2012 Valentine’s Day Mathacre

Creekview High School

Junior Varsity Written Test

1. Dr. Eddy is 3 times as old as Tom. Five years ago, Tom was twice as old as Josh. In 10 years, Dr. Eddy will be twice as old as Josh. How old is Dr. Eddy now?

a) 25b) 22.5c) 7.5d) 6.25e) None of the above

Answer: BSolution: E=3a 2J+10=3(2J-5) 2(6.25)+10=22.5 a=2J-5 2J+10=6J-15 E=2J+10 4J-15=10

4J=25 J=6.25

2. What is the product of the prime factors of 20! minus the difference of the greatest prime number and the lowest?

a) 9699690b) 9699707c) 9699673d) 8699673e) None of the above

Answer: C Solution: prime factors: 2,3,5,7,11,13,17,and 19 [(2)(3)(5)(7)(11)(13)(17)(19)]-(19-2) 9699690-17

9699673

3. Condense:

3 log5 z+2 log5w+log5 x

4−

log5 y

3

a) log 2(z3w2 4√ x¿ 3√ y )¿

b) log5(z3w3 4√x

3√ y)

c) log5(z3w2 4√x

3√ y)

d) log5 ¿¿

e) None of the above

Answer: CWith this problem, it has to be broken down step by step using the rules of logarithms. First, every part of the equation has a log base of 5 so A is eliminated. Next, numbers are only moved to the front of logs when they are previously an exponent. So, z in this case would be raised to the third and w is squared. Another rule to recall is that when expanding numbers that are previously a root the system is then divided by that number. So since the log base of 5 was taken out already that leaves the fourth root of x and it is the same for y except it is the third root. Finally, recall that with expanding multiplication turns to addition while division turns to subtraction. Therefore, the first three logs are multiplied times each other while being divided by the last giving you answer C.

4.

4

4

In the figure above, the diameter of the circle is 4. What is the area of the shaded region?

a) π−34

b)14+ π

2

c)12+ π

8

d)34−π

e) None of the above

Answer: E

5. If N= 2010 x 2011 x 2012 x2013+1, which of the following statements is true?l. N is a prime number.ll. N-1 is divisible by 120lll. N is an odd number

a) ll and lllb) l and llc) llld) le) ll

Answer: AWe know that any even number multiplied by an odd number is going to be even. So, 2010x2011 is even. Same goes for when even numbers are multiplied together. Therefore when 2010, 2011, 2012, and 2013 are multiplied together the answer is even. Looking at the end numbers we can also predict that the last digit is going to be zero so it is indeed divisible by 120 or the odds are very high. Because we know that without one added the last digit is 0, the value for N is actually odd. Then, we know that N is not a prime number because many possible factors besides 1 are found with the numbers multiplied together to get N.

6. Find the area of the triangle whose vertices are (2,3),(5,-2),(-1,-5).

a)652

b)452

c)25

d) 412

e) 392

Answer to question 6: E

Set up a 3x3 matrix. [ 2 3 15 −2 1

−1 −5 1] and solve for determinate then multiple times ½ for area. Which

comes out to 39/2.

7. If j=√−1,what is the value of ( 1+ j1− j )

2013

?

a) 1b) – jc) jd) 1e) 0

Answer: B

Substituting i into the equation it comes out to a negative. Any negative raised to an odd power is going to be a negative.

8. A cube is inscribe in a sphere of radius r. Find the ratio of the volume of the sphere to the cube’s.

a)2π

√3

b) √32 r

c) √3 π2

d)3π

√2e) None of the above

Answer: C

The volume of a sphere is V= 43π r3

and the volume of a cube is V=s3 so the ratio of sphere to cube’s

volume will be 43π r3

s3 . Since the diagonal of the cube is the diameter of the sphere s√3=2 r . so s=2 r

√3

Plug into ratio formula:

43π r3

( 2r√3 )

3 This turns out to be √3π /2

9. What digit does 20122012 end in?

a) 0b) 3c) 1d) 4e) None of the above

Answer: DWhen squaring 2012 each time it turns out that with one it ends in 2, 2 it ends in 4 and goes on like this: 8,6,2,4,8,6,2,4,8 and so on. All ending in either 2,4,6 or 8. Therefore, because 2,6, and 8 are not a choice, you can conclude that 4 is the correct answer. (you can also know that any even number times another even number is going to be even)

