MPM1D Exam Review

MPM1D Exam Review #1: Polynomials & Exponents

1. Write in expanded form, then evaluate.

a) 53 b) ( )24− c) ( )24− d) 3

54⎟⎠

⎞⎜⎝

⎛ e) 4

62⎟⎠

⎞⎜⎝

⎛−

2. Evaluate.

a) ( )325 − b) ( ) ( )3354 −−−− c) ( )[ ]6312 −− d) ( )

2

2

3222

−

−− e)( ) ( )

1553

2733 2

4

3

−

−+

−

+

3. Evaluate the expression 27279 23 −+− xxx for 3=x 4. Write each as a single power, then evaluate.

a) 53 333 ×× b) 3

5

66 c) ( )432xy when 2=x and 1−=y d) ( )4255

5. Classify the following polynomials by the number of terms. a) 254 zy − b) 2222 352 baabba −+ 6. Identify the like terms in each of the following: a) 2222 2,5,8,3,3,3 yxyxxyxyxyx b) 2,2,2,2 23 bbb 7. Simplify the following: a) ( ) ( )yxyx 5243 −++ b) ( ) ( ) ( )baabba −+++− 35732 c) ( ) ( )yxyx 3752 −−+ d) ( ) ( )prqrqp 52736 +−−+− e) ( ) ( )4635 −++ xx f) ( ) ( )2534 −−− yy 8. Use the distributive property to expand and simplify:

a) ( )253 −y b) ( )27 2 +aa c) ( )[ ]34523 −+ p

d) ( )423 −x e) ( )42 +x f) ( )423 +− x g) ( )3437 2 +−− xx

h) ( )5328 2 +− xx i) ( )xx 356230 2 + j) ( ) ( )2222 5234 nmmnnmnm −+−−−

9. In a soccer league, teams are awarded 3 points for a win and 1 point for a tie. a) Write an expression to determine a team’s total points. b) Use your expression to find the number of points earned by a team that has 5 wins and 2 ties.

10. Simplify by collecting like terms. a) xx 365 ++ b) 23 +− xx c) 362 22 ++− xx d) xx 235 −++− e) xx 564 −+− f) xxxx 323 33 ++− g) 725534 22 −−−++ xxxx h) yxyx 6453 +++ i) abababa 353546 323 −+−+−

j) 2222 42223 ywywywyw +−−−+ k) mdmd 5435 −−+

11. Simplify each of the following.

a) ( ) ( )2345 baba × b) 7

56

ddd c)

( )( )25

36

yy

12. Expand and simplify. a) ( ) ( )264523 22 +−+−+ xxxx b) ( ) ( )2623625 22 +−−+− xxxx c) ( ) ( )yxyx 635324 −−+ d) ( )[ ] ( )32432 −−+ aaaaa

13. a) Determine an expression for the volume of a cube with side lengthx .

b) Suppose we doubled the side length of the cube. Determine a new expression for the volume of the cube, in terms ofx . c) How many times greater is the larger cube than the smaller cube? Determine how many times bigger the following cube is than the first: d) If the sides were tripled. e) If the sides were made four times longer.

14. Determine a simplified expression for the Perimeter and Area of the following:

15. a) Determine a simplified expression for the area of the shaded region.

b) Determine an expression for the total edging (both inside and outside).

16. Matt is building a dock at his cottage. The length of the dock is 3 metres longer than twice the width. a) Draw a diagram of the dock and label the width and length with algebraic expressions. b) Find simplified expressions for both the Area and Perimeter of the dock. c) Find the Perimeter and Area of the dock if the width is 2 metres.

Exam Review #1 – Solutions

1a) 243 b) 16 c) -‐16 d) 12564 e)

811

2a) -‐40 b) 7 c) 64 d) 2 e) -‐8 3) 0 4a) 6561 b) 36 c) 256 d) 1953125 5a) binomial b) trinomial

6a) 3xy2 and 8xy2, 5x2y and 2yx2 b) none 7a) 5x -‐ y b) 10a + 3b c) -‐5x + 8y d) p – 8q +5r e) 111 −x f) 1−−y 8a) 15y – 6 b) 7a3 + 14a

c) 60p – 39 d) 126 −x e) 82 +x f) 126 +− x g) 212821 2 −+− xx h) 402416 2 +− xx i) xx 69012880 2 + j) 22 343 nmnm +− 9a) P = 3W + 1T b) P = 17 10a) 68 +x b) 22 +x c) 33 2 −x d) 2−−x e) 24 +− x f) xx 25 3 + g)

