DP Mathematics: Applications and Interpretation Questionbank

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.

Find $f\left(1\right)$.

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

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

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

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

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

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

Kaelani has $450 \text{PGK}$.

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

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

Write down the coordinates of the local minimum point.

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

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

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 $n$.

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.

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

Find the value of $k$.

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

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

Find the smallest possible value of $k$.

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

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

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.

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

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

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

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

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.

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

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.

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

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

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

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

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

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.

Hence or otherwise, solve the inequality .

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

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

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

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.

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

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

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

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

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

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 $x=1$.

Sketch the graphs on the same set of axes.

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

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

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

Explain why $f$ is an even function.

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

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

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

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

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

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.

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

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

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

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.

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

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

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

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

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

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 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.

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

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

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

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

Find $\alpha$ and β in terms of $k$.

Show that $\alpha$ + β < −2.

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

find the value of $b$.

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

State the range of $f$.

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

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 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.

Write down the value of $c$.

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

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

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 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).

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.

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.

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.