niryuu/sni-each-converted huggingface dataset:
If \(a - 5 = O\), what is the value of \(a + 5\) ?
(A)-10 (B)-5 (C)0 (D)5 (E)10
Output: E
The sum of two consecutive even integers is \(t\). In term of \(t\), what is the value of the greater of the two numbers?
(A)\(t + 2\) (B)\(t + 1\) (C)\(\frac { t } { 2 } \) (D)\(\frac { t + 1 } { 2 } \) (E)\(\frac { t + 2 } { 2 } \)
Output: E
If \(3y^4 + xy - x^2 = y^2 - 25\) and \(y = 0\), how many possible values of \(x\) are there?
(A)0 (B)1 (C)2 (D)3 (E)4
Output: C
For all real numbers x and y, let \(\star\) be defined by \(x \star y = x^2 - y\). What is the value of \(4 \star (3 \star 2)\)?
(A)1 (B)9 (C)15 (D)24 (E)40
Output: B
If \(x + 2y = 14\), \(y = z + 2\), and \(z = 4\), then what is the value of x?
(A)2 (B)4 (C)6 (D)8 (E)10
Output: A
\(S = \ { 3, 6, 12\ } \) From the given above, how many different values can be created by forming fractions \(\frac { a } { b } \) such that \(a\) and \(b\) are distinct elements of the set \(S\)?
(A)2 (B)3 (C)4 (D)5 (E)6
Output: C
\begin{gather*} 3, 8,13,18,23,28, 33, 38 \end{gather*} If from the sequence of numbers above, a new sequence is created by increasing each odd-valued term by 3 and decreasing each even-valued term by 2, the sum of the terms of this new sequence is how much greater than the sum of the terms of the original sequence?
(A)4 (B)6 (C)8 (D)10 (E)12
Output: A
If \(3x < 0 < y\), which of the following is greatest?
(A)0 (B)y (C)\(-3x\) (D)\(-(3x + y)\) (E)\(-(3x - y)\)
Output: E
If \(k\) is at least 20 more than \(\frac { 1 } { 3 } \)of \(p\), which of the following describes this relationship?
(A)\(k \leq p + \frac { 60 } { 3 } \) (B)\(k \leq \frac { p } { 3 } + 60\) (C)\(k \leq p + \frac { 20 } { 3 } \) (D)\(k \geq p + \frac { 60 } { 3 } \) (E)\(k \geq \frac { p } { 3 } + 60\)
Output: D
If \(x^{-\frac{2}{3}} = \frac{1}{36}\), then what does x equal?
(A)\(-6\) (B)\(\frac{1}{6}\) (C)6 (D)18 (E)216
Output: E
If \(r = -2\) and \(s = 5\), what is the value of \(r^2 (2r + s)\)?
(A)\(-36\) (B)\(-28\) (C)\(-4\) (D)4 (E)36
Output: D
The average rate of change of a function f between a and b is defined by \(C(x) = \frac { f(b) - f(a) } { b - a } \). If \(f(2) = 6\) and \(f(5) = 15\), what is the average rate of change between 2 and 5?
(A)3 (B)4 (C)5 (D)7 (E)9
Output: A
If set S has the property that x is in S, and \(x^2 - 2x + 1\) is also in S, which of the following sets could be S?
(A)1,2,3 (B)0,1,2 (C)-1,0,1 (D)0,2 (E)1,2
Output: B
Which of the following sets of numbers has the property that the sum of any two numbers in the set is also in the set? I. \(\{ 1,2,3,...\}\) II. \(\{ 1,3,5,...\}\) III. \(\{ 10,20,30,...\}\)
(A)I only (B)II only (C)III only (D)I and III only (E)II and III only
Output: D
For all positive values of \(p\) and \(q\), let \(p \circledast q\) be defined by \(\frac { pq } { p + q } \). If \(p \circledast q = \frac { 1 } { p } \circledast \frac { 1 } { q } \), which of the following must be true?
(A)\(p = q + 1\) (B)\(p = q\) (C)\(p = -q\) (D)\(pq = 1\) (E)\(pq = p + q\)
Output: D
If 125 percent of x is 150, what is x percent of 75?
