Assessment 2 RMO 10th Class

ASSESSMENT #2 RMO Batch/10th

Directions:

1. This exam consists of 30 questions in Part A and 2 questions in Part B. Please check tomake sure that you have all the pages.

2. Allot your time accordingly. This is a 90-minute test. Do not spend too much time onany one problem. If a question seems to be too difficult, make your best possible guess.Your score will be the number of correct responses.

3. On the scantron form provided for you, darken in the space corresponding to the correctanswer. Please mark all answers carefully and erase completely when changing an answer.Mark only one answer for each question. Only those answers on the answer sheet willbe counted.

Please do NOT open the test until you are told to do so.

1. How many positive integers below 2009 are divisible by both 4 and 6?

A. 502 B. 334 C. 167 D. 83 E. None of these.

2. What is the distance between the points (4,12) and (-2,4)?


3. What term fits in the blank space in the sequence 1/2, 4, 1/8, 16, , 64, ?

A. 1/24 B. 1/32 C. 1/40 D. 1/42 E. None of these.

4. Write an equation in point-slope form of the line that contains the point (3, 5) and hasthe slope 1.

A. y + 3 = (x 5) B. y 3 = (x+ 5) C. y + 5 = (x 3)

D. y 5 = (x+ 3) E. None of these.

5. If the measure of an angle is 78 less than the measure of its complement, what is themeasure of the angle?


6. The length of one side of a rectangle is 8 cm and its perimeter is 40 cm. Find the area ofthe rectangle.

A. 320 cm2 B. 256 cm2 C. 96 cm2 D. 48 cm2 E. None of these.

7. Given the diagram below, what is the measure of 2?

54

2

1 345

A. 63 B. 71

C. 81 D. 77

E. None of these.

1

8.a2 =

A. a B.1

aC. |a| D. a2 E. None of these.

PART A

9. Evaluatec4d4e5

c1(d2e5)3.

A. c3d2e20 B. c5d2e15 C. c3d10e10

D. c5d2e20 E. None of these.

10. In a movie, an FBI agent has to stop a ticking time bomb within 30 seconds by cuttingone wire from a group of 4 indistinguishable red wires and another wire from a group of5 indistinguishable green wires. There is exactly one combination that would stop thebomb from exploding. What is the probability (mathematically, not by the movie script)that the FBI agent will succeed if he or she randomly cuts a red wire and a green wire?

A. 5% B. 20% C. 25% D. 45% E. 100%

11. Write an equation in point-slope form of the line that contains the points (2, 6) and(1, 3).

A. y 3 = 1 (x+ 1) B. y 6 = 1 (x 2) C. y 6 = 3(x 2)

D. y + 3 = 1 (x 1) E. None of these.

12. What is the slope of the line 42x+ 9y = 7?

A. 6 B. 314

C. 314

D.14

3E. 14

3

13. Solve 7x1 = 494x for x.


14. Simplify15z3 z2 11z 3

3z2 2z 1 .

A. 5z 3 B. 5z + 3 C. 15z + 3 D. 3z + 5 E. None of these.

15. Neerajs mother is three times as old as Neeraj, and twelve years from now she will be twice as old as Neeraj is then. How old is Neeraj now?


2

16. What is the last digit of 72009?

A. 1 B. 3 C. 5 D. 7 E. 9

17. A rock and a half rock weigh a pound and a half. The rock and the half rock weigh thesame. How much would half the whole rock and the whole of the half rock weigh?

A. 21

2B. 1

1

8C. 1

7

2D. 2

1

4E. None of these.

18. Which of the following is a common factor of x2 10x+ 21 and x2 + 2x 15?

A. x+ 5 B. x 3 C. x+ 3 D. x 7 E. None of these.

19. Fred and Desi left Steamtown Mall at 9:00 am and began walking in opposite directions.At 1:00 pm that same day they were 20 miles apart. If Fred walks 0.5 mph slower thanDesi, what is Desis speed of walking?

A. 2.25 mph B. 2.75 mph C. 3.5 mph D. 5 mph E. None of these.

20. Expand and simplify the expression (a + b)7 + (a b)7.

A. 2a7 + 42a5b2 + 70a3b4 + 14ab6 B. 2a7 + 21a5b2 + 35a3b4 + 7ab6

C. a7 + 21a5b2 + 35a3b4 + 7ab6 D. 2a7 + 42a5b2 + 35a3b4 + 14ab6

E. None of these.

3

21. Simplify128a7b3c4 (extract all perfect roots).

A. 8a2 b c2ab B. 8|a3| |b| c22ab C. 8|a3| |b| c22

D. 8|a7| |b3| c22a E. None of these.

22. If 3b2 = 10b 48, then b =

A.8

3or 6 B. 3

8or 6 C. 3

8or

1

6D.

