Linear Equations & Graphs - IB Math AI Question Bank

Question 1

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

The cost, $C$ dollars, of a taxi journey of $d$ kilometres is modelled by

C=2.80d+4.50

A

State the cost per kilometre and the fixed initial charge.

[1]

B

Calculate the cost of a journey of $12$ kilometres.

[1]

C

A journey costs $32.50. Find the distance travelled.

[2]

Question 2

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

A straight line $L$ passes through the points $A(-3,7)$ and $B(5,-1)$ .

A

Find the gradient of $L$ .

[2]

B

Find the equation of $L$ in the form $y=mx+c$ .

[2]

C

Write down the coordinates of the $x$ -intercept of $L$ .

[1]

Question 3

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

A straight access ramp rises vertically by $0.72\ \text{m}$ over a horizontal distance of $9.60\ \text{m}$ . The ramp has a constant gradient.

A

Calculate the gradient of the ramp.

[2]

B

Express the gradient as a percentage.

[1]

C

A second ramp has the same gradient and rises vertically by $1.20\ \text{m}$ . Calculate its horizontal distance.

[2]

Question 4

SL • Paper 1

Easy

Calculator Permitted

SL • Paper 1

Easy

Calculator Permitted

An exchange office uses a linear model to convert an amount $x$ in US dollars to an amount $y$ in euros. Under this model, $40$ US dollars converts to $36$ euros and $125$ US dollars converts to $112.50$ euros.

A

Determine the gradient of the linear model.

[2]

B

Find the equation of the model in the form $y=mx+c$ .

[2]

C

Calculate the number of euros received for $250$ US dollars.

[1]

Question 5

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The equation of a line $L$ is

3x+4y-24=0

A second line $N$ passes through $P(2,1)$ and is perpendicular to $L$ .

A

Find the gradient of $L$ .

[1]

B

Find the coordinates of both axis intercepts of $L$ .

[2]

C

Find the equation of $N$ in the form $y=mx+c$ .

[3]

Question 6

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

The line $L$ has equation

(k-1)x+2y-6=0

where $k$ is a constant. The line $L$ is parallel to the line $y=-3x+4$ .

A

Write down the gradient of $L$ in terms of $k$ .

[1]

B

Find the value of $k$ .

[2]

C

Hence find the coordinates of the two axis intercepts of $L$ .

[2]

Question 7

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

For a product, the supply price $P$ dollars and demand price $P$ dollars are modelled in terms of the quantity $q$ by

P=0.4q+12

and

P=-0.2q+48

respectively.

A

Interpret the gradient of each line in this context.

[2]

B

Determine the quantity for which the supply price equals the demand price.

[2]

C

Find the corresponding price.

[1]

Question 8

SL • Paper 1

Medium

Calculator Permitted

SL • Paper 1

Medium

Calculator Permitted

A temperature sensor has a linear calibration model $R=mT+c$ , where $T$ is the actual temperature in $^\circ\text{C}$ and $R$ is the sensor reading. When $T=5$ , the reading is $11.8$ . When $T=25$ , the reading is $51.4$ .

A

Find the value of $m$ .

[2]

B

Find the calibration model.

[2]

C

The sensor reading is $37.5$ . Calculate the actual temperature.

[2]

Question 9

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The points $A(-2,3)$ and $B(6,-1)$ are endpoints of a line segment. The line $L$ is the perpendicular bisector of $AB$ .

A

Find the midpoint of $AB$ .

[2]

B

Find the gradient of $L$ .

[1]

C

Find the equation of $L$ in the form $y=mx+c$ .

[2]

D

Write the equation of $L$ in general form.

[1]

Question 10

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The height $H$ metres of a road above sea level is modelled using two straight-line segments. For $0\leq x\leq500$ , where $x$ is horizontal distance in metres,

H=0.06x+120

The second segment passes through $(500,150)$ and $(900,166)$ .

A

Find the gradient of the second segment.

[2]

B

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

[2]

C

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

[2]

Question 11

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The line $L$ has equation

ax+3y-12=0

where $a>0$ . The line $L$ is perpendicular to the line $2x-y+5=0$ .

A

Determine the value of $a$ .

[2]

B

Find the two axis intercepts of $L$ .

[2]

C

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

[2]

Question 12

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The lines

L_1:y=(p-1)x+2

and

L_2:3x+y=14

intersect at the point $P$ , whose $x$ -coordinate is $3$ .

A

Find the value of $p$ .

[2]

B

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

[2]

C

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

[2]

Question 13

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

Consider the lines

L_p:(p+1)x-2y+4=0

and

M:3x-6y-5=0

where $p$ is a real constant.

