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IBMathsDPAnalysis & Approaches (AA)HLExam Questions2. FunctionsLinear Functions & Graphs

Linear Functions & Graphs (DP IB Analysis & Approaches (AA): HL): Exam Questions

4 hours31 questions
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Hard
Very Hard
1a
Sme Calculator3 marks

The equation of a line l subscript 1 is 2 x minus y plus 6 equals 0.

For the line l subscript 1 find:

(i) the y-intercept

(ii) the x-intercept

(iii) the gradient.

How did you do?

1b
Sme Calculator3 marks

A new line, l subscript 2 comma intersects the x-axis at (4, 0) and is perpendicular to l subscript 1.

Find:

(i) the gradient of the line l subscript 2

(ii) the equation of the line l subscript 2 Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

How did you do?

2a2 marks

The coordinates of point A are open parentheses 2 comma space 8 close parentheses and the coordinates of point B are open parentheses negative 8 comma space 2 close parentheses. M is the midpoint of open square brackets A B close square brackets.

Find the coordinates of M.

How did you do?

2b2 marks

The line l subscript 1passes through A and B.

Find the gradient of l subscript 1.

How did you do?

2c3 marks

Find the equation of the line l subscript 1. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

How did you do?

3a2 marks

The coordinates of point A are open parentheses 1 comma space 7 close parentheses and the coordinates of point B are open parentheses 5 comma space 5 close parentheses. M is the midpoint of open square brackets A B close square brackets.

Find the coordinates of M.

How did you do?

3b2 marks

The line l subscript 1 passes through the points A and B.

Find the equation of l subscript 1. Give your answer in the form of space y equals m x plus c.

How did you do?

3c3 marks

A new line, l subscript 2, is the perpendicular bisector to open square brackets A B close square brackets.

Find the equation of l subscript 2. Give your answer in the form of y equals m x plus c.

How did you do?

4a2 marks

Plumber A charges a fixed fee of $25 plus $15 per hour.

Using t for the number of hours a job takes, and C subscript A for the total cost of a job, in dollars, from Plumber A, write down an equation connecting t and C subscript A.

How did you do?

4b2 marks

A job takes the plumber seven hours.

Calculate the total cost of the job.

How did you do?

4c2 marks

Plumber B charges a fixed fee of $20 plus $16 per hour.

Using t for the number of hours a job takes, and C subscript B for the total cost of a job, in dollars, from Plumber B, write down an equation connecting t and C subscript B.

How did you do?

4d3 marks

Determine which plumber would be the cheapest for a job taking six hours.

How did you do?

5a2 marks

The diagram below shows the line l subscript 1, which intersects the y-axis at Aopen parentheses 0 comma 10 close parentheses  and the x-axis at Bopen parentheses 5 comma 0 close parentheses.

q5-2-1-medium-linear-functions-and-graphs

Find the equation of l subscript 1. end subscript Give your answer in the form of y equals m x plus c.

How did you do?

5b
Sme Calculator2 marks

Find the length of [AB].

How did you do?

5c2 marks

A second line, l subscript 2 is parallel to l subscript 1 and intersects the x-axis at Copen parentheses 8 comma 0 close parentheses.

Find the equation of l subscript 2 Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

How did you do?

5d1 mark

Find the coordinates where l2  intersects the y-axis.

How did you do?

6a
Sme Calculator4 marks

Photocopy shop A charges $122 for 115 copies, and $190 for 200 copies.

Assuming a linear relationship, find

(i) the price for 180 copies

(ii) how many copies could be made for $385.20.

How did you do?

6b
Sme Calculator3 marks

Photocopy shop B charges $0.82 per copy and a fixed fee of $25.50.

State which photocopy shop is cheaper to make 220 copies.

How did you do?

7a
Sme Calculator2 marks

A family can be supplied with electricity by two companies that have different pricing structures:

                      Company A: Fixed fee of $25/month and $0.2 per kWh consumed.

                      Company B: Fixed fee of $10/month and $0.22 per kWh consumed.

Determine the equation of the cost function for both companies, where the total monthly cost y is a function of the monthly electricity consumption x in kWh.

How did you do?

7b
Sme Calculator4 marks

Calculate the monthly energy consumption that results in the same monthly cost from both companies.

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8a2 marks

Ardie’s monthly expenditure, C space open parentheses x close parentheses comma is a linear function of his monthly income, x. Ardie’s monthly expenditure is $1000 when his monthly income is $1200 and his monthly expenditure increases by $60 for every $150 increase in his monthly income.

