Find the gradient of the second segment.

Find an equation for the second segment in the form $H=mx+c$.

State whether the two segments join at the same height and whether they are parallel. Justify both answers.

Determine the value of $a$.

Find the two axis intercepts of $L$.

Calculate the area enclosed by $L$ and the coordinate axes.

Find the value of $p$.

line $N$ passes through $P$ and is perpendicular to $L_1$. Find the equation of $N$.

Find the coordinates of the $x$-intercept of $N$.

Write down the gradient of each line.

Determine the value of $p$ for which the lines are parallel.

Determine the value of $p$ for which the lines are perpendicular.

Find the gradient of $L$.

Find the midpoint of $PQ$ in terms of $a$.

Find the equation of $N$ in terms of $a$.

Find $a$ if $N$ passes through the origin.

Determine the coordinates of $A$ and $B$.

Find the equation of $L$ in general form.

Find the coordinates of the point where $N$ intersects $L$.

Find the gradient of the road.

Find the equation of the path through $R$.

Find the coordinates of $S$.

Calculate the length of the path from the station to the road.

Calculate the volume of water in the tank after $45$ minutes.

The pump must be stopped before the volume falls below $200$ litres. Determine the latest whole number of minutes after draining begins at which the pump may be stopped.

lighthouse is located on the $y$-axis and on the rescue route. Find its coordinates.

navigation channel has equation $y=2x-9$. Determine whether the channel is perpendicular to the rescue route. Justify your answer.

The total cost of staging the performance is modelled by $C=9.75n+2400$. Find the number of tickets for which revenue equals cost.

The theatre can sell only a whole number of tickets. Determine the minimum number of tickets that must be sold for the revenue to be at least $5000.

Find an equation for $H$ in terms of $x$.

tunnel entrance is at a height of $600$ metres. Determine its horizontal distance from the station according to this model, giving your answer in kilometres.

Calculate the pressure when the sensor voltage is $4.80$ volts.

second sensor has equation $v=0.0625p-0.150$. Explain whether the two calibration lines ever intersect.

Determine the amount of storage for which the two companies charge the same monthly amount.

Storage amounts cannot be negative. Hence determine which company is cheaper for all possible storage amounts, and justify your answer.

Find the coordinates of the point $T$ where the two paths meet.

One coordinate unit represents $250$ metres. Calculate the length of the new path from $S$ to $T$.

flood gate must be closed when the water level reaches $2.80$ metres. Determine the time at which it should be closed.

The harbour manager predicts a water level of $3.30$ metres at $15{:}00$. Determine whether this prediction is consistent with the model. Justify your answer.

For $k=-\dfrac17$, independently of your answer to part (ii), find the coordinates of the intersection of $L_k$ and $M$.

Explain why there is no value of $k$ for which $L_k$ is the same line as $M$.

Hence find the coordinates of the orthocentre $H$ of triangle $ABC$.

Verify that $H$ also lies on the altitude through $B$.

Find the coordinates of the point $F$ where the perpendicular line meets $L$.

Hence find the coordinates of $P'$.

Determine the coordinates of the point where the drainage channel meets the upper edge.

The drainage channel is instead allowed to begin at $D(d,0)$, where $d$ is real. Determine the range of values of $d$ for which the perpendicular channel meets the line segment $AB$.

State which plan is cheaper for $n=250$, and calculate the difference in cost.

Plan B changes its variable rate from $0.20$ to $k$ dollars per gigabyte, while its fixed cost remains $6$. Determine the greatest value of $k$ for which Plan B is no more expensive than Plan A for every $n$ in $0\leq n\leq300$.

regulation permits a maximum uphill gradient of $6.5\%$ and a maximum downhill magnitude of $5.5\%$. Determine whether both sections comply.

stricter design requires the magnitude of every gradient to be at most $5\%$, while retaining the same vertical rise and fall. Determine the minimum total horizontal distance required and explain why the existing route cannot meet this design.

Determine the point where the boat reaches the coastline and calculate the distance travelled.

point has general coordinates $(a,b)$. Hence show that its perpendicular distance from the coastline is
$\frac{|3a+4b-40|}{5}$

Eliminate $d$ to express $r_1$ as a linear function of $r_2$.

For general gauges $r_1=ad+b$ and $r_2=cd+e$, where $c\neq0$, find the direct linear relationship between $r_1$ and $r_2$. State the conditions for the two gauges to give equal readings for every depth.

For a document containing $400$ words, determine which service is faster and by how many minutes.

Service A reduces its variable processing rate from $0.14$ to $k$ minutes per word, keeping its fixed time at $3$ minutes. Determine the greatest value of $k$ for which Service A is no slower than Service H for every document with $0\leq w\leq500$.

Find the equation of $N$ and the coordinates of its two intercepts.

line through the origin is perpendicular to $N$. Find its point of intersection with $N$ and explain why this point is the midpoint of the two intercepts.

Show that $L_1$ and $L_2$ are perpendicular.

Calculate the area of the triangular reserve.

Determine the value of $k$ and hence find the coordinates of $C$.

Explain why $CA=CB$ without calculating both distances separately.

third vehicle travels through $T$ on a path perpendicular to $L_B$. Find the equation of this path in general form with integer coefficients.

Determine whether the third vehicle's path is parallel to $L_A$. Justify your answer.

At $t=2$, determine whether the two parallel fronts are distinct or coincident.

At $t=-7$, independently of the perpendicular case in part (a)(ii), find the coordinates of the intersection of the two wave fronts.

Find the coordinates of the two points at which the inspection flight crosses the corridor boundaries.

Hence show that the perpendicular width of two general parallel lines $y=mx+c_1$ and $y=mx+c_2$ is
$\frac{|c_1-c_2|}{\sqrt{1+m^2}}$
and calculate the width of the survey corridor.

Find the raw reading for which the laboratories give the same corrected reading, and state that corrected reading.

Laboratory B changes its calibration to $y=0.75x+p$. Determine the range of values of $p$ for which the two calibration lines intersect at a raw reading in $0\leq x\leq60$.

third cyclist has position $x_C=vt+b$, where $v>2.8$ and $b\leq40$. Determine conditions on $v$ and $b$ for cyclist C to meet cyclist A during the first $120$ seconds, including at $t=0$.