A

Write down the gradient of each line.

[2]

B

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

[2]

C

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

[2]

Question 14

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

The points $P(a,2a+1)$ and $Q(a+4,2a-3)$ lie on a line $L$ , where $a$ is a real constant. A line $N$ is perpendicular to $L$ and passes through the midpoint of $PQ$ .

A

Find the gradient of $L$ .

[2]

B

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

[1]

C

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

[2]

D

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

[1]

Question 15

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

A line $L$ has a positive $x$ -intercept $A$ and a positive $y$ -intercept $B$ . The midpoint of $AB$ is $(3,4)$ . A line $N$ passes through the origin and is perpendicular to $L$ .

A

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

[2]

B

Find the equation of $L$ in general form.

[2]

C

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

[3]

Question 16

HL • Paper 1

Medium

Calculator Permitted

HL • Paper 1

Medium

Calculator Permitted

On a coordinate map, a straight road has equation

4x-3y+12=0

A station is located at $R(6,2)$ . A straight path from $R$ meets the road at a right angle at the point $S$ . One coordinate unit represents one kilometre.

A

Find the gradient of the road.

[1]

B

Find the equation of the path through $R$ .

[2]

C

Find the coordinates of $S$ .

[2]

D

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

[2]

Question 17

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A water tank is being drained at a constant rate. After $12$ minutes, it contains $860$ litres of water. After $32$ minutes, it contains $610$ litres. The volume $V$ litres after $t$ minutes is modelled by a straight line.

A

I.

Calculate the gradient of the line and interpret its meaning in this context.

[2]

II.

Find an equation for the model in the form $V=mt+c$ .

[2]

B

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

[2]

C

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.

[2]

Question 18

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

On a coordinate map, a ferry travels along a straight route from port $A(-4,7)$ to port $B(8,1)$ . Each coordinate unit represents one kilometre. A rescue vessel follows a route parallel to $AB$ and passes through $R(2,-3)$ .

Coordinate map of ports A and B with ferry route and rescue point R.

A

I.

Calculate the gradient of the ferry route.

[2]

II.

Find the equation of the rescue route in the form $y=mx+c$ .

[2]

B

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

[2]

C

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

[2]

Question 19

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A theatre sells tickets for a performance. When $120$ tickets are sold, the total revenue is $3060. When $260$ tickets are sold, the total revenue is $6210. The revenue $R$ dollars is modelled as a linear function of the number $n$ of tickets sold.

A

I.

Calculate the gradient of the revenue model and state its units.

[2]

II.

Find the linear model for $R$ in terms of $n$ .

[2]

B

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

[2]

C

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.

[2]

Question 20

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A section of a mountain railway rises at a constant gradient. Its height is $428$ metres above sea level at a horizontal distance of $1.20$ kilometres from the station and $512$ metres above sea level at a horizontal distance of $2.60$ kilometres. Let $H$ be the height in metres and $x$ the horizontal distance in metres.

A side-view diagram of a straight mountain railway incline, showing horizontal distance x and vertical height H. The two measured locations are marked, without displaying a calculated gradient.

A

I.

Calculate the gradient of the railway.

[2]

II.

Express this gradient as a percentage.

[2]

B

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

[2]

C

A 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.

[2]

Question 21

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

A laboratory sensor produces a voltage $v$ volts that changes linearly with pressure $p$ kilopascals. The calibration line has general equation
$5p-80v+24=0$

A

I.

Write the equation in the form $v=mp+c$ .

[2]

II.

Interpret the gradient and the vertical intercept in this context.

[2]

B

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

[2]

C

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

[2]

Question 22

SL • Paper 2

Medium

Calculator Permitted

SL • Paper 2

Medium

Calculator Permitted

Two companies offer monthly cloud-storage plans. For $x$ gigabytes of storage, Company A charges $A=0.18x+7.50$ dollars. Company B charges $16.50 for $30$ gigabytes and $28.50 for $90$ gigabytes, with its charge modelled linearly.

Monthly charge vs storage for two cloud plans.

A

I.

Find the gradient of Company B's model.

[2]

II.

Find Company B's charge $B$ in terms of $x$ .

[2]

B

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

[2]

C

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

[2]

Question 23

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

A straight cycle path passes through $P(-1,6)$ and $Q(5,9)$ . A rest station is located at $S(7,2)$ . A new path from $S$ will meet the cycle path at a right angle.

Coordinate grid showing the cycle path through P and Q and the rest station S, with equal visual scale on both axes.

A

I.

Find an equation of the cycle path in the form $y=mx+c$ .