(A)70 (B)90 (C)120 (D)150 (E)185
Output: B
If \(p > 0\), then 5 percent of 7 percent of \(4p\) equals what percent of \(p\) ?
(A)14\% (B)7\% (C)1.4\% (D)0.14\% (E)0.07\%
Output: C
If 299 is divisible by \(2B\), and \(B\) represents a positive digit, what is the value of \(B\) ?
(A)2 (B)3 (C)4 (D)6 (E)9
Output: B
If \(a < b < c < 0\), which of the following must be true? I. \(a - b < c\) II. \(-a > -c\) III. \(a + b < b + c\)
(A)I only (B)III only (C)I and II only (D)II and III only (E)I, II, and III
Output: D
\(A = \{ 2, 4, 6 \}\) \(B = \{ 1, 2, 3, 4 \}\) Set \(A\) and \(B\) are shown above. If a is \(a\) member of set \(A\) and \(b\) is a member of set \(B\), how many different values are possible for \(ab\)? If the statement above is true, which of the following must also be true?
(A)6 (B)8 (C)10 (D)12 (E)14
Output: B
If a number is increased by 45 and the result is multiplied by 2, the result is 18 less than the original number. In which of the following equations does \(x\) represent the number?
(A)\(x + 2 \times 45 = 18 - x\) (B)\(2x + 45 = x - 18\) (C)\(2(x + 45) = x + 18\) (D)\(2(x - 45) = x - 18\) (E)\(2(x + 45) = x - 18\)
Output: E
If \(\frac { 2x + 10\sqrt { x } + 12 } { \sqrt { x } + 3 } = 3\sqrt { x } \), what is the value of \(x\)?
(A)0 (B)2 (C)4 (D)9 (E)16
Output: E
The average (arithmetic mean) of x and y is 7, and the average of x, y, and z is 10. What is the value of z?
(A)23 (B)17 (C)16 (D)11 (E)3
Output: C
If \(0.4x + 2 = 5.6\), what is the value of x?
(A)0.9 (B)1.4 (C)9 (D)14 (E)90
Output: C
If a, b, and c are positive and \(a^ { -2 } b^2c^2 > a^ { -3 } b^3c^2\), which of the following must be true? I. \(a > b\) II. \(a > c\) III. \(b > c\)
(A)I only (B)II only (C)III only (D)I and III only (E)I, II and III
Output: A
Which of the following is an equation equivalent to the statement ``12 less than the product of 3 and b is 9''?
(A)\(3(b - 12) = 9\) (B)\(12 - 3b = 9\) (C)\(3b - 12 = 9\) (D)\(12b - 3 = 9\) (E)\(12 + 3b = 9\)
Output: C
If \(f(x) = |x - 4|\) , which of the following has the LEAST value?
(A)\(f(8)\) (B)\(f(4)\) (C)\(f(0)\) (D)\(f(-4)\) (E)\(f(-8)\)
Output: B
What is the value of \(2x^2 - 3x - 7\) when \(x = -5\) ?
(A)28 (B)42 (C)58 (D)78 (E)108
Output: C
For any two distinct negative integers c and d, which of the following must be less than \(cd\) ?
(A)\(cd + 1\) (B)\((-cd)^2\) (C)\(d(c - 2)\) (D)\((c - 1)(d - 1)\) (E)\((c + 1)(d + 1)\)
Output: E
Which of the following expressions has the greatest value?
(A)\(4 \times 4 \div 4 + 4\) (B)\(4 \div 4 \times 4 + 4 \) (C)\(4 \times 4 - 4 \times 4\) (D)\(4 \div 4 + 4 \times 4\) (E)\(4 + 4 \times 4 - 4\)
Output: D
Sequence Y: 5, 10, 15, ... Sequence Z: 3, 6, 12, ... The first term in Sequence Y is 5, and each term after the first is 5 more than the preceding term. The first term in sequence Z is 3, and each term after the first is 2 times the preceding term. What is the least value of x such that the xth term of Sequence Z is more than three times the xth term of Sequence Y?
(A)5 (B)6 (C)7 (D)8 (E)9
Output: B
Set C is made up of a series of consecutive integers whose sum is a positive even number. If the smallest number in the set is \(-2\), what is the least possible number of integers that could be in the set?