22 −+− xx h) yx 117 + i) 35711 23 −+− baba j) 22 3yw + k) md 2− 11a) a8b6 b) d4 c) y8

12a) 7187 2 −− xx b) 2482 ++− xx c) yx 427 +− d) 4a3 + 22a2 +3a 13a) 3xV = b) 38xV = c) 8 times d) 27 times e)

64 times 14) Perimeter = 210 +x Area = xx 34 2 + 15a) Area = 9x b) Edging = 1212 +x 16a) b) Area = xx 32 2 + Perimeter = 66 +x c) Area = 14m2 Perimeter = 18m

3x 2x

2x

2x + 12x+7

2x

x - 1 x

MPM1D Exam Review #2 – Equations

1. Solve. a) 1152 =+x b) 853 −=−− y c) 32410 −=+ f d) 346 −=+− u

e) 8411 +−= p f) 5x + 4 = 2x + 13 g) -‐3r + 7 = -‐5r + 3 h) 108835

−=+ nn

i) aaaa 73624 −+=−− j) 11337 +−=+− dd k) ( ) 1376

23

+=+x

l) 375=

−b m) ( ) ( ) 11521234 +−−=+− xx n) 1326 −= m

o) ( ) 32531

−=+ pp p) 5

1235 +=

− xx q) ( ) ( )3212

53

−=+ vv

r) 4432

71 xx −

=−+ s)

222

613

47

3123 −

+−+

=−

−−

+xxxxx

t) Complete formal checks for f), h), and q).

2. Isolate the indicated variable. a) maF = for m b) 2rA π= for r c) ( )wlP += 2 for w

d) bmxy += for m e) 2

2atvtd += for a

3. Tyler has $23.65 to spend. He wants to buy a book that costs $5.95 and spend the rest on magazines that cost $2.95 each. a) Write an equation that models the number of magazines that Tyler can purchase. b) Solve the equation. 4. In a triangle, the largest angle is twelve times larger than the smallest angle, while the smallest angle is three times smaller than

the middle angle. Find the measure of the angles. 5. The sum of three consecutive integers is 24. Find the three integers.

6. The sum of three consecutive even integers is 42. Find the three integers.

7. The sum of three consecutive odd integers is -‐33. Find the three integers.

8. One number is three more than four times another. If the sum of the two numbers is 23, then find both numbers.

9. One number is six more than six times another. If you subtract the larger number from the smaller number the result is -‐41.

What are the two numbers?

10. The total age of three cousins is 20. Suresh is half as old as Hakima and four years older than Saad. How old are the cousins?

11. Two friends are collecting pop tabs for charity. Chris has 250 more pop tabs than Jeremy. Together they have collected 880 pop tabs. How many tabs has each person collected?

12. Jacob, Garrett, and Liam are raising money for a marathon. Jacob raises twice as much as Liam, while Liam raises $50 less than

twice as much as Garrett. Together, they raised $2475. How much did each person raise?

13. The length of a banquet hall is double its width. The perimeter of the banquet hall is 192m. a) Find the length and width of the banquet hall. b) How much shorter would it be to walk across the diagonal of the banquet hall instead of around the perimeter?

14. The formula ( )3295

−= FC is used to convert Fahrenheit temperatures to Celsius.

a) Determine the Celsius temperature when 90=F . b) Solve for F in terms of C. c) Determine the Fahrenheit temperature when 25=C .

15. When you multiply a number, x, by k, add n, and divide by r, the result is w.

a) Write an equation that represents this relation. b) Solve the relation for x.


