math wizard 2010(review)

8/13/2019 Math Wizard 2010(REVIEW)

http://slidepdf.com/reader/full/math-wizard-2010review 1/12

Eliminations

1.

What are the last two digits in the expansion of

76

53

2?

2. Find the inverse Laplace transform of 3 2

5 2

5 8 4

s F s

s s s

.

3. If 9

3 9

x

x f x

, find

1 2 2009

2010 2010 2010 f f ... f

.

4. Evaluate: 0

2 1 xt cos t e dt

5. What is the sum of the sequence of 2468, 1012, 1416,…, 8688, 9092, 9496?

6. If1 4 3

3

k k

, find10

10

1k

k .

7. How many ways can the AAC choose the quizmasters for four quarter rounds if each quarter rounds

requires a male and a female quizmaster and that there are 6 females and 5 males to choose from?

8. If the diagonal increases at a rate of 2.5 m/s, at what rate is the area of the shaded region changing

when the side of the square is 8 cm? Assume that the figure maintains its proportion as it magnifies.

9. Find ABE ADE ADC A : A : A when 120m A ,m BAD m DAE m EAC .

10. Given that 1 2 AF : FB DE : EA : , what part of parallelogram ABCD is shaded?

11. The radius of the circle C is twice the radius of circle B. Find the area of the shaded region if the

radius of circle B is 1 cm.



12. Evaluate: 2 234

4 3

x cos y y sin x dx dy

13. Solve the differential equation

2 3

3 2

2 3 5 1

dy d y d y y x x x

dx dx dx .

14. Evaluate:

1

2009 2010

2010 2009 x x

x

15. Find all the values of x such that the determinant of the transpose of the matrix

1 2

3 5

0 1

x

A x

x

is equal to the determinant of the inverse of A . Note that A is non-singular.

16. Solve for the particular solution of 2 7 3 0 x'' x' x with initial boundary conditions

0 3 0 0 x ,x' .

17. Find the sum of the coefficients of the general equation of the ellipse bounded by the lines

5 1 8 y x , y x , y x and 2 y x .

18. Find the value of x such that the determinant of the matrix

2

2

1 1

1 1 1

1 1 1

x x x

x x

x

is a maximum.

19. What is the probability of getting a sum from 10 to 18 from tossing three dice?

20. Find the directional derivative of x f x, y e cos y at point 2 0 , , in the direction of 2 3i j .

21. Find the maximum value of 5 2 2 2 sin x cos x sin x cos x .

22. Let A be a continuous function. If

2 2

2 2

2 3 9 4

3 2 2

df df x y

dx dy

df df x xy y

dx dy

, find f x, y .

23. In right triangle ABC, 18 AB and 24 BC . BE is an altitude to side AC, DE is an altitude to side

AD, DG is an altitude to side AG and FG is an altitude to side AF. If this continues infinitely, find the

area of the shaded region.



9. (4) An 8-cm earthworm and a 10-cm centipede are racing each other. If their speed and 1.5 and 1

cm/sec, respectively, and if their tails are aligned at the start of the race, how long for the

earthworm to overtake the centipede?

Answer. 20 sec

10. (6) Solve the system of equation:

3 2

2 0

2 3

x y z

x z

x y z

Answer. 7 9 14

13 13 13 x, y,z , ,

11. (6) Find the solution for the equation: 2 2 2tan x sin x

if 0 2 x ,

Answer.3 2 4 5

02 2 3 3 3 3

x , , , , , ,

12. (6) 10% of the homes in Bay, Laguna are insured against flood. If four homes are chosen at random,

what is the probability that at least three of the four homes are insured of flood?

Answer. 0.0037%

13. (6) A number N gives a remainder of 3 when divided by 8 and a remainder of 7 when divided by 9.

Find the remainder when N is divided by 72.

Answer. 43

14. (6) Lexy has 20 watermelon seeds and 4 pots of different sizes. To prevent the seeds from

overcrowding, pot 1,2,3 and 4 have a maximum capacity of 9, 7, 11 and 14, respectively. How many

ways can Lexy plant his seeds if he uses all his pots?

Answer. 82015. (10) In the figure below, the radius of B is half that of A, radius of C is half that of B, radius of D is half

that of C, and so on. Determine the area of the shaded region.

