1.
If (x^2 * y) / (2z) = 2x, what is y in terms of x and z if x, y, and z are non-zero real numbers?
Correct Answer
D. (4z)/x
Explanation
(x^2*y)/2z = 2x
x^2 * y = 4xz
y = (4z)/x
2.
A number is given as 3513N, where N is a positive integer that fills in the units digit. Which of the following values of N would result in 3513N being a multiple of 9?
Correct Answer
D. 6
Explanation
For a number to be a multiple of 9, the sum of its digits must be a multiple of 9. Since 3 + 5 + 1 + 3 = 12, N must be 6.
3.
X $$ Y is defined as (X + Y)*(X - Y). Which of the following represents X^2 $$ Y^2
Correct Answer
B. X^4 - Y^4
Explanation
Since X $$ Y = X^2 - Y^2, X^2 $$ Y^2 = (X^2)^2 - (Y^2)^2 = X^4 - Y^4
4.
Which of the following number is in the domain of f(x) = squareroot(x-2)/(x-7)
Correct Answer
D. 2
Explanation
Since the denominator is (x-7), x cannot be 7.
Also, since the numerator is a square root function, x-2 must necessarily be greater than or equal to 0, so x must be >= 2. Therefore, only 2 applies.
5.
If |x| = |y|, which of the following is necessarily true?
Correct Answer
E. |x| - |y| = 0
Explanation
Since |x| = |y|, x could be equal to y or -y. Therefore, choices A-D cannot apply because only one case can hold true to the equation.
Choice E, |x| - |y| = 0 is obvious because if |x| = |y|, it is like saying: |x| - |x| = 0, which is always true.
6.
Let X be the number of prime numbers on [1,10]. Let Y be the number of composite numbers on [1,10]. What is X+Y?
Correct Answer
D. 9
Explanation
Since [1,10] includes the numbers from 1 to 10, you must simply count all the primes and composites on 1 to 10; that is, you count all the numbers on 1 to 10 except for 1, because 1 is neither prime nor composite. Therefore, 10-1 = 9.
7.
Solve for x: 2x + 3 = 4x - 3
Correct Answer
C. 3
Explanation
2x + 3 = 4x - 3
3 = 2x - 3
6 = 2x
3 = x
8.
If a @ b = 2(a + |b|), what is the value of 2 @ (-2 @ -1)?
Correct Answer
E. 8
Explanation
First, find the value included in the parentheses first.
-2 @ -1 = 2(-2 + |-1|) = 2(-2 + 1) = 2(-1) = -2
Second, find the value of 2 @ (...)
2 @ -2 = 2(2 + |-2|) = 2(2 + 2) = 2*4 = 8
9.
John is 4 years older than Marie. In ten years, he will be twice as old as Marie. How old is Marie now?
Correct Answer
A. 6
Explanation
Set up a system of equations:
J + 4 = M
J + 10 = 2 * M
Substitute equation 1 for J in 2
(M - 4) + 10 = 2 * M
M + 6 = 2 * M
6 = M
10.
Let f(x) = x^2 - 2x. What is the value of f(2)?
Correct Answer
A. 0
Explanation
Substitute 2 for x in the original function:
2^2 - 2 * 2 = f(2)
4 - 4 = 0
f(2) = 0
11.
Which of the following is not a true statement?
Correct Answer
D. Every number that is odd is not divisible by two
Explanation
3 is not divisible by 2, but 3 is neither even nor composite.
12.
If f(x - 3) = g(x + 3), what does f(x) equal?
Correct Answer
E. G(x+6)
Explanation
Let x - 3 = u
Then x + 3 = u + 6 (x + 3 + 3)
f(u) = g(u+3) in general.
So f(x) = g(x+3+3) = g(x+6)
13.
If |x| = |y| and |x| = z, which of the following is always true?
Correct Answer
B. Z = |y|
Explanation
Only choice B will always be true by the transitive property.
Since |x| = |y|, and |x| = z, |y| = z.
14.
If x/y = z^2 and y/(xz) = z, and x, y, and z are not equal to 0, which of the following must be true?