(A)4 (B)5 (C)6 (D)7 (E)8
Output: E
If \(4x + 2 = 26\), then \(4x + 8 =\)
(A)32 (B)34 (C)36 (D)38 (E)40
Output: A
If \( 6\frac { a } { b } + 3 = 21\) , what is the value of \(\frac { a } { b } \) ?
(A)-4 (B)2 (C)3 (D)4 (E)6
Output: C
If \(a < b < 0 < c\), which of the following must be true? I. \(-b > -a\) II. \(a + c < b + c\) III. \(a + b < c\)
(A)I only (B)II only (C)III only (D)II and III only (E)I, II, and III
Output: D
Of the following numbers, which is least?
(A)\(2 + \frac{1}{2}\) (B)\(2 - \frac{1}{2}\) (C)\(\frac{1}{2} - 2\) (D)\(2 \times \frac{1}{2}\) (E)\(2 \div \frac{1}{2}\)
Output: C
If the set of four integers \(n\), \(2n\), \(n + 2n\), and \(n + 4n\) consists only of prime numbers, then the set is called a "prime convergence". How many such sets exist?
(A)None (B)One (C)Three (D)Four (E)More than four
Output: A
If the fraction \(\frac{1}{21}\) equals the repeating decimal 0.0476190476190..., what is the 51st digit after the decimal point of the repeating decimal?
(A)0 (B)1 (C)4 (D)6 (E)7
Output: E
If \(\frac { 1 } { k } > \frac { 1 } { 2 } \), then which of the following must be true?
(A)\(k > 2\) (B)\(k \leq 2\) (C)\(1 < k < 2\) (D)\(0 < k < 2\) (E)\(0 \leq k \leq 2\)
Output: D
If \(k\) is a positive integer, which of the following is equivalent to \(3^k + 3^k + 3^k\)?
(A)\(3^ { 3k } \) (B)\(3^ { k + 2 } \) (C)\(3^ { k + 1 } \) (D)\(9^ { 2k } \) (E)\(9^ { 3k } \)
Output: C
The average (arithmetic mean) of 7,20, and x is 20. What is the value of x ?
(A)20 (B)27 (C)32 (D)33 (E)40
Output: D
If \(4(x + y) - 2(x + y) = 12\), then \(x + y =\)
(A)2 (B)4 (C)6 (D)8 (E)10
Output: C
If \(p < q < 0\), which of the following must be less than \(\frac { p } { q } \)?
(A)\(-p\) (B)\(-q\) (C)\(pq\) (D)2 (E)1
Output: E
If \(ab \neq 0\), which of the following must equal \(\frac { a } { b } \) ?
(A)\(\frac { a^2 } { b^2 } \) (B)\(\frac { a^3 } { b^3 } \) (C)\(\frac { 1 } { a } b\) (D)\(\frac { a } { 5 } \frac { b } { 5 } \) (E)\(\frac { a + 1 } { b + 1 } \)
Output: D
If \(x = 2y - 5\) and \(z = 16y^3\), what is \(z\) in terms of \(x\)?
(A)\(\frac { (x + 5)^3 } { 2 } \) (B)\((\frac { x + 5 } { 2 } )^3\) (C)\(2(x + 5)^3\) (D)\(4(x + 5)^3\) (E)\(8(x + 5)^3\)
Output: C
\begin { gather* } 4, 11, 18, 25, ...\end { gather* } The first four terms in a sequence are shown above. Each term after the first is found by adding 7 to the preceding term. Which term in the sequence is equal to \(4 + 40 \times 7\) ?
(A)The 18th (B)The 36th (C)The 39th (D)The 40th (E)The 41st
Output: E
If \(x = 5 + y\) and \(x = 5 - y\), what is the value of \(y\) ?
(A)0 (B)\(\frac { 1 } { 2 } \) (C)1 (D)\(\frac { 3 } { 2 } \) (E)5
Output: A
In the equation \(y = kx\), \(x = 4\) when \(y = 6\) and k is a constant. What is the value of y when \(x = 16\)?
(A)1 (B)2 (C)10 (D)14 (E)24
Output: E
If \(2^a = k\), \(k^b = m\), and \(m^c = 16\), what is the value of \(a \dot b \dot c\) ?
