1.
If A $ B = A * B - ( A + B), what is 3 $ (2 $ 1)?
Correct Answer
A. -5
Explanation
First, use the order of operations to find the parenthetical value of (2 $ 1), which is 2*1-3 or -1
Next, plug in -1 for (2 $ 1) to find 3 $ -1 = 3 *-1 - (3 - 1) = -3 - 2 = -5
2.
How many numbers less than 1000 are divisble by 3?
Correct Answer
D. 333
Explanation
Every multiple of three is divisible by 3, so divide 1000 by 3 and take the whole part = 333.3333 = 333
3.
If a^b = c^d, which of the following is not necessarily true?
Correct Answer
D. A = c
Explanation
A, B, C, and E can be proven by substituting a^b for c^d. However, D is not always true. For example, 2^3 = 8^1, but 2 is not equal to 8.
4.
Which of the following lines does not intersect y = 5 x + 2?
Correct Answer
E. 5x - y = -29
Explanation
You are looking for a line whose slope is equal to the slope of the given line, or 5. Slope in standard form = -A/B, so E is the answer (-5/-1 = 5 = slope)
5.
A regular polygon has 9 sides. What is the degree measure of the angle, within the polygon, between any two sides?
Correct Answer
D. 140
Explanation
First, you must figure the sum of the measures of the internal angles, which is equal to (n-2)180 or (9-2)180 = 7 * 180. Then, you should divide this measure by the number of sides to find the length of each angle. (7 * 180)/9 = 7 * (180/9) = 7 * 20 = 140 degrees
6.
A cubic box, X, has sides of length n. Another cubic box, Y, has sides of length 2n. How many boxes X could fit into a single box Y?
Correct Answer
C. 8
Explanation
The volume of Y is equal to (2n)^3 = 8n^3. The volume of X is equal to n^3. So, to find the number of X's that could fit into Y, you should divide the volume of Y by that of X, or 8n^3/n^3, which is equal to 8.
7.
A number is called "round" if it contains at least one zero as a digit. How many three-digit numbers are "round?"
Correct Answer
B. 171
Explanation
An easy strategy here is to just count the number of "round" numbers between 100 and 199 and multiply that number by 9 for every hundreds digit (100, 200, 300... to 900). So, there are 19 "round" numbers through 100-199 (100, 101, 102 ... 109 = 10 + and 110, 120, 130... 190 = 9). 19 * 9 = 171
8.
How many ways can Pete, Mary, Sue, and Joe stand in a line if Joe and Sue cannot stand next to each other?
Correct Answer
C. 12
Explanation
Step 1: Total Arrangements
There are 4 people, so without any restrictions, they can be arranged in 4 factorial (4 x 3 x 2 x 1) ways, which equals 24 ways.
Step 2: Arrangements Where Joe and Sue Are Together
Treat Joe and Sue as one unit. This changes the problem to arranging 3 units: {Joe-Sue}, Pete, and Mary.
These 3 units can be arranged in 3 factorial (3 x 2 x 1) ways, which equals 6 ways.
Joe and Sue can switch places within their unit, and this can happen in 2 factorial (2 x 1) ways, which equals 2 ways.
Therefore, Joe and Sue can be together in 6 x 2 = 12 ways.
Step 3: Valid Arrangements (Joe and Sue Not Together)
Subtract the number of ways Joe and Sue are together from the total arrangements: 24 (total ways) - 12 (Joe-Sue together) = 12 ways
9.
A square, X, has sides of length n. Another square, Y, has sides of length 1.5n. How many X can fit into a single Y?
Correct Answer
D. 2.25
Explanation
The number of X's that fit into Y is equal to the ratio of area of Y to that of X. Area of Y = (1.5n)^2 = 2.25. The area of X = n^2. 2.25n^2/n^2 = 2.25
10.
A triangle has sides of length 7, 11, and X. Which of the following cannot be X?
Correct Answer
A. 2
Explanation
The sum of lengths of two sides of a triangle cannot be less than the length of the third side. So, 2 + 7 = 9, but 9 < 11
11.
If |a| < |b|, and a > b, which of the following is necessarily true?
Correct Answer
E. None of the above
Explanation
None of the given options are necessarily true under the conditions that the absolute value of a is less than the absolute value of b, and a is greater than b. Each statement can be disproven with counterexamples. These conditions do not support the conclusions drawn in the statements about absolute values and arithmetic operations.
12.
Six children sit at a circular table. In how many orders can they sit at the table?
Correct Answer
E. 120
Explanation
At a circular (ring) table, the order ABCDEF = BCDEFA = CDEFAB, and so on. So, you need to find the total number of orders and divide this number by 6 to recognize the fact that the table is circular. The total number of orders is 6!, and 6!/6 = 5! = 5*4*3*2*1 = 120.
