Solving system of linear equations and polynomial equations (using GDC): notes and practice questions

A chemical compound is dissolving in a solvent. Initially, there are 225 grams of the compound. After 15 minutes, 144 grams of the compound remain undissolved.

The mass of the undissolved compound, M grams, remaining after t minutes, can be modelled by the differential equation $\frac{dM}{dt} = -k\sqrt{M}$, where k is a positive constant.

(a) Show that $M = (15 - \frac{t}{5})^2$.

(b) Calculate the time it takes for the entire compound to dissolve.

(a) A space probe, 'Voyager Alpha', is launched from a space station. The position of the probe at time $t$ days after launch is given by the vector

$\mathbf{r} = \begin{pmatrix} 10 \\ 20 \\ 5 \end{pmatrix} + t \begin{pmatrix} 80 \\ 40 \\ 10 \end{pmatrix}$

Distances are measured in thousands of kilometres.

Find the position vector of the probe $3$ days after launch.

(b) A second space probe, 'Explorer Beta', is launched from a different station. The position vector of this probe is given by the vector

$\mathbf{s} = \begin{pmatrix} -50 \\ -30 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 70 \\ 50 \\ 12 \end{pmatrix}$

Determine if the two flight paths intersect and, if so, state the point of intersection.

(c) The two probes were launched at the same time, so $\lambda = t$.

State, with a reason, whether the two probes actually collide.

(d) Calculate the distance between the two space stations (the initial launch points).

(e) Calculate the shortest distance that ever exists between the two probes and the time when this occurs. Assume $t \ge 0$ days.

Consider if a rocket height is, H in cm, is modelled with respect to time, t in seconds, according to the following equation:

$H(t) = at^{3} + bt^{2} + ct + d$

If the model passes through the following points:

$a$ Find the value of $d$.

$b$ Set three equations in terms of $a,b$ and $c$.

$c$ Find the values of $a,b$ and $c$.

The "LearnFast" online learning platform charges a monthly subscription fee of $\$40$ and an additional $\$7$ per premium course. If a student enrolls in a minimum of $10$ premium courses in a month, a one-time discount of $\$25$ is applied to their total bill.

This can be modelled by the following function, $L$, which gives the total cost when enrolling in a minimum of $10$ premium courses at LearnFast:

$L(x) = 7x + 15, x \ge 10$

where $x$ is the number of premium courses a student enrolls in.

Find the total cost of enrolling in $20$ premium courses at LearnFast.

Find $L^{-1}(99)$.

Another online learning platform, "SkillUp", charges a flat rate of $\$8$ per premium course, with no monthly subscription or discounts. A student must enroll in a minimum of $10$ premium courses for a direct comparison with LearnFast's discounted model.

The total cost at LearnFast is cheaper than SkillUp when $x > k$.

Find the minimum integer value of $k$.

In a controlled chemical experiment, the initial temperature, $x$ (in degrees Celsius), of a reactant mixture is adjusted before a reaction begins. The adjusted temperature, $f(x)$, is then used to determine the reaction rate, $g(f(x) )$.

The adjustment function is given by $f(x) = x - 3$, for $x \in \mathbb{R}$.

State the range of $f(x)$.

The overall reaction rate, $gf(x)$, as a function of the initial temperature $x$, is modelled by $gf(x) = 4x^2 - 24x + 30$, for $x \in \mathbb{R}$.

State the range of $gf(x)$.

The experiment is designed to achieve a specific 'target state' when the adjusted reaction rate $f(gf(x) )$ equals 0. Solve the equation $fgf(x) = 0$.

Determine the function $g(x)$.

A new manufacturing plant, 'InnovateTech', began production. On its first day, the plant produced 1200 units. Due to an optimized workflow, the production increased by 80 units on each subsequent day.

Calculate the number of units produced by InnovateTech on day 15 of its operation.

Another plant, 'QuantumFab', also started production on the same day, producing 1000 units. QuantumFab increased its production by 3% of the previous day's output on each subsequent day.

Calculate the number of units produced by QuantumFab on day 15 of its operation.

On which day, $n$, will QuantumFab's production exceed InnovateTech's production for the first time?

(a) A local community group, "Green Spaces", is fundraising for a new park playground. They received an initial donation of £750 and plan to collect £200 each month from local businesses. Their goal is to raise £12,000.

Determine how many more months it will take the group to fundraise for their goal.

(b) To speed up the fundraising, the group decides to increase the monthly contribution by a fixed amount of £$k$ each month, starting from the original £$200$, so they can reach their £$12\,000$ goal within 18 months.

Find the smallest value of $k$ (rounded to the nearest dollar) so that the group can achieve their goal.

(c) If the "Green Spaces" group continues to increase their monthly contributions at the rate found in part (b) and spends none of it, they aim for a larger milestone of £$25\,000$ to include additional facilities.

Determine after how many more months (from the end of the 18-month period in part (b) ) they will have at least £$25\,000$ fundraised.

An architect is designing a parabolic arch for a new building. The shape of the arch can be modelled by the function $f(x) = ax^2 + bx + c$, where $x$ and $f(x)$ are measured in metres. The highest point of the arch (the vertex) is at $(2, 4)$. One end of the arch is at the origin $(0, 0)$, and the other end is at $(p, 0)$.

(a) Find the value of $p$.

(b) Find the value of

(i) $a$.

(ii) $b$.

(iii) $c$.