Correct Answer
A. |x| = |y|
Explanation
Equation 1: x/y = z^2
Equation 2: y/(xz) = z
x / y = z^2
y / x = z^2
So x / y = y / x
So the two have the same absolute value:
x^2 = y^2
x = + or - y
15.
(5 * 5 + 3) / 2 + 7 / 7 = ?
Correct Answer
D. 15
Explanation
5 * 5 = 25 + 3 = 28
28/2 = 14
14 + 1 = 15
16.
How many ways can four students sit around a circular table?
Correct Answer
C. 6
Explanation
The number of arrangements in a circle is given by (n-1)!, so (4-1)! = 3! = 3*2*1 = 6.
17.
Solve for x: 2^(x+4) = 4^(x-2)
Correct Answer
E. 8
Explanation
First, we want the bases to be the same:
2^(x+4) = 2^2^(x-2) = 2^(2(x-2))
2^(x+4) = 2^(2x-4)
x + 4 = 2x - 4
4 = x - 4
x = 8
18.
A set of integers includes: 44, 66, 88, and x. If the mean of the scores is 80, what is x?
Correct Answer
D. 122
Explanation
(44 + 66 + 88 + x)/4 = 80
44 + 66 + 88 + x = 320
x = 320-88-66-44
x = 122
19.
John is older than Maria. Maria is younger than Sue. Terry is the oldest of the four of them. Who is the youngest?
Correct Answer
B. Maria
Explanation
J > M
M < S
T > (J,M, and S)
So,
M < (J, S, T)
Maria is the youngest.
20.
If y = x^2 + 3, at how many points does y=0?
Correct Answer
A. 0
Explanation
Since x^2 is positive always, x^2 + 3 is always positive, so y will never be 0.
21.
If the length of the sides of a cube is tripled, by what factor is the volume of the cube increased?
Correct Answer
E. 27
Explanation
V = s^3
so V of the new cube = (3s)^3 = 3^3*s^3 = 27s^3
27s^3/s^3 = 27
22.
If x = y, which of the follow is necessarily true?
Correct Answer
A. X^2 = y^2
Explanation
x = y. You can square both sides of an equation and retain equality, so
x^2 = y^2
23.
If f(x) = g(x) + h(x), which of the following is not always true for all f, g, and h that are defined at x?
Correct Answer
C. F( f(x) ) = f ( g(x) ) + f( h(x) )
Explanation
Although f,g,h are defined at x, f(f), f(g), and f(h) are not always defined at x. The answer is C.
24.
Which of the following will be equal to the 1 times the sign (positive or negative) of x?
Correct Answer
D. |x|/x
Explanation
The quotient |x|/x is equal to one times the sign of x.
25.
A fair coin is flipped three times. What is the chance that no head will appear?
Correct Answer
A. 1/8
Explanation
The chance of no heads appearing is the same as the chance of all tails appearing, which is: (1/2)^3 or 1/8
26.
A train leaves the train station at 10:00 AM travelling 100 mph due west. Another train leaves another station along the track travelling the opposite direction. If the trains meet at 11:00 AM and travelled 100 miles, what is the speed of the other train?
Correct Answer
C. 100 mpH
Explanation
The total time is 1 hr. The trains meet in 1 hr and both travel 100 miles, so the speed of both is 100 mph.
27.
If a block has dimensions of 3, 4, and 5 units, how many 2 by 2 by 3 blocks can fit in the block?
Correct Answer
A. 5
Explanation
V(big block) = 3*4*5 = 60
V(small block) = 2*2*3 = 12
V(big)/V(small) = 5 small blocks.
28.
If x < |y|, which of the following is always true?
Correct Answer
D. -y < x < y
Explanation
By definition of absolute value x < + or - y
x < y or x < - y
x < y or x > -y
-y < x < y
29.
A six-sided dice is rolled twice. What is the chance that the sum of the two rolls will be exactly 10?
Correct Answer
A. 1/12
Explanation
10 = 5 + 5, 4 + 6, 6 + 4
The probability of any given combination is (1/6)^2 = 1/36
There are three combinations
3*1/36 = 3/36 = 1/12.
30.
In a class election between Ricky, Susie, and Mikey, Ricky earns 20% of the vote and Mikey earns 30% of the vote. There are 24 members of the class. How many votes did Susie receive?
