Extended Questions (Section B HL Only) (DP IB Analysis & Approaches (AA): HL): Exam Questions

The function $f$ has a derivative given by $f\text{'}\left(x\right)=\frac{1}{3x\left(k-x\right)}$, for $x\in \mathrm{ℝ},x\ne 0,x\ne k$, where $k$ is a positive constant.

The expression for $f\text{'}\left(x\right)$ can be written in the form $\frac{a}{3x}+\frac{b}{k-x}$, where $a,b\in \mathrm{ℝ}$.

Find $a$ and $b$ in terms of $k$.

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

$R$ is the population of rabbits on an island. The rate of change of the population can be modelled by the differential equation

$\frac{\mathrm{d}R}{\mathrm{d}t}=\frac{3R\left(k-R\right)}{4k}$

where $t$ is the time measured in years, $t\ge 0$, and $k$ is the maximum population that the island can support.

The initial population of the rabbits is 20.

By solving the differential equation, show that $R=\frac{20k{\mathrm{e}}^{\frac{3}{4}t}}{k-20+20{\mathrm{e}}^{\frac{3}{4}t}}$.

After two years, the population of rabbits has risen to 70.

Find $k$.

Find the value of $t$ at which the population of rabbits is growing at its fastest rate.

A particle is moving in a vertical line, and its acceleration, $a$ m s⁻², at time $t$ seconds, $t\ge 0$, is given by $a=\frac{1-v}{2}$, where $v$ is the velocity in metres per second and $v<1$.

The particle starts at a fixed origin O with initial velocity $v_0$ m s⁻¹.

By solving a suitable differential equation, show that the velocity of the particle at time $t$ is given by $v\left(t\right)=1-{\mathrm{e}}^{-\frac{t}{2}}\left(1-v_0\right)$.

The particle moves down in the negative direction until its displacement from O reaches a minimum. Then the particle changes direction and starts moving up, in the positive direction.

(i) If the initial velocity of the particle is $-3$ m s⁻¹, find the time at which the minimum displacement of the particle from O occurs, giving your answer in exact form.

(ii) If $T$ is the time in seconds when the displacement reaches its smallest value, show that $T=2\ln \left(1-v_0\right)$.

(i) Find a general expression for the displacement of the particle from O, in terms of $t$ and $v_0$.

(ii) Hence find an expression for the minimum displacement of the particle, in terms of $v_0$.

Let $v\left(T-k\right)$ represent the velocity of the particle $k$ seconds before the minimum displacement, and $v\left(T+k\right)$ the velocity of the particle $k$ seconds after the minimum displacement.

(i) Show that $v\left(T-k\right)=1-{\mathrm{e}}^{\frac{k}{2}}$.

(ii) Given that $v\left(T+k\right)=1-{\mathrm{e}}^{-\frac{k}{2}}$, show that $v\left(T-k\right)+v\left(T+k\right)\le 0$.

Paola is modelling a small vase from her house for her maths project. To model the edge of the vase in cross-section, she decides to use a function $f$ of the form

$f\left(x\right)=\frac{q{\mathrm{e}}^{\frac{x}{2}}}{2+{\mathrm{e}}^x}$

where $x\in \mathrm{ℝ},x\ge 0$ and $q\in {\mathrm{ℝ}}^{+}$.

The function and the vase are represented in the diagrams below.

The vertical height of the vase, OB, is measured along the $x$-axis. The radius of the vase’s opening is OA, and its base radius is BC.

To model the vase, the region enclosed by the graph of $y=f\left(x\right)$, the $x$-axis, the $y$-axis and the line $x=\ln 43$ is rotated 360° about the $x$-axis.

Show that the volume of the solid of revolution formed is $\frac{14q^2\pi }{45}$ units³.

The volume of the actual vase is 100 cm³.

Use this information to find the value of $q$.

Find the cross-sectional radius of the vase

(i) at its base,

(ii) at its widest point.

Paola wants to investigate how the cross-sectional radius of the vase changes.

Sketch a graph of the derivative of $f$, and use it to find the value of $x$ at which the cross-sectional radius of the vase is decreasing most rapidly.

The points $\mathrm{A}\left(2,3,0\right)$, $\mathrm{B}\left(-2,4,1\right)$, $\mathrm{C}\left(1,-1,3\right)$ and $\mathrm{D}\left(5,-2,2\right)$ lie on the plane ${\Pi }_1$ and form a parallelogram, where [AB] and [CD] are one pair of parallel sides and [BC] and [AD] are the other pair of parallel sides. Each unit on the coordinate grid is equivalent to 1 cm in length.

Find the vector product of $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{AC}}$.

Hence, or otherwise, find the Cartesian equation of the plane ${\Pi }_1$.

A second plane ${\Pi }_2$ contains the point with position vector $\left(\begin{array}{c}5\\ -3\\ 5\end{array}\right)$ and also the line L, which has vector equation $\mathit{r}=\left(\begin{array}{c}6\\ 1\\ 2\end{array}\right)+\lambda \left(\begin{array}{c}4\\ -1\\ -1\end{array}\right).$

Show that ${\Pi }_1$ and ${\Pi }_2$are parallel.

A parallelepiped is a 3D object made up of six faces that are parallelograms lying in pairs of parallel planes. EFGH is a parallelogram on ${\Pi }_2$ that is congruent to ABCD, and the points A, B, C and D on ${\Pi }_1$ are joined to the points E, F, G and H respectively on ${\Pi }_2$ to form a parallelepiped.

Given that the coordinates of E are $\left(3,6,0\right)$, find the coordinates of H.

The volume of a parallelepiped can be found using the formula $\left|\left(\mathit{a}\times \mathit{b}\right)·\mathit{c}\right|$, where $\mathit{a}$, $\mathit{b}$ and $\mathit{c}$ are vectors corresponding to three edges meeting at a single vertex of the parallelepiped.

Show that the volume of the parallelepiped ABCDEFGH is 40 cm³.