DP Mathematics: Applications and Interpretation Questionbank

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

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

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

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 $p$.

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

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 $g\left(0\right)$.

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

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

Write down the value of $a$.

Find the value of $b$.

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.

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

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

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

State the range of $g$.

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.

Find the coordinates of P.

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

Determine the value of $k$.

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 $a$, $b$, $c$, $d$, $e\in \mathbb{Z}$.

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

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

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 $y=8x^3-12x^2+16x-24$.

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 $\lambda$.

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 $d$.

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

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.

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 $g$.

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

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

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

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

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

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

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

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

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

Find the period of $g$.

Find the range of $g$.

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 $g^{-1}\left(x\right)$.