(c) Write down the equation of the axis of symmetry of the arch.

The concentration of a reactant A, in mg/L, in a chemical reaction over time $t$ (in minutes) is modelled by the function:

$C(t) = -t^3 + 15t^2 - 48t + 100$, for $0 \le t \le 10$.

(a) Find the coordinates of the local minimum point of the concentration.

(b) Find the coordinates of the local maximum point of the concentration.

(c) Find the set of values of $t$ for which the concentration of reactant A is above $100$ mg/L.

A new fertilizer is being tested on a specific plant species. The concentration of a key nutrient, $C$, in the plant (in mg/L) is modelled by the function $C(x) = 5 - \frac{30}{x-2}$, where $x$ is the amount of fertilizer added (in grams). The amount of fertilizer used is restricted to $-4 \le x \le 8$, where $x \neq 2$ (due to experimental constraints).

Find the range of the nutrient concentration $C(x)$.

Determine the amount of fertilizer $x$ that results in a nutrient concentration of $-1$ mg/L. Give your answer in the form $C^{-1}(-1)$.

(a) An engineer is designing a section of a roller coaster track. To ensure a smooth ride, the track must follow a specific curve. Three sensor readings are taken at different horizontal positions along this section of the track, giving the following height coordinates: $(1, 4)$, $(2, 7)$, and $(3, 14)$.

Assuming the track follows a quadratic path of the form $y = ax^2 + bx + c$, find the equation of this quadratic curve.

(b) A fourth sensor reading is taken at a horizontal position of $x=4$, recording a height of $y=28$. Determine if this new point lies on the quadratic curve found in part (a).

(c) Due to the discrepancy, the engineer decides that a cubic function, $y = Ax^3 + Bx^2 + Cx + D$, would be a better fit for the track. Find the equation of the cubic function that passes through all four points: $(1, 4)$, $(2, 7)$, $(3, 14)$, and $(4, 28)$.

Let the function $C(w)$ represent the cost in dollars per display stand, where $w$ is the width of the stand in meters.

$C(w) = \frac{500}{w^2} + 1.2$ for $5 \le w \le 15$.

Find the range of $C$.

The function $C^{-1}$ is the inverse function of $C$.

Find $C^{-1}(15)$.

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

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

The concentration of a certain chemical, $C$, in a solution over a period of time can be modelled using the function $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$, where $t$ is the time in hours after the experiment begins.

Sketch the graph of $C(t) = -0.005t^3 + 0.1t^2 - 0.2t + 5$ for $0 \le t \le 20$.

Find the concentration after $2$ hours.

Find the concentration after $15$ hours.

Find the maximum concentration and the time in hours at which this occurs.

Find the minimum concentration and the time in hours at which this occurs.

Find the times in hours when the concentration is $6$ mol/L.

(a) An architect is designing the entrance to a new tunnel. The shape of the entrance is modelled by a quadratic curve, with the base of the tunnel on the x-axis. The curve has end points (0, 6) and (10, 6), and its vertex is (5, 9). Distances are measured in metres.

The quadratic curve can be expressed in the form $y = ax^2 + bx + c$ for $0 \leq x \leq 10$.

(a.i) Write down the value of $c$.

(a.ii) Hence, form two equations in terms of $a$ and $b$.

(a.iii) Hence, find the equation of the quadratic curve.

(b) Calculate the area of the tunnel entrance.

A drone launches a package, and its trajectory is modelled by the equation $h(x) = -0.015x^2 + 0.6x + 5$, where $h(x)$ is the height of the package in metres and $x$ is the horizontal distance in metres from the launch point.

On paper, sketch the graph of the path that the package flies for $x \ge 0$. Clearly indicate the initial height, the maximum height, and the horizontal distance when it lands.

Find the height of the package when it has travelled a horizontal distance of $15$ metres.

Find the maximum height of the package.

Find the horizontal distance at which the package lands on the ground, and explain what this value represents in the context of the problem.

A biologist is studying the growth of a bacterial colony in a petri dish. The population of the colony, $N$, can be modelled by an exponential function

$N(t) = Ae^{kt}$

where $t$ is the time in hours since the start of the experiment, and $A$ and $k$ are constants.

At the start of the experiment, the bacterial colony has a population of 250 cells. After 4 hours, the population has grown to 800 cells.

Write down the value of $A$.

Find the value of $k$.

The number of visitors (in hundreds) to a new eco-tourism resort $t$ months after its opening is modelled by the function $N(t) = -0.001 t^3 + 0.045 t^2 - 0.375 t + 10$.

Sketch the graph of $N$ against $t$ for the first $40$ months, clearly indicating any intercepts and local extrema within this domain.

Find the maximum number of visitors (to the nearest whole number) during the first $40$ months.

Find the time(s) when the number of visitors is above $1200$. Give your answer in months, correct to two decimal places.

The height of a diver above the water surface after jumping from a diving board is modelled by the function

$h(t) = -4.9t^2 + 5t + 10$

where $h(t)$ is the height in metres and $t$ is the time in seconds after the diver leaves the board.

(a) Write down the height of the diving board above the water surface.

(b) Find the value of $t$ when the diver enters the water. Give your answer to three significant figures.

(c) State an appropriate domain for $t$ in this model.

(a) When the profit is zero, find the possible number of units produced, $x$.

(b) Determine the positive values of profit, $P$, for which there is only one positive value of $x$ (units produced).