Systems of linear equations (by hand and with GDC): notes and practice questions

A biologist is studying a species of fish. The length, $L$ cm, and weight, $W$ g, of each fish in a sample are recorded.

The lengths of the fish are summarized in the following box and whisker diagram.

Find the largest value of $L$ that would not be considered an outlier.

The regression line of $W$ on $L$ is $W = 25L - 150$. The regression line of $L$ on $W$ is $L = 0.03W + 10.5$.

One of the fish in the sample weighs 200 g. Estimate the length of this fish.

Find the mean weight of the fish in the sample.

A biologist records the number of eggs in the nests of a certain species of bird. The results for 25 nests are shown in the following frequency table.

It is given that the mean number of eggs per nest is 2.6.

(a) Find the value of p and the value of q.

The biologist enters the data into a competition. The competition score is calculated using the formula $S = 10x - 5$, where $x$ is the number of eggs in a nest.

(b) Find the mean competition score.

Two drones, Drone X and Drone Y, have position vectors with respect to an origin O given respectively by

$\boldsymbol{r}_X = \begin{pmatrix} 10 \\ -2 \\ 2 \end{pmatrix} + t \begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}$

$\boldsymbol{r}_Y = \begin{pmatrix} -1 \\ -2 \\ 9 \end{pmatrix} + t \begin{pmatrix} 3 \\ 2 \\ -1 \end{pmatrix}$

where $t$ represents the time in minutes and $0 \le t \le 3$.

Entries in each column vector give the displacement east of O, the displacement north of O and the distance above sea level, all measured in kilometres.

(a) Find the three-figure bearing on which Drone Y is travelling.

(b) Show that Drone X travels at a greater speed than Drone Y.

(c) Find the acute angle between the two drones' lines of flight. Give your answer in degrees.

The two drones' lines of flight cross at point P.

(d) (i) Find the coordinates of P.

(ii) Determine the length of time between the first drone arriving at P and the second drone arriving at P.

(e) Let $D(t)$ represent the distance between Drone X and Drone Y for $0 \le t \le 3$.

Find the minimum value of $D(t)$.

The following diagram shows the graph of $y = f(x)$, where $f(x) = \frac{Cx + D}{x+2}$, for $x \in \mathbb{R}$, $x \neq -2$ and $C, D \in \mathbb{Z}$.

The graph has a y-intercept at $(0, -3)$ and an x-intercept at $(3, 0)$.

(a) Find the value of $C$ and the value of $D$.

(b) Describe a sequence of transformations that maps the graph of $g(x) = \frac{1}{x}$ onto the graph of $y = f(x)$.

Consider the system of equations

$\begin{cases} kx + 4y = 2 \\ x + ky = -1 \end{cases}$

where $k \in \mathbb{R}$.

(a) Find the values of $k$ for which the system has a unique solution and find this solution in terms of $k$.

(b) For the system of equations, find the value of $k$ for which there is:

(i) no solution;

(ii) infinitely many solutions. For this case, write down the general solution.

A team of architects is designing a new building and needs to define a support beam's orientation. The beam must be perpendicular to two existing structural walls, Wall A and Wall B. The equations of the planes representing these walls are given by:

Wall A ($\Pi_A$): $x + 2y - z = 7$

Wall B ($\Pi_B$): $2x - y + 3z = 1$

Find a Cartesian equation of the plane ($\Pi_C$) that represents the orientation of the support beam, given that it passes through the origin (0, 0, 0).

Find the coordinates of the point where Wall A, Wall B, and the support beam's plane ($\Pi_C$) intersect.

Consider the non-zero vectors $\vec{u}$ and $\vec{v}$. Let $\theta$ be the angle between $\vec{u}$ and $\vec{v}$.

Using the definitions of $\vec{u} \cdot \vec{v}$ and $\vec{u} \times \vec{v}$ in terms of $|\vec{u}|$, $|\vec{v}|$ and $\theta$, show that $(\vec{u} \cdot \vec{v})^2 + |\vec{u} \times \vec{v}|^2 = |\vec{u}|^2|\vec{v}|^2$.

A triangle PQR has vertices P(1, 0, 1), Q($a, b$, 2) and R(4, 1, 1), where $a, b \in \mathbb{Q}$.