Answer.48

5

sq. units

Clincher

(4) Among ten rebuilt transmissions, three were not properly rebuilt and has internal defect and will fail in

its first few miles of operation. If four were randomly selected, what is the probability that exactly one is

defective?

Answer. 1/2

Quarterfinals 2



1. (2) Every three balls simultaneously drawn from an infinite number of balls assured Shelly that he

draws at least one green ball. What is the probability that exactly one green ball is drawn in 5

successive draws of one ball each?

Answer.211

243

2. (2) Exactly one of George, Parker and Jeo is lying.

George: Jeo is taller than Parker.

Parker: I am taller than George.

Jeo: I am the tallest.

Who is the tallest?

Answer. Parker

3. (4) Find the point on the graph of2 2

4 4 x y that is nearest to the line 2 3 12 0 x y .

4. (4) Evaluate:3

1

Arc tan r dr

Answer.3 1

23 4 2

ln

5. (4) Let S be the set of all permutations of a, b, c, d and e. Find the 37th

word if the elements of S are

arranged alphabetically.

Answer. bdace

6. (4) Iael, Ram and Joey are all party goers. 18 parties were attended by exactly 2 of them, and 20

were attended by all three of them. If Iael attended 2/9 of the total number of parties and 3 more

than the number of parties he was not able to attend to. _____?

Answer. 52%

7. (6) Find x such that

0 1 0 0 1 0 1 0

1 0 0 0 0 0

0 0 0 1 1 0 0

0 0 1 0 0 1 0

x x x

x

x

.

Answer. 0 2 ,

8. (6) If 3cot , find

2

2 4

4 2

sin sin tan

cos cos Arc sin sec

.

Answer. 3 3 3

9. (6) A quiz has 3 multiple choice items, each with 3 choices and 2 true or false items. If Sheldon is sure

with 2 multiple choice items and had to guess everything else. What is the probability that he gets a

perfect score?

Answer. 1/4

10. (8) Given the figure below, find m AD m CD .



Answer. 80/3

11. A seedling grows by 9k of its current height every 6 months for the first 2 years. After which it grows

by2

9k of its current height every 9 months for the next 4 years. If its height today is 4096 units and

its height 6 years ago was 1 unit, find k .

12. (10) Let H be an operational symbol such that 0

k ex

k H f x lim f x e dx

where f x is

defined for 0 x . Solve for H f x when f x sin x cos x .

Quarterfiinals 3

1. (2) Determine the sum of the digits of100 10

10 100 ?

Answer. 720

2. (2) Three points are on a circle and divides it into non-overlapping arc with measures 8 7 , and

3 cm. Find the length of the shortest chord that can be drawn by connecting any 2 of the 3 points.

Answer. 9 cm

3. (2) Three congruent circles are positioned tangent to each other such the centers form an equilateral

triangle. Find the total area of the circle.

Answer. 48 sq units.4. (2) Given the table below, find the probability that the respondent is a male given that the

respondent answered yes.

FEMALE MALE

NO 2 3

YES 10 8

MAYBE 13 14

Answer. 4/9

5. (2) Lita, Rita and Tita said exactly one of the following statements: I have a green chair.

Lita is on my left. Someone to my right is on a red chair. Without knowing who said which statement,identify who sits on a blue chair?

Answer. Lisa

6. (4) Let 23 108 324 f x x x . Find

2

2

x f x

x

when f x is at its minimum.

Answer. 3

7. (4) Evaluate:

22

0

2

ie

ied d

Answer. 15



8. If 1 2 2 1 2 3 4 2 1 2n n n f x ,x ,...,x x x x x ... x x and 2

2n

i j

f H f x M

x x

, find

the cum of the eigenvalues ofT H H .

Answer. 0

9. (4) Hyun Joong made a right triangular hat with legs of length 0.5 m and 1.2 m. Geum Jan Di wanted

a smaller hat. Hyun Joong cut the hat into two and produced another 2 right triangular hats. Find the

area of the smaller hat.

Answer. 15/338 sq m

10. (6) Find y R such that

1 3

3 4

3 5

1 3

2 5

2

0

3

0

1

x x

yx x

x yx

x yx

x x

has a unique solution?

Answer. 0 1 R ,

11. (6) In a solar system with infinite planets, let the sun be the origin. If the nth planet is at3

nn,

and planet x is located such that it is collinear with the sun and the 4th

planet and is 217 units

away from the 3rd

, what is the radius of planet x’s orbit?

