of $m$ when $p=0$.

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

the number of new people infected on day 6.

Hence find the area enclosed by the exponential function, the $x$-axis, the $y$-axis and the line $x=4.4$.

Determine the two positions where the path of the arrow intersects the path of the ball.

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

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

Find the gradient of $L$.

Hence or otherwise, find the obtuse angle formed by the tangent line to $f$ at $x=8$ and the tangent line to $f$ at $x=2$.

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

Find the value of $k$.

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

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

Find the value of $a$.

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

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

Find the cost of producing 70 shirts.

Find f'(x)

Find the value of k.

Find the value of $a$.

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.

Prove that $p\left(x\right)$ has only one real zero.

Write down the transformation that will transform the graph of $y=p\left(x\right)$ onto the graph of $y=8x^3-12x^2+16x-24$.

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.

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

Hence or otherwise, solve the inequality .

Sketch the graphs $y=\text{si}{\text{n}}^{3}x+\text{ln}x$ and $y=1+\text{cos}x$ on the following axes for 0 < $x$ ≤ 9.

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

Solve $(g\circ f)(x)=0$.

Another function, $g$, can be written in the form $g\left(x\right)=a\times f\left(x+b\right)$. The following diagram shows the graph of $g$.

Write down the value of a and of b.

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

Write down the range of $f$.

Sketch the graph of $y=f(x)$ showing clearly the minimum point and any asymptotic behaviour.

Sketch the graph of $y=f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.

Find the equation of $N$. Give your answer in the form $ax+by+d=0$ where $a$, $b$, $d\in \mathbb{Z}$.

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

Find the value of $k$.

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

Find the value of $p$;

Find the coordinates of P.

Sketch the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$ on the same axes, clearly stating the points of intersection with any axes.

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

Find the area of triangle $\text{AOB}$ in terms of $k$.

Find $f\left(1\right)$.

The graph of $f$ has a local minimum at point $\text{A}$.

Find the coordinates of $\text{A}$.

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

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

Consider the function $g\left(x\right)=3-x$.

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

Find an expression for $P$ in terms of $d$.

Find the equation for Jorge’s model.

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

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.

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

the initial dose of the drug.

Comment on the validity of your answer to part (b).

Find the value of $k$.

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

Write down the amplitude of the function.

Comment on the appropriateness of modelling this scenario with a geometric sequence.

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.

State one criticism of this interpretation.

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

$a$.

$d$.

Find the equation of the straight line, giving your answer in the form $\text{ln}m=ap+b$, where $a\text{,}b\in \mathbb{R}$.

Write down two more equations for $a$, $b$ and $c$.

Find the equation of the regression line of $\text{ln}\left(T-25\right)$ on $t$.

her maximum speed, in ms^–1.

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

Write down a reason why this estimate is not reliable.

State the domain of $P$.

Write down the value of $h$.

Write down the maximum amount of yeast in this solution.

On the grid above, sketch the graph of f ⁻¹.

Find the value of $a$.

Hence find the value of a and of b.

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

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

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

Find the $x$-intercept of the graph of $f\left(2x\right)$.

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

Calculate how much extra money Jashanti needs.

Find $\left(f\circ g\right)\left(\left(x\text{,}y\right)\right)$.

Find $\left(g\circ f\right)\left(\left(x\text{,}y\right)\right)$.

On the same axes, sketch the graph of $f\left(-x\right)$.

Use the model to find the volume of the barrel.

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

Write down the number of solutions to $f(x)=-6$.

Consider the function $f\left(x\right)=x^4-6x^2-2x+4$, $x\in \mathbb{R}$.

The graph of $f$ is translated two units to the left to form the function $g\left(x\right)$.

Express $g\left(x\right)$ in the form $a{x}^4+b{x}^3+c{x}^2+dx+e$ where $a$, $b$, $c$, $d$, $e\in \mathbb{Z}$.

Sketch the graph of $f(x)$, indicating on it the equations of the asymptotes, the coordinates of the $y$-intercept and the local maximum.

