2012 Mathacre JV Written Test

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

Answer: B
Solution: 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) 9699690
b) 9699707
c) 9699673
d) 8699673

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:

a)
b)

c)
d)

Answer: C
With 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)

b)
c)
d)

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 120
lll. N is an odd number

a) ll and lll
b) l and ll
c) lll
d) l
e) ll

Answer: A
We 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)
b)
c)
d)
e)

Answer to question 6: E

Set up a 3x3 matrix. and solve for determinate then multiple times ½ for area. Which

comes out to 39/2.

7. If ?

a)
b) –
c)
d) 1
e) 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)
b)
c)

d)

Answer: C
The volume of a sphere is and the volume of a cube is so the ratio of sphere to cube’s

volume will be Since the diagonal of the cube is the diameter of the sphere

Plug into ratio formula: This turns out to be

9. What digit does end in?

a) 0
b) 3
c) 1
d) 4

Answer: D
When 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) 22
b) 23
c) 28
d) 31
e) 21

Answer: E
Key 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 C

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.
B
A

E

C

D

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

I. Triangle ADE is similar to triangle ACB.
II. AE = AB
III. AE • ED = AB • AD
IV. AB2 + DC2 = AD2 + BC2
a) l only
b) l and lll
c) l,ll, and lV
d) ll and lV
e) none of the above

Answer A
Solution: Because it is given that we know that the two triangles have one angle in
common. Then segment DB bisects therefore since they are
both right angles, the measures of 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

a)
b)
c)
d)
e) Not possible

Answer: D
take the log base 2 of both sides

Cannot factor simply so plug into
And you get (10 +/- √70)/3

14. Find the average of the coordinates of the focus of the parabola

a) 3/8
b) 5/8
c) -3/8
d) -11/8

Answer: D
vertex: (-2, 1) focus: (-2, -3/4)
4p = 7 average: (-2+-3/4)/2
p = 7/4 = -11/8
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) 72
b) 90
c) 67
d) 74

e) 65

Answer: A
There 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) 11
b) 9
c) 7
d) 6
e) 10

Answer: C
If 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%.

17. Which of the five fractions is the largest?

a)

b)

c)

d)

e)

Answer: D
The fractions are all of the form and n is equal to five consecutive odd numbers. So since 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.

B

E 110°
F

A D C

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

a) 45°
b) 80°
c) 110°
d) 70°

Answer: D
It is 70° becasuse you know the angles of figure AEFD equals to 360° and bisectors create right triangles.
becasuse it is supplementary to and since ∆BDC is 90°, and
so you add 110°+90°+90°= 290° and subtract from 360° and the difference is 70°.

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

a) 119
b) 129
c) 115
d) 101

Answer: B

Working 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

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

Answer: A
You 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) 90
b) 89
c) 91
d) 92

Answer: C
You 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 (

a) 6
b) 6.5
c) 2

d) 3
e) 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) ½
b) 2
c) 1
d) 1.75

Answer: B
This 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.
which comes out to 2.

24. Y varies directly with 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)
b)
c)
d)
e)

Answer: D
The equation is so to get z by itself you rewrite it as and since we know that k is equal to
2 and x is equal to 3y+4 we can substitute those into the equation. but this is not the

simplest form you can expand and so the answer is z=

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) 82
b) 75
c) 61
d) 68

Answer: A
Solution:
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.

2012 Mathacre JV Written Test

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2012 Mathacre JV Written Test