1a) 3 b) 1 c) 221

− d) 43 e)

43

− f) 3 g) -‐2 h) 1954 i)

45 j) -‐2 k)

934

− l) 26 m) -‐5 n) 221 o)

514 p) -‐28 q) -‐27

r) 3273

s) 3513

2a) aFm = b)

πAr = c) lPw −=

2 d)

xbym −

= e) ( )2

2t

vtda −= 3a) m95.295.565.23 += b) 6

4) 10°, 50°, 120° 5) 7, 8, 9 6) 12, 14, 16 7) -‐13, -‐11, -‐9 8) 4, 19 9) 7, 48 10) 4, 8, 16 11) 315, 565

12) $375, $700, $1400 13a) 64m, 32m b) 24.4m 14a) 32.2 b) 3259

+= CF c) 77 15a) r

nkxw += b)

knwrx −

=

MPM1D Exam Review #3 – Relationships 1. Jacob did a science experiment to test Beer’s Law, which states that as the

concentration of a solution increases, the absorption of light increases as well. The data he collected is shown to the right.

a) Identify the independent and dependent variables. b) Create a scatter plot to represent the relationship between concentration and

light absorption. c) How does the value of the dependent variable change as the independent variable

increases? d) Identify any outliers in the data. 2. Data was collected to study whether students’ test marks were related to the number

of hours of television they watched the night before the test. Data was collected for 10 students.

a) Make a scatter plot of the data. b) Describe any trends in the data. c) Draw a line of best fit on your scatter plot. d) Determine the equation of your line of best fit. e) Using the line of best fit, predict the test score for a student that watched 6 hours of television. f) Using the line of best fit, predict the amount of TV watched by a student that scored 70%. 3. A driver leaves home at 8:00 AM and drives 120km by 9:30 AM. From then until

10:30 AM, she travels another 50km. She drives an additional 200km by 12:30 PM. She then stops for 30 minutes, before driving back home. She arrives home at 5:30 PM. She travels at a constant speed during each period of time. a) Draw the distance-‐time graph that represents this trip. b) Determine the driver’s speed over each section of the trip.

4. Justin takes a bus from the airport to his hotel.

Use the distance-‐time graph to the right to describe Justin’s trip.

Exam Review #3 -‐ Solutions 1a)increases, light absorption increases. d) (140, 39.7) 3b) 80km/h, 50km/h, 100km/h, 0km/h, 82.2km/h

Concentration (µM/L) Absorption (%) 0 0 20 4.7 40 9.6 60 13.5 80 18.2 100 21.4 120 27.3 140 39.7 160 34.2 180 37.9 200 42.6

TV Watched (h) 2 4 0 3 2 2 1 3 1 2 Test Mark (%) 82 64 84 70 74 76 85 73 94 90

D E F

H

B

A

C

G

MPM1D Exam Review #4 – Linear Relations

1. Alexei works during the weekends at a restaurant, earning $10.50 per hour. His pay varies directly with the time, in hours. a. Choose appropriate letters for variables and make a table of values showing Alexei’s pay for 0h, 1h, 2h, 3h, and 4h. b. Graph the relationship. c. Write an equation to represent this relationship.

2. Matt bikes 50km to a friend’s home. The distance, d, in kilometres, varies directly with the time, t, in hours. a. Find an equation relating distance and time if it takes 1.5 hours to travel 24 kilometres. b. What does the rate of change represent in this problem? c. Use the equation to determine how long it will take Matt to reach his friend’s house.

3. Loreena has $50 in her piggy bank, but withdraws $2.50 each day to buy a chocolate milk and muffin at the cafeteria. Create a table of values, a graph, and an equation to represent the amount of money remaining in her piggy bank each day.

4. The amount of juice varies directly with the amount of water used to prepare it. Tommy used 2L of water to make 2.5L of juice.

a. Explain why it makes sense for this relation to be a direct variation. b. Write an equation relating the amount of juice made to the amount of water needed.

5. Identify each of the following as direct variation, partial variation, or neither.

a) 25 += xy b) dC ⋅= π c) xy −=1 d) 52 −= xy e) 61

32

+−= xy

6. a) Complete the table of values on the right, given that y varies partially with respect to x.

b) Identify the initial value and constant of variation from the table, and use them to write an equation in the form bmxy += .

7. A company is having business cards printed. The cost to design the cards is $25. There s an additional

charge of $0.04 per business card printed. a) Identify the fixed cost and variable cost. b) Write an equation representing this relationship. c) Use your equation to determine the total cost of 500 business cards. d) Use your equation to determine the number of business cards that can be purchased for $225.