Correct Answer
C. 12
Explanation
First, you will need to know Susie's percentage: 100% - 30% - 20% = 50%
Second, you should multiply that percentage by the total number of votes: 50% * 24 = .5 * 24 = 12
31.
If n is an integer, q is a rational number, and r is an irrational number, which of the following statements is incorrect?
Correct Answer
D. R - q is a rational number
Explanation
If a rational is subtracted from an irrational, the result is always an irrational. So, D is not true.
32.
What is the slope of the line given by 5x - 2y = 0?
Correct Answer
B. 5/2
Explanation
Put it Slope intercept form, so:
5x - 2y = 0
2y = 5x
y = (5/2)x
y = mx + b
m = 5/2
33.
The letters of the word: "alphabet" are placed into a hat and scrambled. What is the probability of choosing a vowel at random from the hat?
Correct Answer
C. 1/3
Explanation
The vowels are: a,e,i,o, and u. "alphabet" has 2 a's and 1 e, or 3 vowels total out of 9 letters. 3/9 = 1/3.
34.
If Mike can buy four apples for one pear and four pears for one peach, and if one peach is equal to eight bananas, how many bananas will Mike receive for one apple?
Correct Answer
A. 1
Explanation
4 apples = 1 pear
4 pears = 1 peach
1 peach = 8 bananas
So,
1 apple = 1/4 pear
1/4 pear = 1/8 peach
1/8 peach = 1 banana.
So,
1 apple = 1 banana
35.
If Pete can run five miles in one hour, how many hours would it take him to run 25 miles?
Correct Answer
C. 5
Explanation
5 miles / 1 hr = 5 mph
25 miles / 5 mph / 5 miles.
36.
Which of the following numbers is not a possibility value of the function f(x) = x^2 - 1
Correct Answer
E. -2
Explanation
x^2 is always positive, so the minimum point of f is at f(x) = -1 or x = 0. So, f can never be less than -1, so E is not possible as -2 < -1
37.
If a number is divisible by 2 but not divisible by 3, what can be concluded about the number?
Correct Answer
D. It will leave a remainder when divided by 6
Explanation
A number that is only divisible by 2 and not 3 is not divisible by 6. Therefore, it leaves a remainder when divided by 6.
38.
In football, a team earns 6 points for a touchdown and can earn 0, 1, or 2 extra points after each touchdown. Which of the following scores is not possible after earning three touchdowns?
Correct Answer
E. 25
Explanation
A team can earn between 6 and 8 points per touchdown. So, after three, they could score between 18 points and 24 points. 25, however, is not possible.
39.
A teacher tells her students that on a 100-question test, the answer to every third question is C and the answer to every fourth question is B. Based on this, how many questions can be answered using her suggestion?
Correct Answer
C. 50
Explanation
Since every third and fourth question is covered, two out of every four questions are covered. That means 1/2 of 100 or 50 questions total are answerable with her suggestion.
40.
Twice the sum of two different numbers is equal to one-third of the product of the squares of the numbers. Which of the following equations represent this statement?
Correct Answer
D. 2a+2b = (1/3)a^2*b^2
Explanation
The sum of two different numbers is a + b
Twice that sum is 2(a+b) = 2a+2b
Product of the squares of two different numbers = a^2*b^2 = (ab)^2
One-third of the product = (1/3)(ab)^2
Only D works out.
41.
If |x| > |y|, and x < y, which of the following is always true?
Correct Answer
D. X is negative
Explanation
For |x| > |y| and x < y, the magnitude of x must be greater than y but x must be less than y. This can only occur where x is negative.
42.
How many ways can the letters in the word "WORD" be arranged?
Correct Answer
C. 24
Explanation
The number of ways that four things can be arranged is 4! = 4*3*2*1 = 24.
43.
What is the value of n! divided by (n-3)! ?
Correct Answer
D. N*(n-1)*(n-2)
Explanation
n! = n*(n-1)*(n-2)*(n-3)...
(n-3)! = (n-3)*(n-4)*(n-5)...
by division: n!/(n-3)! = n*(n-1)*(n-2)