10. How many integers between 1 and 500 are perfect squares?

a) 22b) 23c) 28

B

A

D

C

E

d) 31e) 21

Answer: EKey word is BETWEEN. So it does NOT include 1. Therefore there is: 4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,364,400,441,484. Which is 21. (many people are gonna choose a if they didn’t catch that trick)

11. Which of the following statements below is a logical inference from the following assertions:

l. For a person to be either conceited or annoying it is necessary that they be smart or dramatic.ll. Drake is not smart.lll. Drake is annoying.

a) Drake is conceited.b) Drake is awesome.c) Drake is dramatic.d) Both B and Ce) None of the above

Answer: C Because drake is not smart, he cannot be conceited…so A is eliminated, but because he is annoying he has to be either smart or dramatic…since we know he is not smart he has to be dramatic. Choice C. we cannot assume he is awesome with the give evidence so choice C is the best answer.

12.

Let ABCD be a cyclic quadrilateral (meaning all vertices lie on a circle) with diagonals AC∧BD as shown in the figure. Focus on point E on the diagonal BD so that ∠ DAE equals ∠ BAC and ∠DCB∧∠ADC are right angles. Which of the following statements is true?

I. Triangle ADE is similar to triangle ACB.

II. AE = ABIII. AE • ED = AB • AD IV. AB2 + DC2 = AD2 + BC2a) l onlyb) l and lllc) l,ll, and lVd) ll and lVe) none of the above

Answer ASolution: Because it is given that ∠DAE=∠BAC we know that the two triangles have one angle in common. Then segment DB bisects ∠ ADC∧segment AC bisects∠DCB therefore since they are both right angles, the measures of ∠ ACB∧∠ADE areequal . Meaning that ∆ADE and ∆ACB have two angles measures in common and because all triangles add up to 180 degrees, they have 3 angles in common. Therefore, they are similar triangles. Then, because nothing else can be proven without further evidence, the answer choice is A.

13. Find the sum of the solutions of 23 x2

=324x−2

a) 20+√946

+ 20−√946

b) 10+2√103

+ 10−2√103

c) 10+√943

+ 10−√943

d) 10+√703

+ 10−√703

e) Not possible

Answer: D23 x2

=324x−2take the log base 2 of both sides

1 (3 x2 )=5 (4 x−2 )3 x2=20 x−103 x2−20 x+10=0

Cannot factor simply so plug into x=−b±√b2−4ac2a

And you get (10 +/- √70)/3

14. Find the average of the coordinates of the focus of the parabola x2+4 x+7 y−3=0

a) 3/8b) 5/8c) -3/8

d) -11/8e) None of the above

Answer: Dx2+4 x=−7 y+3 vertex: (-2, 1) focus: (-2, -3/4)x2+4 x+4=−7 y+3+4 4p = 7 average: (-2+-3/4)/2( x+2 )2=−7 y+7 p = 7/4 = -11/8

( x+2 )2=−7 ( y−1 ) focus: (-2, 1-7/4)

15. The pages of a book are number 1 through 374. How many times does the digit 7 appear in this numbering?

a) 72b) 90c) 67d) 74e) 65

Answer: AThere is a ones unit of 7 for every 10 numbers, so from 1 to 374 there are 37 sets of 10 so 37 7s. Then there are 10 tens digits of 7 in every set of 100 (70-79) so that from 1 to 374 contains 3 sets. So 30 tens of 7, but final from 370-374 contains 5 more tens digits of 7. So, 37+30+5=72 times

16. Robin Hood is an archery contests. If he is down on his luck and missed the target on his first shot and hits the target on the next three shots, what is the least number of consecutive hits he must achieve following the first four shots in order to hit the target on more than nine tenths of his shots?

a) 11b) 9c) 7d) 6e) 10

Answer: CIf Robin Hood missed 1 out of 4 shots in the beginning that means he hit the target 75% of the time. (3/4). We want it to be the least amount of shots to hit the target on more than 90% accuracy though. If he hits the target the next 11 shots he would have 93% accuracy (14/15) but it isn’t the SMALLEST amount. If he hits the target the on the next 9 shots he has about 92.8% accuracy (13/14), but this is yet again not the smallest amount. 10 is the same because it is between 9 and 11. Then, if he hits the target 6 times after the first 4 that would mean he had exactly 90% accuracy. Since the problem asks for smallest amount that is MORE THAN 90% it cannot be the answer so 7 is the answer. Bc 10/11 shots is the smallest amount over 90%.