Write an expression connecting Ardie’s monthly expenditure, C space open parentheses x close parentheses comma with his monthly income, x.

How did you do?

8b
Sme Calculator2 marks

Calculate Ardie’s monthly expenditure when his monthly income is $1885. Give your answer to the nearest dollar.

How did you do?

8c
Sme Calculator2 marks

Find Ardie’s monthly income when his monthly expenditure is $1070. Give your answer to the nearest dollar.

How did you do?

9a1 mark

The diagram below represents a mountain with a west facing slope and an east facing slope labelled W and E respectively.

Horizontal scale: 1 unit represents 100 m.

Vertical scale: 1 unit represents 100 m.

q9-2-1-medium-linear-functions-and-graphs

Find the gradient of the west facing slope.

How did you do?

9b
Sme Calculator6 marks

The gradient of the east facing slope in the diagram is  negative 3 over 10.

Find the total distance to hike over the mountain in km.

How did you do?

9c1 mark

Suggest a reason as to why the actual total distance hiked may be greater than the distance found in part (b).

How did you do?

10a4 marks

The straight lines l subscript 1 and l subscript 2 are shown in the diagram below. l subscript 1 intercepts the x-axis at open parentheses 17 comma space 0 close parentheses and the y-axis at open parentheses 0 comma 17 close parentheses, and l subscript 2  intercepts the x-axis at open parentheses 2 comma space 0 close parentheses and the y-axis at open parentheses 0 comma space minus 1 close parentheses.

q10-2-1-medium-linear-functions-and-graphs

Giving your answer in the form y equals m x plus c, find:

(i) the equation of l subscript 1

(ii) the equation of l subscript 2.

How did you do?

10b
Sme Calculator4 marks

Find the area of the shaded region.

How did you do?

11a
Sme Calculator3 marks

A line passing through the origin O, is perpendicular to a line with equation x thin space plus space y space equals space 16. The two lines meet at point R. P is a point on open parentheses O R close parentheses such that OP : PR = 3 : 1.

Find the equation of the perpendicular line and hence, the co-ordinates of point R.

How did you do?

11b
Sme Calculator2 marks

Find the coordinates of P.

How did you do?

1a1 mark

The diagram below shows the line l with the equation 2 x minus 4 y plus 6 equals 0.

Point P has coordinates open parentheses negative 1 comma 5 close parentheses point Q has coordinates open parentheses 7 comma space 5 close parentheses and point R has coordinates open parentheses 3 comma space 3 close parentheses. M is the midpoint of [PQ].

q1-2-1-hard-linear-functions-and-graphs

Write down the coordinates of M.

How did you do?

1b2 marks

Show that Q lies on l.

How did you do?

1c2 marks

Show that R lies on l.

How did you do?

1d2 marks

Find the area of PQR.

How did you do?

2a2 marks

The line l subscript 1 passes through the points open parentheses 1 comma space 7 close parentheses and open parentheses 5 comma space 5 close parentheses.

Find the equation of l subscript 1. Give your answer in the form of y equals m x plus c.

How did you do?

2b2 marks

A new line, l subscript 2, is perpendicular to l subscript 1 and passes through the point open parentheses 4 comma space 8 close parentheses.

Find the equation of l subscript 2. Give your answer in the form of y equals m x plus c.

How did you do?

2c2 marks

The point Z is the intersection of l subscript 1 and l subscript 2.

Find the coordinates of Z.

How did you do?

3a3 marks

Point A has coordinates open parentheses x comma space 4 close parentheses and point B has coordinates open parentheses 9 comma space y close parentheses. M is the midpoint of open square brackets A B close square brackets and has coordinates open parentheses negative 2 comma space 7 close parentheses.

Find the value of x and y.

How did you do?

3b2 marks

The line l subscript 1 passes through A and B.

Find the equation of l subscript 1. Give your answer in the form a x plus b y plus d equals 0 comma where a comma space b and d are integers.

How did you do?

4a2 marks

The line l subscript 1 passes through A open parentheses negative 3 comma space minus 5 close parentheses and B open parentheses negative 1 comma space minus 7 close parentheses .

Find the equation of l subscript 1. Give your answer in the form y equals m x plus c.

How did you do?

4b2 marks

Point C is such that B is the midpoint of open square brackets A C close square brackets.

Find the coordinates of C.

How did you do?

4c2 marks

Point D is such that C is the midpoint of open square brackets A D close square brackets.