8. Calculate the slope of each line segment.

9. Calculate the slope of the line through each pair of points. a) (-‐2, 5) and (4, -‐8) b) (-‐7, 8) and (4, 8) c) (32, 630) and (58, 1020)

10. Dalton was biking towards his home at a constant speed. After two hours of cycling he was 55km from home. After 4.5 hours of

cycling he was 17.5 km from home. How fast is he cycling? 11. For safety reasons, an extension ladder should have a slope between 6.3 and 9.5 when it is placed against a wall. Determine if

the following ladders are within the safe range. a) A ladder reaches 4m up the wall, while the foot of the ladder is 0.5m from the wall. b) A ladder reaches 3m up the wall, while the foot of the ladder is 0.6m from the wall.

x y 0 5 1 9 2 3 17 4 37

12. Determine whether each relation is linear or non-‐linear. If the relation is linear, determine an equation to represent the relation.

a) x y b) x y c) x y d) x y 0 5 0 14 0 21 3 -‐15 1 11 1 8 3 13 4 -‐12 2 17 2 3 6 5 5 -‐9 3 23 3 -‐1 9 -‐3 6 -‐6 4 29 4 -‐4 12 -‐11 7 -‐3

13. The distance-‐time graph to the right shows two cyclists that are travelling at the same time.

a) Calculate the speed of each cyclist. b) What does the point of intersection of the two lines represent?

Exam Review #4 – Solutions 1a) 0, $10.50, $21.00, $31.50, $42.00 c) y=10.50x 2a) d=16t b) speed c) 3.125 hours 3) Equation: 5050.2 +−= dA 4a) (0,0) is part of the relation, meaning that 0L of water will make 0L of juice b) WJ 25.1= 5a) partial b) direct c) partial d) neither e) partial 6a) (2, 13), (4, 21), (8, 37) b) initial value: 5, constant of variation: 4, equation: 54 += xy 7a) fixed

cost: $25, variable cost: $0.04 per card b) 2504.0 += nC c) $45 d) 5000 8) AB: 23 CD:

41

− EF: 0 GH: undefined 9a) 613

− b) 0 c)

15 10) 15 km/h 11a) safe b) unsafe

12a) linear, 56 += xy b) non-‐linear c) linear, 2138

+−= xy d) linear, 243 −= xy 13a) A: 10km/h B: 20km/h b) The point

where cyclist B passes cyclist A.

MPM1D Exam Review #5 -‐ Linear Relations

1. Match each linear equation with the graph that best suits it. 2. Identify the slope and y-‐intercept for each of the relations

shown below. Then, write the equation of each line.

3. Convert the following to slope-‐intercept form. Identify the slope and y-‐intercept.

a) 0852 =−+ yx b) 1937 =− yx c) ( )1523 +−=− xy

4. Convert the following into standard form.

a) 73 +−= xy b) 211

53

−= xy c) ( )4327 −=− xy

5. Determine the x-‐intercepts, y-‐intercepts, and slope of each of the following.

a) 1425

−−= xy b) 6045 −=− yx c) 03367 =−− yx

6. Graph the following lines on the same grid.

a) 8−=y b) 7=x c) 154

+= xy d) 1243 =− yx e) 4)6(3 ++−= xy

7. Determine whether the points A(2, -‐6), B(-‐3, 10) and C(-‐1, 1) are on the line 024 =−+ yx .

8. Find the equation of the line (in slope-‐intercept form) with a slope of 34

− that passes through the point (5, 7).

9. Find the equation of the line, in point-‐slope form, of the line that has a slope of 118

and passes through the point (-‐9, 12).

10. Determine the equation of the line that is perpendicular to the line 01106 =−+ yx and has the same y-‐intercept as

13 =+ yx . 11. Find the equation of the line that is parallel to the line 012 =−− yx and passes through (8, -‐7).

12. Find the equation of the line perpendicular to 01234 =+− yx that has the same x-‐intercept as 01893 =+− xy .

13. Find the equation of the line the passes through:

a) (3, -‐3) and (-‐3, 5) b) (5, 2) and (5, -‐3) c) (8, -‐5) and (2, 7)

i) iv) iii)

ii)

a)

b)

c)

d)

e)

14. Holly downloads music from a site called MyTunes, which chargers a monthly membership fee plus an amount for each song

downloaded. A 3-‐month record of her site activity is shown below. a) Use two points from the table to determine an equation relating Holly’s monthly

bill the number of songs she downloaded. b) Verify that the third point in the table satisfies your equation.