Answer. 16 units

12. (6) Among all prime numbers less than 20, 2 were chosen at random. What is the probability that

the sum of these numbers is divisible by 3? SetAnswer. 3/7

13. Find the fundamental solution set of cot x cscx secx tanx .

Answer.3 7

4 4 ,

14. Four colored codes can be generated by 7 colors of the rainbow. But blue, indigo, and violet are the

only colors that can be repeated in a code. How many codes can be generated?

Answer. 1273

15. (10) A triangular pyramid of volume 512 cubic meters is to be constructed by folding th equilateral

triangle such that the lateral faces and base re congruent. If this triangle is cut into 16 congruent

pieces each of which is folded to form a pyramid similar to the original, find the volume of one piece

of new pyramid.

Answer. 8 cubic meters

Clincher

1. (4) Find the remainder when 2 4 6 2006 2008! ! ! ... ! ! is divided by 105.

Answer. 11

2. (4) There are 4 red and 6 yellow M & M’s. Lenard got 2 out of the bag and ate them. He then got oneM & M out of the bag and placed it on his hand. Find the probability that the M & M is red.



Answer. 2/5

3. Find x when 2 2ln x ln x .

Answer.2

1 ,e

Quarterfinals 4

1. (2) If x, y Z and 2 3 8748 x y , find

3 x y ?

Answer. 15

2. Geum Jan Di stands 8 m west of the light house. If Hyun Joong sees the light at his south, 480 m

away at an angle of elevation of 60 , find the distance between the two.

Answer. 250 m

3. Two cards are successively drawn from a standard deck of cards without replacement. Find the

probability that the second card drawn is red if the first is not a diamond.

Answer. 77/155

4. The day after the week before the first day of the month of December is Thursday. On what day of

the week will December 25 fall on the same year?

Answer. Saturday

5. (4) If 1

f xtan x

, approximate1

4 f

to two decimal places.

Answer. 1.25

6. Evaluate:20 2

p dq

pq q

7. In how many ways can ten girls sit on a circular table if Laura, and Mara wants to sit together and if

Diane, Jan and Tamara are inseparable?

Answer. 8640

8. A chemist needs a mixture of Chemicals A, B and C. The mixture must be 30% Chemical A, 40%

Chemical B and 30% Chemical C. Initially, he has 600 ml of Chemical A and 400 ml each of Chemicals

B and C but he does not have any measuring device to obtain the amount of chemicals he needed.

What he does have is the knowledge that Chemical A, B and C evaporates at the rate of 3, 4 and 5

ml/ min. At 1:02pm, he started exposing Chemical A to sunlight. Later, he plans to start exposing

chemicals B and C to sunlight as well. At what time must he expose the 2 chemicals to get an amountof each of the needed ratio at the same time?

Answer. 2:42 pm

9. (6) Find x such that

0 3 2 0 1 1

3 0 2 1 0 0 2 16

0 0 0 3 2 0 3 1

0 2 4 1 1 0 1

x x

x

x

x

.

Answer. 4

10. (6) A disco ball hangs on the ceiling with a point z on its surface 2 m away from Ana at an angle of

depression of 60 . The ball rotated on its vertical axis by 180 and the angle of depression from



point z to Ana is 30 . The ball rotated on its horizontal axis by 180 and the point z is back to its

original location. Find the area of the sphere.

Answer. 4 sq m

11. (8) If 16 AB

and AB CD

, find the area of the shaded region.

Answer. 32 sq units

12. 30 students belong in exactly one category. 8 are GC, 9 are HMWBC, 6 are NF and 7 are PF. NF repel

GC, GC repel PF, PF repel HMWBC while HMWBC repel all 3 others. How many groups of 6 can be

formed such that no one from a group repels anybody within the group?

Answer. 1828

13. (10) If 8 xdy y edx with 0 7 y , find

5

5d y x

dx.

Answer. 8 sinh x

Semifinals 1

1. Charlie is thinking of the smallest multiple of 72 whose digits are 1 and 2. What is the number?

Answer. 122,112

2. (2) Evaluate:12 12 8 8

cos sin cos sin

Answer. 1 2

4

3. (2) A circle has a radius of 2 units. Find how fast is the area of a sector changes per sec if the angle

increases by 2 units/ min.

