MPM 2D REVIEW FOR FINAL EXAM
1.For each of the following problems:
i. Determine the unknown quantities and represent these with appropriate variables.
ii. Determine the equations of the linear system that models the problem. Do not solve.
a) Bart’s cell phone package costs $15 per month and an additional $0.10 per minute. Zoe pays $10 per month and $0.12 per minute. Determine a linear system that could be used to find out when their monthly bills would be the same.
2.Lloyd invested $1200 in an RRSP. He invested part in a technology fund that pays 15% per year and the remainder in bonds that pay 5% per year. Determine a linear system that could be used to find the amount he invested in each part if he earned $160 in interest for the year.
3. Monique is starting her own on-line business selling jewellery that she makes. It cost her $3000 to buy a new computer, and her monthly expenses for Internet service and materials average about $175 per month. If her revenue for each month averages about $420, how long will it be before she starts to turn a profit?
4.Solve the system of equations graphically.
y = –2x + 8
3x – 4y = 12
5.For the following system of equations
i. determine, without solving, the number of solutions this system has. Justify your answer.
ii. verify that your answer in (i) is correct by solving the system using substitution.
2x + y = 12
15x + 3y = 18
6.Solve the system of equations using elimination.
6x – 8y = –58
4x + 5y = 13
7.The following table shows the revenue generated from sales of skis and snowboard equipment in Canada from 1994 to 1998 in thousands of dollars.
Year / Ski Equipment Sales ($) / Snowboard Equipment Sales ($)1994 / 60 000 / 29 700
1995 / 45 910 / 21 640
1996 / 41 530 / 39 970
1997 / 32 100 / 31 590
1998 / 41 100 / 49 000
(Source: Statistics Canada)
a) Create a single scatter plot for both data sets.
b) Draw the lines of best fit and determine the equation of each line.
c) Which set of data is closer to being linear? Explain.
d) Which set of data is changing at a faster rate? Explain how you know.
e) Use the model to determine when snowboard sales surpassed ski sales in Canada.
f)If you were an owner of a sports store, how would you use this information?
8.A straight path, in a park, is to be built between points A(–35, 32) and B(29, –16), where the units represent metres.
a) How long is the path?
b) A bench is to be placed halfway along the path. Find the coordinates of the point at which the bench will be placed.
9.A fish catching a small insect on the surface of a still pond causes a circular ripple. The radius of the circle increases at a constant rate of 4 cm/s.
a) Write an equation that describes the circular ripples exactly 1 s after the fish catches the insect.
b) Write an equation that describes the ripple exactly 5 s after the fish catches the insect.
10.Triangle ABC has vertices at A(1, 1), B(4, 5), and C(9, –5).
a) Find the lengths and slopes of its sides.
b) Determine what type of triangle it is.
11.Determine the slope of median AP for the triangle with vertices A(-3,5), B(8,-7) and C(-12,-9).
12.In a Star Trek episode, the evil Borg spacecraft and the star ship Enterprise are approaching an alien planet at the same speed. If the Borg is at (153,-251) and the Enterprise is at (-189,239), who will be first to reach the planet? (Show your work to justify your answer)
13.i) Plot both pairs of points on the grid provided.
Join the line segments, and label each end point.
a)A( -1, 5 ) , B( 3, 7 )
b)C( 9, -4 ) , D( -5, -2 )
ii) Calculate the slope of each
line segment.
iii) Measure the length of each line segment. Answer to 1 decimal place.
iv) Determine the slopes of the two lines perpendicular to these two line
segments.
14.Write the equation of the circle with centre at ( 0, 0 ) and the given radius, r.
a) r = 50 b) r = 0.34
15.Find the radius of the circle with centre ( 0, 0 ) that passes through ( 9, -12 ).
16.Calculate the midpoint of each line segment. No decimal answers.
a) L( -2, 6 ) N( -3, -3 ) b) E( 7, -8 ) , F ( -7, 8 )
17.Triangle ABC has vertices A( 9, -2 ), B( 4, -5 ) C ( 5, 4 ). Calculate the
centroid of the triangle.