Explain why $f$ is not a function for $-\frac{3\pi }{4}⩽x⩽\frac{\pi }{4}$.

Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y=-x$.

This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of revolution.

Explain why $f$ is an even function.

Hence, show that there are no solutions to $g^{\prime }(x)=0$;

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.

Given that the length of $\text{AM}$ is $\sqrt{45}$, find the area of triangle $\text{ANC}$.

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

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

Find the coordinates of A.

Describe the transformation by which $f\left(x\right)$ is transformed to $g\left(x\right)$.

Find the value of $d$.

Find the minimum value of $\omega$.

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

Find $f\text{'}\left(p\right)$ in terms of $k$ and $p$.

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

Find the value of $K$.

Write down the minimum number of pizzas that can be ordered.

Find the range of $h$.

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

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

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

Find the range of $f$.

Find the value of $k$.

Write down the gradient of the line through $\text{B}$ and $\text{D}$.

Find the equation of the line through $\text{B}$ and $\text{D}$. Give your answer in the form $ax+by+d=0$, where $a, b$ and $d$ are integers.

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.

Find the period of the function.

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

Find the equation of the line that the road follows.

Write down one feature of this graph which suggests a cubic function might be appropriate to model this scenario.

Write down the value of $d$.

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 the number of lines parallel to $L$ that are tangent to $f\left(x\right)$. Justify your answer.

Solve $f\left(x\right)=f^{-1}\left(x\right)$.

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

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

an appropriate range for $h\left(t\right)$.

$b$.

Identify which parameter will change.

By considering the graph of $h\left(t\right)$, determine the length of time during one revolution of the Ferris wheel for which the chair is higher than the cowboy statue.

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

Write down the coefficient of determination.

Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.

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

Write down, in the form $y=mx+c$, the equation of $L_2$.

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

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.

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

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

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

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

The maximum capacity of the container is 8000 fruit flies.

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

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

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

Find the $y$-intercept of the graph of $f\left(4x\right)$.

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

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

Write down the equation of the vertical asymptote.

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

Hence solve  in the range 0 < $x$ ≤ 9.

State with a reason whether or not $f$ and $g$ commute.

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

Draw the line $y=-6$ on the axes.

Show that $\frac{1}{x+1}-\frac{1}{x+2}=\frac{1}{x^2+3x+2}$.

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

Sketch the graph of $y=g\left(t\right)$ for t ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.

Show that the $x$-coordinate of the minimum point on the curve $y=f(x)$ satisfies the equation $\tan x=2x$.

Sketch the graphs on the same set of axes.

Explain why the inverse function $f^{-1}$ does not exist.

State the range of $f$.

Solve the inequality $\left|3x\arccos (x)\right|>1$.

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

While in Kota Kinabalu, Criselda spent $440 \text{MYR}$. She returned to the Currency Exchange counter and changed the remainder of her $\text{MYR}$ into $\text{USD}$.

Calculate the amount of $\text{USD}$ she received.

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

Write down the equation of $T$.

For $a=-\frac{\pi }{2}$, sketch the graph of $y=g\left(x\right)$. Indicate clearly the maximum and minimum values of the function.

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

Find the maximum speed of the ball.

State the range of $g$.

The tangent to $y=f\left(x\right)$ at P passes through the origin (0, 0).

Determine the value of $k$.

Show that $b\approx 0.00939$.

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

Solve $f\left(x\right)=0$.

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

State, in the context of the question, what the value of $8.50$ represents.

Find the value of $a$.

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

State a mathematical reason why Professor Wei might believe this.

Estimate the value of Roger’s laptop after $5$ years.

Write down the range of $f^{-1}\left(x\right)$.

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.

Write down the value of $A$.

Find the gradient of the line through $\text{A}$ and $\text{C}$.

Write down the $x$-coordinate of point $\text{D}$.

maximum value of $h$.

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

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

State an appropriate domain for $t$ in this model.

Find the value of $k$.

find the value of $a$ and of $b$.

the maximum height of the object.

Show that $\text{ln}\left(T-25\right)=bt+\text{ln}a$.