[2]

II.

Find an equation of the new path through $S$ .

[2]

B

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

[2]

C

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

[2]

Question 24

SL • Paper 2

Hard

Calculator Permitted

SL • Paper 2

Hard

Calculator Permitted

During a six-hour observation period, the water level beside a harbour wall is modelled by a straight line. At $09{:}00$ , corresponding to $t=0$ , the level is $1.35$ metres. At $11{:}30$ , corresponding to $t=2.5$ , the level is $2.10$ metres.

A

I.

Calculate the rate of change of the water level, in metres per hour.

[2]

II.

Find a linear model for the water level $L$ metres after $t$ hours.

[2]

B

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

[2]

C

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.

[2]

Question 25

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The line $L_k$ has equation
$(k+2)x+(k-1)y-12=0$
where $k$ is a real constant. The line $M$ has equation $2x-3y+5=0$ .

A

I.

Find the value of $k$ for which $L_k$ is parallel to $M$ .

[2]

II.

Find the value of $k$ for which $L_k$ is perpendicular to $M$ .

[2]

B

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

[2]

C

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

[2]

Question 26

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The vertices of triangle $ABC$ are $A(-2,1)$ , $B(6,3)$ and $C(1,8)$ . An altitude of a triangle passes through a vertex and is perpendicular to the opposite side.

Coordinate-plane diagram of triangle ABC with labelled vertices.

A

I.

Find the equation of the altitude through $C$ .

[2]

II.

Find the equation of the altitude through $A$ .

[2]

B

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

[2]

C

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

[2]

Question 27

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

A digital image uses coordinates measured in pixels. Under a reflection in the line $L$ , points on $L$ remain fixed. A point $P(8,7)$ is reflected in the line $L$ to a point $P'$ .

Coordinate diagram showing line L and point P for reflection.

A

I.

Find the gradient of $L$ and the gradient of a line perpendicular to $L$ .

[2]

II.

Find the equation of the perpendicular line through $P$ .

[2]

B

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

[2]

C

Hence find the coordinates of $P'$ .

[2]

Question 28

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A flood barrier has a straight upper edge joining $A(0,6)$ to $B(18,0)$ , where coordinates are measured in metres. A drainage channel begins at $D(6,0)$ and meets the upper edge at a right angle.

Coordinate diagram of the flood barrier upper edge and a perpendicular drainage channel.

A

I.

Find the gradient of $AB$ .

[1]

II.

Find the equation of the upper edge in the form $y=mx+c$ .

[2]

B

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

[3]

C

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 $AB$ .

[3]

Question 29

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A company compares two monthly cloud-storage plans. For $n$ gigabytes of additional storage, the costs in dollars are
$C_A=18+0.12n$
and
$C_B=6+0.20n$

Monthly cost graphs for Plans A and B versus additional storage.

A

I.

Interpret the gradient of the graph of $C_A$ .

[1]

II.

Find the coordinates of the intersection of the two cost graphs.

[2]

B

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

[3]

C

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 $0\leq n\leq300$ .

[3]

Question 30

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A mountain route consists of two straight sections. The first joins $A(0,120)$ to $B(800,168)$ and the second joins $B$ to $C(1400,138)$ . Horizontal distance and height are measured in metres.

A

I.

Calculate the gradient of each section.

[2]

II.

Express the gradients as percentage gradients and interpret their signs.

[2]

B

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

[2]

C

A 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.

[3]

Question 31

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A straight coastline is represented by
$3x+4y=40$
A rescue boat is at $P(12,9)$ . One coordinate unit represents one kilometre. The boat travels along the shortest straight route to the coastline.

Coastline, rescue boat at P, and the perpendicular rescue route.

A

I.

Find the gradient of the coastline.

[1]

II.

Find an equation of the boat's route.

[2]

B

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

[3]

C

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

[3]

Question 32

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two gauges measure the depth $d$ metres of liquid in a tank. Their readings are modelled by
$r_1=1.2d+4,\qquad r_2=0.9d+10$

Two straight calibration lines showing gauge reading against liquid depth.

A

I.

Interpret the gradient of the graph of $r_1$ .

[1]

II.

Find the depth at which the two gauges give the same reading.

[2]

B

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

[3]

C

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.

[3]

Question 33

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The estimated processing times, in minutes, for two translation services handling $w$ words are
$T_H=12+0.08w$
and
$T_A=3+0.14w$

Processing time against word count for services H and A.

A

I.

Interpret the vertical intercept of the graph of $T_H$ .

[1]

II.

Find the word count for which the estimated processing times are equal.