13.
If a two-sided coin is flipped three times, what is the probability that at least one head will show up?
Correct Answer
E. 7/8
Explanation
The probability of one head showing up is equal to 1 - P(All tails), and the probability of all tails is (1/2)^3 = 1/8, so 1- 1/8 = 7/8
14.
If m & n = (m + n)^(m - n), what is 2 & (2 & 2)?
Correct Answer
B. 3
Explanation
First, do what is in the ( )'s. (2 & 2) = (2 + 2) ^ ( 2 - 2) = 4^0 = 1.
Next, plug in 1 for (2 & 2) to get 2 & 1 = (2 + 1)^(2 - 1) = 3^1 = 3.
15.
A rectangle's length is twice its width. If the perimeter of the rectangle is 30 meters, what is the area of the rectangle?
Correct Answer
D. 50 square meters
Explanation
First, let's define the width of the rectangle as w meters. According to the problem, the length (l) is twice the width, so we can express this as: l = 2w
The formula for the perimeter (P) of a rectangle is given by: P = 2l + 2w Substituting the given perimeter and the expression for l, we get: 30 = 2(2w) + 2w 30 = 4w + 2w 30 = 6w w = 5 meters
Now that we know the width, we can find the length: l = 2w = 2*5 = 10 meters
The area (A) of the rectangle is calculated by multiplying the length and width: A = l * w = 10 * 5 = 50 square meters
16.
A three-digit number is called "big" if any two of its digits are equal. How many three-digit numbers are "big?"
Correct Answer
D. 252
Explanation
Start with 100-199 and then multiply the result by 9 for each hundreds digit. 100, 101, 110, 111, 112, 113, ... 119, 121, 122, 131, 133, 141, 144, 151, 155... = 2 + 10 + 2(8) = 12 + 16 = 28. 28 * 9 = 252
17.
If 2 ^ (4x + 3) = 4 ^ (x - 1), what is x?
Correct Answer
B. -5/2
Explanation
If a ^ b = a ^ c, then b = c. However, 2 does not equal 4, so we should make the two equal. 4 = 2 ^ 2, so 4 ^ (x - 1) = (2 ^ 2) ^ (x - 1) = 2 ^ 2(x-1) by exponent rules. So since 2 = 2, 4x + 3 = 2(x-1) -> 4x + 3 = 2x - 2 -> 2x = - 5 -> x = -5/2
18.
Which of the following values of x is not in the domain of the function y = x / (x^2-2x+1)
Correct Answer
E. 1
Explanation
Any time the denominator is equal to 0, that value of x is not included in the functional value. So, x / (x^2-2x+1) = x / (x-1)^2. (x-1)^2 = 0 only when x = 1.
19.
If a + b = y, what is a^2 + 2ab + b^2?
Correct Answer
C. Y^2
Explanation
a^2+2ab+b^2 = (a+b)^2 = y^2
20.
The number 100 has two trailing zeros. How many trailing zeros does 100! have?
Correct Answer
D. 24
Explanation
Every trailing zero indicates a factor of 10 or 5 * 2. Since there are many more factors of 2 than 5's, you should count the number of factors of 5 there are in 100! Since it is one big product (100 * 99 * 98...), count: 5, 10, 15, 20, 25 (5 * 5), 30, 35, 40, 45, 50 (5*5), 55, 60, 65, 70, 75 (5*5), 80, 85, 90, 95, 100 (5*5). The total number of 5's is 24.
21.
Which of the following statements is always true?
Correct Answer
D. |a^2 + b^2 - 1|
Explanation
Since |X + Y| is always less than or equal to |X| + |Y|, substitute a^2 for X and b^2 for Y to find |a^2 + b^2| is less than or equal to |a^2| + |b^2|
22.
In physics, force = mass * acceleration. Suppose you have an original force F and new force G in which the mass is increased by a factor of two and the acceleration is increased by a factor of four. What is the ratio of G:F?
Correct Answer
E. 8:1
Explanation
F = ma, G = (2m)(4a) = 8ma; 8ma:ma simplifies to 8:1
23.
If a^2 = b^2, which of the following is/are always true?
I. a = b
II. |a| = |b|
III. |a - b| = 0
Correct Answer
B. II only
Explanation
If a^2 = b^2, then a = + or - b.
So, I is not true because a could be = -b
II is true because |-b| = |b|
III is false because |b - (-b)| = |2b| = 2|b| is not equal to 0.
24.
If (a-b)^2 = (a+b)^2, what is the value of ab?
Correct Answer
C. 0
Explanation
(a+b)^2 = (a-b)^2
a^2 + 2 ab + b^2 = a^2 - 2 ab + b^2
2 ab = -2 ab
4 ab = 0
ab = 0