Sketch the graph of $C$ against $x$, labelling the maximum point and the $x$-intercepts with their coordinates.

predict the temperature of the metal rod after 3 minutes.

Use the trapezoidal rule to find an estimate for the area.

With reference to the shape of the graph, explain whether your answer to part (a)(i) will be an over-estimate or an underestimate of the area.

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

Use all the coordinates in the table to find the equation of the least squares cubic regression curve.

Write down an expression for the area enclosed by the cubic function, the $x$-axis, the $y$-axis and the line $x=4.4$.

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

$L$.

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.

state the value of $a$ and the value of $c$;

find the value of $b$.

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

Find u.

Hence, write down $f^{-1}\left(x\right)$.

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

Find the value of $q$.

(i)     Find the value of $c$.

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

(iii)     Find the value of $a$.

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

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

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

Find the value of A.

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

Write down the amount of interest he earned after 5 years.

Find the value of a.

Hence, show that a = 2.

Find the equation of $L$ in the form $y=mx+c$.

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

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

Find the remainder when $p\left(x\right)$ is divided by $\left(x-3\right)$.

Calculate $f(1)$.

Find $p_{av}$(0.007).

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Sketch the graph of $y=f\left(x\right)$ showing clearly the position of the points A and B.

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

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

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

Write down the domain of the function.

Express $x^2+3x+2$ in the form $(x+h{)}^2+k$.

Hence find the value of $p$ if ${\int }_0^{1}f(x)\text{d}x=\ln (p)$.

Find $\alpha$ and β in terms of $k$.

Show that $\alpha$ + β < −2.

Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.

Draw the line $N$ on the diagram above.

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

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.

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.

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

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

Write down the range of $f$.

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

Determine the first year in which this model predicts the average number of visitors per concert will exceed the total seating capacity of the concert hall.

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 $a$.

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

State, in the context of the question, what the value of $34.50$ represents.

Write down the zero of $f\left(x\right)$.

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

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

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

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

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

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

$x=4$.

$x=6$.

Find the equation of the new model.

Find the period of the function.

minimum value of $h$.

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.

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

Town $\text{C}$ is due north of town $\text{A}$ and the road passes through town $\text{C}$.

Find the $y$-coordinate of town $\text{C}$.

Calculate the pH of the coffee.

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.

An orchestra has a sound intensity of 6.4 × 10⁻³W m⁻² . Calculate the intensity level, $L$ of the orchestra.

Find the coordinates of $\text{X}$.

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

Write down one reason, with reference to the context, to support this decision.

Find the equation of the regression line for $\ln k$ on $\frac{1}{T}$.

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

$k$.

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.

Use the model to calculate the total amount of time when fishing should be stopped.

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.

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

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

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

Find the acute angle between $y=x$ and $L$.

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 total time that the baking tin is in the oven.

(i)     Find $w$.

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

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

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

Find the x-coordinate of P.

Find the value of s.

Calculate the amount of money he has in the account after 5 years.

Using your value of k , find f ′(x).

Find the $x$-coordinate of point A.

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

Find the gradient of $L$.

The random variable $X$ follows a Poisson distribution with a mean of $\lambda$ and $6\text{P}\left(X=3\right)=3\text{P}\left(X=2\right)-2\text{P}\left(X=1\right)+3\text{P}\left(X=0\right)$.

Find the value of $\lambda$.

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

Sketch the graph of $y=\frac{1-3x}{x-2}$, showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.

Show that there is exactly one point of inflexion, B, on the graph of $y=f\left(x\right)$.

Solve the equation $\frac{x}{2}+1=\left|x-2\right|$.

Find $f(2)$.

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

Write down the $x$-coordinate of the point of inflexion on the graph of $y=f\left(x\right)$.

find the value of $f\left(1\right)$.

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

Factorize $x^2+3x+2$.

Show that $g\left(t\right)={\left(\frac{1+t}{1-t}\right)}^2$.

Find the largest possible domain $D$ for $f$ to be a function.

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.

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

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

Write down the initial design fee charged for each order.

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 gradient of $L_1$.