Find the coordinates of D.

How did you do?

5a3 marks

Point A has coordinates open parentheses 11 comma space 12 close parentheses and point B has coordinates open parentheses 4 comma space minus 8 close parentheses. The line l subscript 1 passes through M, the midpoint of open square brackets A B close square brackets. The gradient of l subscript 1 is minus2.

Write down the equation of l subscript 1, giving your answer in the form y equals m x plus c.

How did you do?

5b1 mark

The line l subscript 2 passes through A and has a gradient of 5.

Write down the equation of l subscript 2, giving your answer in the form y equals m x plus c.

How did you do?

5c2 marks

Point C is the intersection of l subscript 1 and l subscript 2.

Write down the coordinates of C, giving the x and y coordinates as fractions.

How did you do?

6a2 marks

Finn borrows $3200 from his parents and decides to pay them back $c in the first month and then $m each subsequent month.

After two months Finn has paid back his parents a total of $1000, this can be expressed as m plus c space equals space 1000. After half a year he still owes his parents $1000.

Write down another equation connecting m and c.

How did you do?

6b2 marks

Find the value of m and c.

How did you do?

6c
Sme Calculator2 marks

Finn’s parents apply a 6.25% net interest to the $3200 total.

Calculate the number of months it takes Finn to pay back his parents.

How did you do?

7a3 marks

The line l subscript 1 has the equation 5 y minus 2 x plus 1 equals 0. Point A has coordinates left parenthesis x comma space 3 right parenthesis and is the intersection of l subscript 1 and l subscript 2. l subscript 2 is perpendicular to l subscript 1.

Write down the equation of l subscript 2, giving your answer in the form y equals m x plus c.

How did you do?

7b
Sme Calculator5 marks

Point B lies on l subscript 2 and is positioned such that AB is 12 units and the x-coordinate for B is less than the x-coordinate for A.

Find the coordinates of B.

How did you do?

88 marks

The points Kopen parentheses 6 comma space y close parentheses and Nopen parentheses x comma space 9 close parentheses lie on the line l subscript 1 where x, y element of straight integer numbers.

KN has a length of 5 units.

Find all the possible values for x and y.

How did you do?

9a2 marks

Point A has coordinates open parentheses 2 comma space 1 close parentheses, point B has coordinates open parentheses 5 comma space 7 close parentheses, and point C has coordinates (8, 1).

Calculate the length of AC.

How did you do?

9b3 marks

Point D lies on the line open parentheses A C close parentheses. The line open parentheses B D close parentheses is perpendicular to the line open parentheses A C close parentheses.

Find the coordinates of D.

How did you do?

9c3 marks

Calculate the area of the triangle ABC.

How did you do?

10a3 marks

The distance between points Popen parentheses 20 comma space 40 close parentheses  and Qopen parentheses x comma space 20 close parentheses is equal to 25 units.

Find the possible values of x.

How did you do?

10b3 marks

It is given that x less than 20.

Point R is such that point Q is the midpoint of the line segment with endpoints P and R.

Write down the coordinates of R.

How did you do?

1a
Sme Calculator3 marks

A cat breeder is measuring the rate at which a kitten grows by measuring its “back-length”, which is the distance from the base of the neck to the base of the tail. At 1 week old the kitten measured 2.42 cm. 8 weeks later the kitten’s back length had doubled.

Using a linear model, find an equation linking , the kitten’s “back-length” in centimetres, to x, the age of the kitten in weeks.

How did you do?

1b
Sme Calculator1 mark

Find the kitten’s “back-length” at birth.

How did you do?

1c
Sme Calculator1 mark

Use the model to find the age, in weeks, of the kitten that has a “back-length” of 7 cm.

How did you do?

1d
Sme Calculator1 mark

This breed of cat is fully grown after 50 weeks.

Find the cat’s “back-length” at 50 weeks.

How did you do?

1e
Sme Calculator2 marks

Comment on the suitability of the linear model used.

How did you do?

2a2 marks

Point P has coordinates open parentheses 6 comma space minus 2 close parentheses, R has coordinates open parentheses negative 2 comma space 2 close parentheses. The equation of the line PQ is y space equals space minus 2 and the equation of the line QR is y equals 2 x plus 6.

Find the coordinates of point Q.

How did you do?

2b
Sme Calculator3 marks

Find the distance of PQ and QR.

How did you do?

2c2 marks

Find the area of triangle PQR.

How did you do?