[2]

B

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

[3]

C

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 $0\leq w\leq500$ .

[3]

Question 34

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The lines $L_1:3x+2y=17$ and $L_2:x-4y=-9$ intersect at $P$ . A line $N$ passes through $P$ and has positive $x$ - and $y$ -intercepts of equal numerical value.

A

I.

Find the coordinates of $P$ .

[2]

II.

State the gradient of $N$ .

[2]

B

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

[2]

C

A 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.

[2]

Question 35

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Three boundary lines of a wildlife reserve are

L_1:y=2x+1,\qquad L_2:y=-\frac12x+11, \qquad L_3:y=-x+5

The three pairwise intersections form a triangle.

Coordinate grid of the three boundary lines forming a triangular region.

A

I.

Find the intersection of $L_1$ and $L_2$ .

[2]

II.

Find the other two vertices of the triangle.

[2]

B

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

[2]

C

Calculate the area of the triangular reserve.

[2]

Question 36

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

The points $A(2,-1)$ and $B(10,5)$ lie on a straight line. A point $C(k,4k-9)$ lies on the perpendicular bisector of $AB$ , where $k$ is a real constant.

A

I.

Find the midpoint of $AB$ and the gradient of $AB$ .

[2]

II.

Find the equation of the perpendicular bisector of $AB$ .

[2]

B

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

[2]

C

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

[2]

Question 37

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two autonomous vehicles move along straight paths on a coordinate grid. Vehicle A travels on $L_A:2x-y-7=0$ . Vehicle B travels on $L_B:px+4y-18=0$ , where $p$ is a constant. The paths intersect at $T$ , whose $x$ -coordinate is $5$ .

A

I.

Find the coordinates of $T$ .

[2]

II.

Determine the value of $p$ .

[2]

B

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

[2]

C

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

[2]

Question 38

HL • Paper 2

Hard

Calculator Permitted

HL • Paper 2

Hard

Calculator Permitted

Two straight wave fronts move across a rectangular simulation grid. At time $t$ seconds, the first front is represented by
$L_t:(t+1)x-2y+6=0$
and the second front is represented by
$M_t:3x+(t-4)y-9=0$

A

I.

Determine the value of $t$ for which the two wave fronts are parallel.

[2]

II.

Determine the value of $t$ for which the two wave fronts are perpendicular.

[2]

B

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

[2]

C

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

[2]

Question 39

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two parallel boundaries of an aerial survey corridor are modelled by
$y=0.4x+8$
and
$y=0.4x-12$
where coordinates are measured in kilometres. An inspection flight follows a straight path through $S(10,0)$ and perpendicular to both boundaries.

Coordinate diagram of two parallel corridor boundaries and a perpendicular inspection flight through point S.

A

I.

State the gradient of the inspection flight.

[1]

II.

Find an equation of the inspection flight.

[2]

B

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

[3]

C

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.

[3]

Question 40

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two laboratories convert a raw instrument reading $x$ into a corrected reading $y$ . Laboratory A's calibration line passes through $(12,10)$ and $(57,40)$ . Laboratory B uses
$y=0.75x-1$

A

I.

Find the gradient of Laboratory A's calibration line.

[1]

II.

Find the equation of Laboratory A's calibration line.

[2]

B

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

[3]

C

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 $0\leq x\leq60$ .

[3]

Question 41

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

During a training session, the positions of two cyclists along a straight track are modelled by
$x_A=2.8t+40,\qquad x_B=3.4t+10$
where $x$ is measured in metres and $t$ in seconds.

Position-time graph for cyclists A and B.

A

I.

Interpret the gradients of the two position-time lines.

[2]

II.

State the initial distance between the cyclists.

[1]

B

A 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 $t=0$ .

[3]

C

A 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 catch cyclist A during the first $120$ seconds.

[3]

Question 42

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

The edge of a roof is modelled by the line segment joining $(0,4)$ and $(10,7)$ , where coordinates are measured in metres. A support beam is placed perpendicular to the roof edge at the point whose horizontal coordinate is $6$ . The other end of the beam lies on the ground line $y=0$ .

Coordinate graph of the roof edge, the perpendicular support beam, and the ground line.

A

I.

Find the gradient of the roof edge.

[1]

II.

Find the equation of the roof edge.

[2]

B

Determine the coordinates of the lower end of the support beam and calculate the beam's length.

[3]

C

A perpendicular beam starts from a general point $(t,0.3t+4)$ on the roof. Deduce an expression for its length and determine the shortest possible beam for $0\leq t\leq10$ .