8

3or

1

6E. None of these.

23. How many liters of water must be added to 15 liters of a 40% acid solution to produce asolution which is 10% acid?


24. If a square of side 8 is inscribed in a circle, what is the area of the circle?

A. 64pi B. 32pi C. 128pi D. 64 E. 32

25. Given a cube of side length 2, with A being one of the vertices and AB, AC and AD arethe three edges at A. Let E, F and G be the midpoints of AB, AC and AD. We thencut the corner containing the vertex A off along the plane determined by E, F and G.What is the volume of the part that was cut off?

B

C

D

E

F

G

A

A.1

2B.

1

3

C.1

4D.

1

5

E.1

6

4

26. A shadow in the sun cast by a tree is 48 ft. At the same time, a shadow cast by a nearbypost is 15 ft. If the line of sight distance from the farthest end of the trees shadow tothe top of the tree is 62 ft., what is the line of sight distance from the top of the post tothe farthest end of the posts shadow? Round your answer to the nearest tenth of a foot.

A. 18.1 ft. B. 19.4 ft. C. 22.9 ft. D. 14.1 ft. E. None of these.

27. Find the surface area of the solid below.

2a

3a

2a

5a

6a

2a

B. 124a2

D. 104a2

A. 96a2

C. 120a2

E. None of these.

28. Alex orders a round 12 in. (diameter) pizza and it is cut into 12 equal pieces where eachcut goes through the center. He eats a quarter of the whole pizza while his wife eats aquarter, his son eats a third, and his daughter eats a sixth. He and his daughter do noteat the crust so they give it to the dog. How many inches of crust (as measured from theoutside) does the dog eat?

A. 5pi in. B. 6pi in. C. 7pi in. D. 8pi in. E. None of these.

5

29. On a line, there are three points J , H , and K (not necessarily in this order). The distancebetween J and H is 4x 15, between H and K is 2x + 3 and between J and K is 48.There is more than one possible value for x. What is the sum of all these possible values?

A. 10 B. 28 C. 17 D. 5 E. 43

30. In the given diagram, the circle has radius equal to 60 while the (short) arc length fromD to E is 80 and the (short) arc length from B to C is 88. What is the measure of BACin radians?

O

A

B

C D

E

A.7

5B.

5

7

C.12

5D.

5

12E. None of these.

(a) A circle of radius 2 and a circle of radius 5are externally tangent to each other at Tand tangent to a horizontal line at pointsP and Q, as shown. If points O and Care the centres of the circles, then O, T , Care collinear and both OP and CQ areperpendicular to PQ. By constructing aline segment passing through O and parallelto PQ, determine the distance betweenP and Q.

P Q

O

CT

(b) A circle of radius 4 and a circle of radius 9are tangent to a horizontal line at pointsD and E, as shown. A third circle can beplaced between these two circles so thatit is externally tangent to each circle andtangent to the horizontal line. If DE = 24,determine with justification, the radius ofthis third circle.

D E

(c) Three circles with radii r1 < r2 < r3 are placed so that they are tangentto a horizontal line, and so that adjacent circles are externally tangent toeach other. F,G,H, I, J, and K are the points of tangency of the circles tothe horizontal line, as shown. The lengths of FG,HI, JK, in no particularorder, are 18, 20 and 22. Determine, with justification, the values of r1, r2 and r3.

KJIHGF

r3r3r2r2r2 r1

r3

r1r1

2.

3

The volume of a cylinder with radius r and height h equals pir2h.The volume of a sphere with radius r equals 4pir3.

(a) The diagram shows a sphere that fits exactlyinside a cylinder. That is, the top and bottomfaces of the cylinder touch the sphere, and thecylinder and the sphere have the same radius, r.State an equation relating the height of thecylinder, h, to the radius of the sphere, r.

rh

(b) For the cylinder and sphere given in part (a), determine the volume of the cylinderif the volume of the sphere is 288pi.

(c) A solid cube with edges of length 1 km is fixed in outer space. Maya, the babyspace ant, travels on this cube and in the space around (but not inside) this cube. If Maya is allowed to travel no farther than 1 km from the nearest point on the cube, then determine the total volume of space that Maya can occupy.

PART B

1.

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Assessment 2 RMO 10th Class

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