3a4 marks

Point P has coordinates (4, 2) and point Q has coordinates (1, 8). l subscript 1 is the perpendicular bisector of [PQ].

Find the equation of l subscript 1, giving your answer in the form y equals m x plus c.

How did you do?

3b2 marks

l subscript 2 passes through Q and has a gradient of minus3.

Find the equation of l subscript 2, giving your answer in the form y equals m x plus c.

How did you do?

3c
Sme Calculator4 marks

Point C lies on l subscript 2 and has a negative x coordinate. The length of QC is 14 units.

Find the coordinates of C.

How did you do?

4a3 marks

Point A has coordinates open parentheses 0 comma space y close parentheses, l subscript 1 passes through points A and B and has the equation 4 x minus 3 y minus 3 equals 0. The length of [AB] is 10 units.

Find the possible coordinates of point B.

How did you do?

4b2 marks

l subscript 2 passes through points C and D, is parallel to l subscript 1 and crosses the x-axis when x space equals 5.

Write down the equation of l subscript 2. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

How did you do?

4c3 marks

Point C has coordinates open parentheses x comma space 4 close parentheses and CD is half the length of AB.

Find the possible coordinates of point D.

How did you do?

5a2 marks

The line l subscript 1 passes through Aleft parenthesis 10 comma negative 12 right parenthesis  and Bleft parenthesis 5 comma space 2 right parenthesis

Find the equation of l subscript 1. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

.

How did you do?

5b3 marks

The line l subscript 2 passes through M, the midpoint of [AB] and has a gradient of 7.

Find the equation of l subscript 2. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

How did you do?

5c
Sme Calculator5 marks

Point C lies on l subscript 2 such that MC has a length of 12 units.

Find the possible coordinates of C.

How did you do?

6a2 marks

Carpenter A charges a fixed fee of $28 plus $22.50 per hour. The carpenter then charges tax on top of the total cost of 21%.

Defining suitable variables, write down an equation to represent the charges, including tax, made by the carpenter.

How did you do?

6b
Sme Calculator4 marks

Carpenter B charges a fixed fee of $35 and $26.50 per hour. These prices already account for tax.

Find the number of hours for which both carpenters charge the same amount.

How did you do?

6c
Sme Calculator2 marks

Tom needs a job doing by a carpenter that he estimates will take 8.5 hours. Carpenter C has given Tom a fixed quote of $250.

Determine which carpenter would be the cheapest option for Tom.

How did you do?

7a1 mark

The line l subscript 1 has equation 6 x plus 4 y minus 21 equals 0 and crosses the x-axis at point Aopen parentheses p comma space 0 close parentheses.

Find the value of p.

How did you do?

7b2 marks

The line l subscript 2 is perpendicular to l subscript 1 and crosses the x-axis at Bopen parentheses 3 p comma space 0 close parentheses.

Write down the equation of l subscript 2. Give your answer in the form a x plus b y plus d equals 0, where a comma space b and d are integers.

How did you do?

7c5 marks

l subscript 2 crosses the y-axis at point C.

Find the area of the triangle OAC, where O is the origin. Give your answer as a fraction.

How did you do?

88 marks

The points Aopen parentheses x comma space minus 5 close parentheses  and Bopen parentheses negative 1 comma space y close parentheses  lie on the line l subscript 1 where x comma space y space element of space straight integer numbers. AB has a length of 13 units.

Find all the possible x and y values.

How did you do?

9a1 mark

A quadrilateral has four vertices with coordinates  straight A left parenthesis negative 1 comma negative 1 right parenthesis comma space straight B left parenthesis 2 comma 2 right parenthesis comma space straight C left parenthesis 4 comma 2 right parenthesis comma space straight D left parenthesis negative 1 comma negative 3 right parenthesis.

Show that [AB] and [CD] are parallel.

How did you do?

9b
Sme Calculator3 marks

Calculate the distance of

(i) AB.

(ii) CD.

How did you do?

9c4 marks

Find the area of the quadrilateral ABCD.

How did you do?

10
Sme Calculator6 marks

The point straight F open parentheses negative 2 comma space 4 close parentheses lies on the line l subscript 1. l subscript 1 crosses the y-axis at point straight G.

Another line, l subscript 2, is perpendicular to l subscript 1 at the point F and crosses the y-axis at the point straight H open parentheses 0 comma space minus 3 close parentheses .

Find the area of the triangle GFH. Give your answer as a fraction.

How did you do?