F

110°

B

A CD

E

17. Which of the five fractions is the largest?

a)860124563581860124563583

b)860124563583860124563585

c)860124563585860124563587

d)860124563587860124563589

e)860124563583860124563584

Answer: D

The fractions are all of the form n

n+2and n is equal to five consecutive odd numbers. So since

nn+2

gets

larger as n gets larger, the second to the last one is the biggest. (or you could notice that ever number up until the last is the same and look at the fractions…short cut!)

18.

In the figure above, CE and BD are angle bisectors of ∆ABC which intersects at point F. If m∠BFC = 110°, find the measure of ∠ A.

a) 45°

b) 80°c) 110°d) 70°e) None of the above

Answer: DIt is 70° becasuse you know the angles of figure AEFD equals to 360° and bisectors create right triangles. ∠DFC=70 ° becasuse it is supplementary to ∠BFC and since ∆BDC is 90°, ∠BCD=20° and ∠EFD=∠BFCso you add 110°+90°+90°= 290° and subtract from 360° and the difference is 70°.

19. Given real numbers a and b, let a●b=a2+ab solve for 3●(4●(2●1))

a) 119b) 129c) 115d) 101e) None of the above

Answer: BWorking from the inside out: 2 is a and 1 is b. plug into equation and you get 6. So then 4 is a and 6 is b. plug into equation again and you get 40. Now 3 is a and 40 is b. plugging into the equation and you finally end up with 129.

20. If log ab=56 , find loga2b3

a) 84b) 23c) 121/3d) 44e) 511

Answer: AYou have to get b by itself by exponentiating with a . so b= a^56 from there using logic you can eliminate answers by plugging b into the new equation.

21.

Within the figure above, assume all the smaller squares are equal, how many squares of all sizes are there?

a) 90b) 89c) 91d) 92e) None of the above

Answer: CYou count the entire thing as a square and all the small ones and then work around row by row combining squares to equal bigger ones.

22. What is the value of (log 624625¿( log623624)¿

a) 6b) 6.5c) 2d) 3e) 4

Answer: E All these logarithms gradually get bigger starting from around 1.00025 till 1.11328. With common knowledge of logarithms and multiplying each one together we can estimate that each one will relatively be in that range. Because of this, as the number grows bigger and the multiplication continues the number will get bigger and by a bigger ratio. 2 would not be a high answer because of this and above 6 is too high because of the range of from 1 to 1.11. We can narrow it down to 3 and 4. Logically speaking we can estimate that the number would be towards the larger size because the logarithm goes till 625 so there is repetitive multiplication. The most logical answer is 4.

23. A square has vertices at (-2,-1), (-2,5),(4,5),and (4,-1). Find the slope of the line through the origin which cuts the area of the square into halves.

a) ½b) 2c) 1d) 1.75e) None of the above

Answer: BThis is a relatively simple problem if read correctly. The key is through the origin. Since the line has to go through the origin, we are going to use the point (0,0), and we know that the it’s going to pass through

the center of the square which is point (1,2) then using those two points we can determine the slope. 2−01−0

which comes out to 2.

24. Y varies directly with x2 and inversely with z. If K is equal to 2 and x is equal to 4 more than 3 times y, solve for z in terms of y. (in the simplest form please )

a) 2 y (3 y+4 )2

b) 4 (3 y2+12 y+4 )y

c) y (3 y2+12 y+4 )

d) 2 ( 9 y2+24 y+16 )y

e) y2(3 y+4 )

Answer: D

The equation is y= k x2

zso to get z by itself you rewrite it as z= k x2

yand since we know that k is equal to

2 and x is equal to 3y+4 we can substitute those into the equation. z=2 (3 y+4 )2

ybut this is not the

simplest form you can expand and so the answer is z= 2( 9 y2+24 y+16 )

y

25. In a college class, 48 of the students are freshman, 44 are engineering majors and some are art majors. If half of the engineering majors are freshman, a fourth of the freshman are art majors, and there are no just art majors, how many students are in the class?

a) 82b) 75c) 61d) 68e) None of the above

Answer: ASolution: If half the engineering majors are freshman then that means that 22 students are freshman and engineers, but that leaves 26 freshman who are not engineering majors. Then a fourth of the total freshman are art majors, or 12 in this case. That leaves 22 just engineering majors. 22+22+26+12=82.