15. Solve the following linear systems graphically. Check your solution.

a) 52 −= xy and 53 +−= xy b) 04 =−+ yx and 022 =+− yx

16. Majida needs to park her car downtown while she takes a course. She can buy a monthly pass for $82, or she can register for $10 and then pay $6 each day to park. Under which conditions should Majida buy the monthly pass? When should she go with the second plan?


1) (a) -‐> (iii), (b) -‐> (vi), (c) -‐> (iv), (d) -‐> (ii), (e) -‐> (i), (f) -‐> (v) 2i) slope: 2, y-‐int: 3, 32 += xy ii) slope: 54

− , y-‐int: -‐2, 254

−−= xy iii)

slope: 0, y-‐int: 8, 8=y iv) slope: undefined, y-‐int: none, 6−=x 3a) 58

52

+−= xy , slope: 52

− , y-‐int: 58 b)

319

37

−= xy , slope: 37, y-‐

int: 319

− c) 513

52

+−= xy , slope: 52

− , y-‐int: 513

4a) 073 =−+ yx b) 055106 =−− yx c) 01332 =+− yx 5a) x-‐int: 528

− , y-‐

int: -‐14, slope: 25

− b) x-‐int: -‐12, y-‐int: 15, slope: 45 c) x-‐int:

733

, y-‐int: 211

− , slope: 67 6)

7) A: yes, B: no, C: no 8) 341

34

+−= xy 9) ( ) 129118

++= xy

10) 135

+= xy 11) ( ) 782 −−= xy 12) 23

43

+−= xy

13a) ( ) 3334

+−−= xy b) 5=x c) ( ) 582 −−−= xy

14a) 50.2110.0 += nC 15a) (2, -‐1) b) (2, 2) 16) If she is going to park more than 12 times, she should buy the monthly pass. If she parks fewer than 12 times, she should go with the second plan. If she parks exactly 12 times, then it doesn’t matter which plan she choose

Month # of songs Monthly Bill January 54 $26.90 February 38 $25.30 March 21 $23.60

MPM1D Exam Review #6 -‐ Angle Geometry

1. Identify a pair of the following angles in the diagram: a) Corresponding Angles (F-‐Pattern) b) Co-‐Interior Angles (C-‐Pattern) c) Alternate Angles (Z-‐Pattern) d) Opposite Angles (X-‐Pattern) e) Supplementary Angles

2. a) Find the measures of the exterior angles of ΔPQR.

b) Find the measure of ‹ABD. a) b)

3. Find the measures of the unknown angles in the diagrams below.

a) b) c)

4. For the diagram below, find the value of ( )ba + . Give your answer as a number of degrees.

5. Determine the measure of ‹EGA in the diagram to the right. 6. The interior angle in a certain parallelogram is four

times larger than the exterior angle adjacent to it. Determine the measure of each interior angle.

7. In triangle ABC, ‹C is 1° more than five times the measure of ‹A,

while ‹B is 21° less than four times the measure of ‹A. Find the measure of all interior and exterior angles.

a b

d c

e f

g h

P

QR

S

T

U30°

85°

A

B CD

25° 75°

x105°

75° 85°

92°

97°

55°

y

50° z

y x

5x 4x

a

b

3x C

F

H

x + 20

G

? B A

E

D

8. The formula for calculating the sum of the interior

angles for an n-‐sided polygon is ( ) 1802 ×−n . a) Explain why 2 is subtracted from n. b) Explain why (n -‐ 2) is multiplied by 180°.

9. Determine the measures of each interior and exterior angle for the following figures.

a) A regular 12-‐gon b) A regular 15-‐gon c) A regular 25-‐gon 10. The sum of the interior angles of a polygon is 2160°. Find the number of sides. 11. Madison designed a tabletop in the shape of a regular pentagon. Mr. Toms suggested that she redesign it in the shape of a

regular hexagon. By how much would each interior angle change? 12. Determine the value of x in the diagram to the right.

13. Determine the values of a, b, c, d, and e

in the diagram below.

14. In triangle QTU, QU = 6 cm, TU = 8cm, and QT = 10 cm.

What is the perimeter of triangle QRS?