[3]

Question 43

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

An access route consists of two straight ramps separated by a level landing. The first ramp rises from $(0,0)$ to $(r,0.9)$ . The second rises from $(r,0.9)$ to a doorway at $(18,1.5)$ . Assume $0<r<18$ , so the landing lies between the start and doorway. Coordinates are measured in metres.

Point	x-coordinate / m	y-coordinate / m
Start of first ramp	0	0
Landing	$r$	0.9
Doorway	18	1.5

A

I.

Write down the gradient of the first ramp in terms of $r$ .

[1]

II.

Write down the gradient of the second ramp in terms of $r$ .

[2]

B

The first ramp must have gradient at most $1:12$ and the second at most $1:15$ . Show that no position of the landing satisfies both requirements.

[3]

C

The doorway may be moved farther horizontally to the right, while remaining to the right of the landing and while both rises remain unchanged. Determine the minimum total horizontal distance needed to satisfy both gradient limits, and state where the landing should be placed.

[3]

Question 44

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

Two parallel cycle lanes lie on opposite sides of a central avenue. The central avenue is
$x+2y=18$
One cycle lane passes through $A(2,3)$ and the other passes through $B(10,9)$ . Both cycle lanes are parallel to the avenue.

Avenue, parallel lanes, and a perpendicular connector.

A

I.

Find the gradient of the central avenue.

[1]

II.

Find the equations of the two cycle lanes in general form.

[2]

B

A connector begins at $A$ and is perpendicular to the cycle lanes. Determine where it meets the other cycle lane and calculate its length.

[3]

C

Generalize this result by giving the perpendicular distance between the parallel lines $ax+by=c_1$ and $ax+by=c_2$ . Use your expression to verify the connector length.

[3]

Question 45

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A straight decorative beam joins the positive $x$ -intercept $(a,0)$ to the positive $y$ -intercept $(0,b)$ . The beam must pass through the fixed point $P(4,3)$ .

Beam joining positive axis intercepts and passing through P.

A

I.

For $a=10$ , find $b$ .

[1]

II.

Hence find the equation of this beam in general form.

[2]

B

Find an expression for $b$ in terms of $a$ for the full family of beams through $P$ , and state the permitted values of $a$ .

[3]

C

The designer requires the beam to have equal positive axis intercepts. Determine the intercepts and the area enclosed by the beam and the coordinate axes.

[3]

Question 46

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A railway is represented in an old coordinate system by
$2x-5y+10=0$
A new mapping system uses coordinates
$X=x+3,\qquad Y=2y-4$

Point	x	y	X	Y
A	0	2	3	0
B	5	4	8	4
C	10	6	13	8
D	15	8	18	12

A

I.

State $x$ and $y$ in terms of $X$ and $Y$ .

[1]

II.

Find the equation of the railway in the new coordinate system.

[2]

B

A maintenance path passes through $(8,4)$ in the new coordinate system and is perpendicular to the railway in that system. Find its equation.

[3]

C

More generally, suppose $X=x+h$ and $Y=sy+k$ , where $s>0$ . Determine how a gradient $m$ changes and hence state the condition on $s$ for perpendicularity to be preserved for every pair of lines.

[3]

Question 47

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A line displayed on a digital image has pixel coordinates satisfying
$y=-0.8x+400$
Physical coordinates, measured in millimetres, are related to pixel coordinates by
$X=0.5x-20,\qquad Y=0.25y+10$

Transformed line and perpendicular line in physical coordinates.

A

I.

Express $x$ and $y$ in terms of $X$ and $Y$ .

[1]

II.

Find the equation of the displayed line in physical coordinates.

[2]

B

In physical coordinates, a second line passes through $(30,50)$ and is perpendicular to the displayed line. Find their point of intersection.

[3]

C

Suppose more generally that $X=px+h$ and $Y=qy+k$ , where $p,q>0$ . Deduce the transformed gradient of a line with pixel gradient $m$ , and determine when every right angle is preserved.

[3]

Question 48

HL • Paper 3

Hard

Calculator Permitted

HL • Paper 3

Hard

Calculator Permitted

A family of possible geological fault lines is modelled by
$L_k:(k+1)x+(k-1)y=12$
where $k$ is a real parameter.

Several members of the fault-line family passing through a common point.

A

I.

For $k\neq1$ , write down the gradient of $L_k$ .

[1]

II.

Find the two axis intercepts of $L_2$ .

[2]

B

Show that every line in the family passes through the same point, and find this point.

[3]

C

Determine the value of $k$ for which $L_k$ is perpendicular to the line $3x-4y=5$ . Comment on the special member $L_1$ .