The vectors $\vec{u}$ and $\vec{v}$ are defined as $\vec{u} = \vec{PQ}$ and $\vec{v} = \vec{PR}$.

It is given that $\vec{u} \cdot \vec{v} = 4$ and the area of triangle PQR is $\frac{\sqrt{14}}{2}$ square units.

Find the value of $|\vec{u} \times \vec{v}|$.

Hence, or otherwise, find the value of $|\vec{u}|$.

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

Consider a new point S, the vector $\vec{w}$ is defined as $\vec{w} = \vec{RS}$.

It is given that $\vec{u} \cdot \vec{w} = 0$ and $\vec{v} \cdot \vec{w} = 0$, and the area of triangle PRS is 10 square units.

Assuming that $a = 2$, find the possible vectors for $\vec{w}$.

The concentration of a certain nutrient in a plant's root system, $N(t)$, after $t$ hours, is modelled by the function $N(t) = \frac{At+B}{2t+D}$ for $t \ge 0$, where $A, B, D$ are constants.

(a) Given that the model predicts a singularity (vertical asymptote) at $t = -3$, determine the value of $D$.

(b) Given that the concentration is $0.5$ units after $1$ hour and $0.8$ units after $4$ hours, determine the values of $A$ and $B$.

A function $f$ is defined by

$f(x) = \begin{cases} a \sin(x) + b, & x \le \frac{\pi}{2} \\ \cos(2x), & x > \frac{\pi}{2} \end{cases}$

where $a, b \in \mathbb{R}$.

(a) Find an equation relating $a$ and $b$, given that $f$ is continuous at $x = \frac{\pi}{2}$.

(b) Given further that $f'(0) = 3$, find the value of $a$ and the value of $b$.

A student, Leo, has taken seven mathematics tests. His scores are recorded as $85, 92, 78, 88, x, y, 85$, where $x$ and $y$ are integers and $x < y$.

The mean of these seven scores is $86$. The range of the scores is $18$.

(a) Find the value of $x$ and the value of $y$.

(b) Find the median and the mode of Leo's seven test scores.

A student is investigating the relationship between the number of hours spent studying for a test ($x$) and the score obtained on the test ($y$). The following bivariate data set was collected, where $p$ and $q$ represent unknown test scores.

The regression line of $y$ on $x$ has the equation $y = 4.8x + 36.4$.

The regression line passes through the mean point $(\bar{x}, \bar{y})$.

Given that $\bar{x} = 7$ hours, verify that the mean test score $\bar{y} = 70$.

Given that the test score $q$ was 4 points higher than $p$, find the value of $p$ and the value of $q$.

A satellite dish is positioned at a ground control station $A(3, 1, 4)$. The dish is designed to track a celestial object whose path can be modelled by a line $L_1$ with vector equation $\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + t \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$, where $t \in \mathbb{R}$.

The plane $\Pi_1$ of the satellite dish contains the line $L_1$ and passes through the ground control station $A$.

Show that the Cartesian equation of the plane $\Pi_1$ is $x + 2y = 5$.

Consider three large display screens in a museum, represented by the planes:

$\Pi_1 : x + 2y = 5$

$\Pi_2 : 3x + ay - z = 7$

$\Pi_3 : 2x - y + z = k$

where $a, k \in \mathbb{Q}$.

For a special holographic effect, the three planes must intersect along a single line.

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

Three planes are defined by the following equations:

$\Pi_1: x+y+z = 2$

$\Pi_2: 2x-y+z = -1$

$\Pi_3: x+2y+kz = 8$

where $k \in \mathbb{R}$.

(a) Find the value of $k$ for which the three planes do not intersect at a unique point.

(b) For $k=1$, find the coordinates of the point of intersection of the three planes.

A structural engineer is designing a framework and needs to define the orientation of certain surfaces. Consider two existing planar surfaces, $\Pi_1$ and $\Pi_2$, with the following Cartesian equations:

$\Pi_1\text{: } 2x + y - z = 5$

$\Pi_2\text{: } x - 3y + 2z = -1$

Find a Cartesian equation of a third planar surface, $\Pi_3$, which is perpendicular to both $\Pi_1$ and $\Pi_2$, and passes through the point $(1, 2, 3)$.

Determine the coordinates of the point where $\Pi_1$, $\Pi_2$, and $\Pi_3$ intersect.