Exam Review #6 -‐ Solutions 1a) a & e b) c & f c) d & f d) e & h e) b & c 2a) P = 95°, Q = 150°, R = 115° b) 155° 3a) x = 95° b) 116° c) y = 50°, x = z = 130° 4) 20° 5) 120° 6) 144°, 144°, 36°, 36° 7) At A: Int. = 20°, Ext. = 160° At B: Int. = 59°, Ext. = 121° At C: Int. = 101°, Ext. = 79° 8) Check your notes 9a) Int. = 150°, Ext. = 30° b) Int. = 156°, Ext. = 24° c) Int. = 165.6°, Ext. = 14.4° 10) 14 11) 12° 12) 107 13) a = 90°, b = 60°, c = 60°, d = 210°, e = 60° 14) 12 cm

x + 4

x - 5

x + 3 x + 10

x - 7

e d

c

b

a

S R

T U

Q

MPM1D Exam Review #7 – Measurement

1. Find the measures of the unknown sides in the triangles below: a) b)

2. A rectangular lot measures 150 m by 200 m. Instead of walking along the outside of the lot, Alexei hops a fence and walks along

the diagonal of the lot. How much distance does he save? 3. Find the area of the following triangles.

a) b)

4. Zach is planning to repaint the two gables on his house.

One of the gables is shown. A can of paint covers 10 m2. How many cans of paint does he need?

5. A school field has the dimensions shown.

a) Calculate the length of one lap of the track. b) If Loreena ran 625 metres, how many laps did she run? c) Calculate the area of the field.

6. An engine seal is circular in shape, with a square

cut-‐out to fit over a shaft, as shown to the right. a) Calculate the area of rubber required to make the seal. b) The inside and outside edges of the seal have a steel wire embedded inside to add strength. How much wire is needed?

p

15 cm

8 cm 9 cm

q

25 cm

10 m 26 m

4 cm

6 cm

13 m

24 m

4 cm

2 cm

7. Determine an expression for the shaded area of each figure. a) b) c)

8. A DVD player came packaged in the box pictured to the right.

a) Find the volume and surface area of the box.

b) Determine the box’s volume if all dimensions are doubled.

9. Matt made the ramp to the right to test his new Slinky.

Find the volume and surface area of the ramp. 10. A spherical storage tank has a diameter of 16m.

a) Find the surface area of the tank. b) A can of paint costs $30 and covers 20m2. How much will it cost to paint the tank? c) A new tank will be built with a surface area of 2000m2. What radius will be required?

11. Chris is organizing a track meet, and he’s rented the pyramid-‐shaped

changing tent on the right for the athletes. Find the tent’s volume, as well as the amount of fabric used to make the tent.

12. A dozen tennis balls, each with a radius of 3.2 cm, are placed

in a box so that they just barely fit. The balls form a single layer that measures 3 balls by 4 balls. How much empty space is left in the box?

13. Sawdust from a woodworking lab is blown into a conical container

for recycling into other products. The container has a radius of 1.5m and a height of 2m. a) Find the area of aluminium needed to make the sides and top of the container. b) How much sawdust can the container hold?

14. A cone is truncated to make a paper cup as shown in the diagram to the right. Calculate the amount of paper needed to make the cup, as well as the amount of water this cup can hold.

15. A square-‐based pyramid has a volume of 100 m3

and a base area of 40 m2. What is its height? 16. A pyramid and a prism, both with the same height,

each have a base area of 64 m2. How do their volumes compare?

cup

40 cm

30 cm

20 cm

DVD Player

1.4 m

1.4 m

2.5 m

10 cm 80 cm

24 cm

17. A triangular piece of cheese has a volume of 146.4 cm3.

Find the thickness, t, of the cheese.

Exam Review #7 – Solutions 1a) 17cm b) 23.3cm 2) 100m 3a) 120m2 b) 12cm2 4) 6 5a) 170m b) 3.7 c) 1530.04m2 6a) 42.27cm2 b) 37.1cm 7a) 22 rRA ππ −= b)

224 rrA π−= c) 224 rrA π−= 8a) 24000cm3, 5200cm2 b) 192000cm3 9) 9600cm3, 5040cm2 10a) 804.25m2 b) $1230 c) 12.6m 11a) 1.50m3, 8.68m2 12) 1498.61cm3 13a) 18.85m2 b) 4.71m3 14a) 147.59cm2, 147.78cm3 15) 2.5m 16) volume of prism is 3 times larger 17) 3.2cm

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