Find the coordinates of point $\text{C}$.

Find the distance between points $\text{M}$ and $\text{N}$.

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 least value of $a$ such that $g$ has an inverse.

Find the range of $g$.

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

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

Sketch the graph of $V=300\sqrt{3}x-\frac{9}{4}{x}^3$, for $0\le x\le 16$.

Find the value of $c$.

Show that the equation of $L_1$ is $kx+p^{2}y-2pk=0$.

The graph of $f$ is translated by $\left(\begin{array}{c}4\\ 3\end{array}\right)$ to give the graph of $g$.
In the following diagram:

• point $\text{Q}$ lies on the graph of $g$
• points $\text{C}$, $\text{D}$ and $\text{E}$ lie on the vertical asymptote of $g$
• points $\text{D}$ and $\text{F}$ lie on the horizontal asymptote of $g$
• point $\text{G}$ lies on the $x$-axis such that $\text{FG}$ is parallel to $\text{DC}$.

Line $L_2$ is the tangent to the graph of $g$ at $\text{Q}$, and passes through $\text{E}$ and $\text{F}$.

Given that triangle $\text{EDF}$ and rectangle $\text{CDFG}$ have equal areas, find the gradient of $L_2$ in terms of $p$.

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

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

Write down the coordinates of the local minimum point.

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

Find the value of $p$.

Find the value of $f^{-1}\left(0\right)$.

Find the range of $f$.

Hence find the maximum height of the tunnel.

Find the equation of the line passing through these two points.

Hence find the equation of the quadratic curve.

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

minimum value of $h$.

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.

Write down three equations that express the information given above.

Determine whether the unknown liquid is more or less acidic than the coffee. Justify your answer mathematically.

A rock concert has an intensity level of 112 dB. Find the sound intensity, $S$.

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

Find the population of the Bulbul birds after 5 days.

Write down the value of $a$.

an appropriate domain for $h\left(t\right)$.

the distance she runs in 20 minutes.

a formula for $m$ in terms of $p$.

It is believed that two variables, $v$ and $w$ are related by the equation $v=k{w}^n$, where $k\text{,}n\in \mathbb{R}$.  Experimental values of $v$ and $w$ are obtained. A graph of $\text{ln}v$ against $\text{ln}w$ shows a straight line passing through (−1.7, 4.3) and (7.1, 17.5).

Find the value of $k$ and of $n$. 

(i)   the gradient of this line in terms of $c$;

(ii)  the $y$-intercept of this line in terms of $A$.

Explain why the number of degrees of freedom is 2.

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

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

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

Write down the value of $c$.

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 $d\in \mathbb{Z}$.

(i)     Write down the value of $k$.

(ii)     Find $g(x)$.

Write down the equation of the vertical asymptote.

Write down the equation of the horizontal asymptote.

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

Karl decides to donate this interest to a charity in France. The charity receives 170 euros (EUR). The exchange rate is 1 USD = t EUR.

Calculate the value of t.

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.

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

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

The graph of $y=f\left(x\right)$ has a local maximum at A. Find the coordinates of A.

Express $g\left(x\right)$ in the form $A+\frac{B}{x-2}$ where A, B are constants.

The function $h$ is defined by $h\left(x\right)=\sqrt{x}$, for $x$ ≥ 0.

State the domain and range of $h\circ g$.

Sketch the graphs of $y=\frac{x}{2}+1$ and $y=\left|x-2\right|$ on the following axes.

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?

Show that $a=8$.

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

Given that $\underset{x\to +\mathrm{\infty }}{lim}(g\circ f)(x)=-3$, find the value of $b$.

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

Find the equation of $L$.

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

Determine the values of $x$ for which $f(x)$ is a decreasing function.

Given that the graphs enclose a region of area 18 square units, find the value of b.

Write down the value of $f(1)$.

Calculate the amount of $\text{MYR}$ that Criselda received.

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

Draw the tangent $T$ on your graph.

For the value of $a$ found in part (b), write down the domain of $g^{-1}$.

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

Find the value of $k$.