Answer. 1/15 sq units/ sec

4. Two cards are simultaneously drawn from a standard deck of cards. What is the probability that

exactly one is a heart and the other is a queen?

Answer. 6/221

5. (2) There are four party girls. Each has 4 bags. In each bag, there are four cats. How many legs are

there in all?

Answer. 264

6. (4) Find the eigenvalues of the given matrix

16 0 0 0

30 13 0 0

4 5 9 0

4 7 19 5

A

.

Answer. 16, 13, 9, 5

7. (4) There are 3 red flags, 2 green, 1 yellow and 2 blue flags. How many ways can we arrange the flags

in a row such that no yellow is adjacent to any blue flag?

Answer. 840



8. (6) If 10q plog p log q , find 3 3

q plog p log q .

Answer. 970

9. (6) Five boys and 3 girls are to be seated on a row. How many ways can they sit if 2 boys are to sit

together and no two girls are adjacent?

Answer. 2880

10. (8) There are eight congruent spheres and a cube. One octant of each sphere is removed and the

remaining part of each sphere is attached to each corner of the cube, i.e., the center of the sphere is

on the vertex of the cube. If the total volume is 343 252 , find the total surface area.

Answer. 294 298 sq units

11. (8) Evaluate:2 3 2010

6 6 6 6 sin sin sin ... sin

Answer. 2 3

12. (10) An octagonal and a heptagon prism have the same regular polygon with side s as its base. If

OP HP V V , and h is the height of the octagonal prism, express the height of the heptagonal prism

in terms of h .

Answer.4 6

3h

Semifinals 2

1. (2) If 2 1

3

x

g x x

, find 1

5 g

.

Answer. 2

2. Find the fundamental solution set of1

csc tancsc tan

.

Answer.3 5 7

4 4 4 4 , , ,

3. MJ wakes up in a bad mood 3/10 of the time. He knows that 7 of his 13 dreams are about his thesis.

If he dreams about his thesis, it is 37% likely that he will wake up with a good mood. What is the

probability that MJ dreamt about his thesis given that he is in a good mood?

Answer. 37/130

4. (2) The UP Math Club will celebrate their 89th

Foundation Day on February 14, 2010. On what day of

the week was UPMC founded?

Answer. Monday

5. (4) Evaluate:

2 7 7 6

0 8 6 9

0 0 11 16

0 0 0 7

Answer. 1232

6. (4) Determine the coefficient of7 x in the expansion of

103

1 x x ?



Answer. 1740

7. (6) Evaluate:2

6

0

/

cos x dx

Answer.5

32

8. (6) Find2 x y such that 63 225 0 x y ,x, y Z , y least possible integer.

Answer. -3

9. Each of two taxis has a maximum capacity of 4 passengers, two in the front and two at the passenger

seats. What is the number of possible seating arrangements of seven friends if one must sit on a

passenger seat?

Answer. 10,080

10. (8) If2 2

4 x y , find the probability that 1 y ?

11. (8) Evaluate:

4 4

2

6

9 9 125

19 9 12

cos sin tan

sin sin tan

Answer.2 3

3

Final Round

1.

Determine the probability that a factor of 2010 is divisible by 2.

2. If1 2

3 Arccot Arctan x

x

, find x .

Answer. 3

3. Find the remainder when 2010

5

4

n

n !

is divided by 24.

Answer. 9

4. Find the area of the region bounded by2

4 y x and its mirror image with respect to the x

axis.

Answer. 64/3

5. A cone is filled up with water at the rate of 10 cubic m/min. If the ratio between its base and height

is 1:3, at what rate does the area of the base increases when the water is 6 m deep?

6. Using only the first seven positive integers, express 2010 by inserting operations between them.

Answer. 1 2 3 4 5 67

7. Evaluate:

5

0

55

r

r r

?

Answer. 7776

8. What is k if the points 1 2 1 1 3 1 3 , , , , ,k , ,k , are collinear?Answer. -3 or 2



9. If

0 1 2 1 0 1 1

0 1 0 1 0 11 1

1 2 0 1 1 0 0

0 2 3 1 0 3 1 2

a

b a ,

b

, find the trace of A B .

Answer. 45/4

10. If6 4

2 3 sin x sin y ,cos x cos y , find sin x y .

Answer.767

36

math wizard 2010(review)

Documents