18.Expand each of the following and simplify.
a) (x + 4)(x – 7)b) (2x – 9)(3x – 5)c) –4(x + 6)2
19.Factor each expression.
a) x2 – 3x – 28b) 81x2 – 25c) 25x2 – 20x + 4d) 24x2 + x – 10
20.Sketch the graph of each quadratic relation. Clearly label the zeros and the vertex.
a) y = (x – 6)(x + 2)b) y = –(x – 1)(x – 9)
21.Solve each of the following. Determine the zeros.
a) x2 + 7x + 12 = 0 b) x2 – 2x = 48c) 5x2 + 4x = 2 – 5x
22.A model rocket is equipped with a motion detector that measures the height of the rocket at 30 sintervals. On a recent flight, the detector recorded the following data.
Time (s) / Height (m)0 / 0
30 / 315
60 / 540
90 / 675
120 / 720
150 / 675
180 / 540
210 / 315
240 / 0
a) Determine the first and second differences.
b) Decide whether the data displays a linear, quadratic, or other relationship.
c) Create a scatter plot of the data and graph the line or curve of best fit.
d) Determine the equation of the axis of symmetry.______
e) Determine the maximum height reached by the rocket.______
f) Derive an equation that models the relationship.
g) The first zero corresponds to the point at which the rocket leaves the ground. Explain the significance of the second zero.
f)How high is the rocket after 1 min 40 s?______
g) At what point in time is the rocket 400 m above the ground?______
23.Find the quadratic relation in vertex form that
a) has its vertex at (5, 8) and passes through the point (11, –46)
b) has zeros –3 and 7 and passes through the point (5, –8)
c) is defined by y = 2x2 + 12x – 17
24.Find the roots of each of the following quadratic equations using the most appropriate algebraic method. Express your answers to two decimal places, where appropriate.
a) 3(t – 7)2 – 15 = 0 b) 2x2 + 4x – 30 = 0 c) 3x2 – 2x – 11 = x2 – 5x
25.A model rocket is launched from a 5 m high pad straight up into the air with an initial velocity of 150 m/s. The height of the rocket h, in metres, is modelled by h = –5t2 + 150t + 5, where t is the elapsed time in seconds.
a) Use the method of completing the square to determine the maximum height attained by the rocket.
b) For what length of time (how long) was the rocket above 730 m? Do NOT guess, show your work.
26.Determine the values of the three primary trigonometric ratios for BAC in each of the following triangles. Indicate BAC using the Greek letter .
a) b)
27. Find the measures of the indicated sides or angles in each of the following diagrams.
a)Find a and b. b) find c.
c) Find d and e. d) find f.
e) Find g and h.
28.Triangle ABC is an isosceles triangle with equal sides of length 20 cm. The angle between the equal sides is 40°.
a) Draw and label a diagram of this information, first.
b) Find the length of the altitude of this triangle.
c) Find the length of the base of this triangle.
29.To determine the height of a bridge support column, a surveyor uses a transit mounted on a tripod. The transit sits vertically 1.5 m above ground level and is placed 20 m from the base of the column. The angle to the top of the column is 60° from the horizontal.
a) Draw and label a diagram of this information, first.
b) Determine the height of the column.
c) Determine the transit’s angle of declination from the horizontal to the base of the column.
30.Rudy placed a mirror on the ground 6 m from the base of a flagpole. He stepped back until he could see the top of the flagpole reflected in the mirror. Rudy is 1.8 m tall and saw the reflection when he was 1.25 m from the mirror. How high is the flagpole?
31.Solve the triangle by finding the unknown measures.
32.Determine the value of x to one decimal place.
a) b)
33.Calculate the unknown measure c using the sine law. Express your answer to 1 decimal place.
a)
34.In ABC, b = 5.4 cm, c = 6.2 cm, and B = 56. Determine the measure of C ( 1 dec. place),
using the sine law. Draw and label a diagram.
35.In PQR, how long is side p to 1 decimal place? Use the cosine law.
36.Solve ABC, given A = 48, c = 15 cm, b = 12 cm. Use the method of your choice.
Answer to 1 decimal place. Draw and label a diagram.