DP Mathematics: Applications and Interpretation Questionbank

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.

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

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

Find the range of $h$.

Hence write down the domain of $h$.

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

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

Find the range of $f$.

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

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

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

Find the range of $f$.

Find the range of $h$.

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

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

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

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

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

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

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

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

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 $(g^{-1}{)}^{\prime }(x)=0$.

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.

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

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

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

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

Write down the range of $f$.

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

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

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.

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 $\frac{x}{2}+1=\left|x-2\right|$.

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.

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

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

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

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

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

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

Write down the range of $f$.

Find the gradient of $L$.

Find u.

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

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

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

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 $f(8)$.

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

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

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.

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 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.

Write down the domain of the function.

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

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

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

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 $f^{\prime }\left(x\right)$.

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 $d\in \mathbb{Z}$.

Find the value of $b$.

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

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.