(A)2 (B)4 (C)5 (D)6 (E)8
Output: C
The initial number of elements in a certain set is p, where \(p > 0\). If the number of elements in the set doubles every hour, which of the following represents the total number of elements in the set after exactly 24 hours?
(A)24p (B)48p (C)\(2p^{24}\) (D)\((2p)^{24}\) (E)\((2^{24})p\)
Output: E
If \(3a + 3b + c = 26\) and \(a + b = 6\), then \(c\) =
(A)8 (B)14 (C)20 (D)32 (E)44
Output: A
If \(2\sqrt{4n^2} + 7 = 39\), what is the value of n?
(A)8 (B)4 (C)\(2\sqrt{2}\) (D)2 (E)\(\sqrt{2}\)
Output: A
If \(\frac{a^2}{2 \times 3} = 6 \times 49\), what is the value of \(a^2\)?
(A)\(6^2\) (B)\(7^2\) (C)\(13^2\) (D)\(24^2\) (E)\(42^2\)
Output: E
When positive integers \(a\) and \(b\) are divided by 7, the remainders are 4 and 2 respectively. If \(ab\) is divided by 7, what is the remainder?
(A)1 (B)2 (C)3 (D)5 (E)6
Output: A
If \(r\) is a positive even integer and \(s\) is a prime integer, then all of the following are factors of \(rs\) EXCEPT
(A)\(r\) (B)\(s\) (C)\(2s\) (D)\(2r\) (E)2
Output: D
If \(|3x - 6| = 36\), what is one possible value of \(x\) ?
(A)-30 (B)-14 (C)-10 (D)0 (E)10
Output: C
If the members of set A are all the odd single-digit positive integers and the members of set B are all the prime integers, which of the following is the set of numbers common to both set A and set B ?
(A)\(\lbrace 1, 3, 5, 7\rbrace\) (B)\(\lbrace 1, 3, 5, 7, 9\rbrace\) (C)\(\lbrace 2, 3, 5, 7\rbrace\) (D)\(\lbrace 3, 5, 7\rbrace\) (E)\(\lbrace 3, 5, 7, 9\rbrace\)
Output: D
If \(4(x + 5) = 24\), then what is the value of x?
(A)1 (B)\(\frac{19}{4}\) (C)6 (D)\(\frac{29}{4}\) (E)19
Output: A
If m and n are positive integers and \(2m + 3n = 15\), what is the sum of all possible values of m ?
(A)2 (B)3 (C)5 (D)6 (E)9
Output: E
If \(2x - 8\), \(x + 4\), and \(3x -13\) are all integers, and \(x + 4\) is the median of these integers, which of the following could be a value of x?
(A)6 (B)8 (C)11 (D)13 (E)15
Output: C
If \(y < 0\) and \((x + 2)(y - 4) = 0\), then \(x\) =
(A)-4 (B)-2 (C)0 (D)2 (E)4
Output: B
If \(f(x) = 3x - 8\) and \(g(x) = \sqrt{2x^2 + 7}\), what is the value of \(f(g(3))\)?
(A)1 (B)3 (C)5 (D)7 (E)9
Output: D
If \(5k + 8 = -7 + 2k\), what is the value of \(3k\) ?
(A)-15 (B)-5 (C)\(\frac { -15 } { 7 } \) (D)5 (E)15
Output: A
\(ts < 0 < t - s\) If the statement above is true, which of the following must also be true?
(A)\(t > s\) (B)\(s > t\) (C)\(t = s\) (D)\(t > 0\) and \(s > 0\) (E)\(t < 0\) and \(s < 0\)
Output: A
If \(m^ { 2a } x m^b = m^6\) and \(\frac { m^a } { m^b } = \frac { 1 } { m^3 } \), what is the value of \(a + b\)?
(A)2 (B)4 (C)5 (D)7 (E)9
Output: C
Consider the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3, ... . What is the sum of the first 100 terms?
(A)100 (B)180 (C)198 (D)199 (E)200
Output: D
If \(\textcircled{x}\) is defined by \(\textcircled{x} = 5x - \frac{x}{2}\), then \(\textcircled{8} = \)
(A)36 (B)38 (C)40 (D)45 (E)48
Output: A
If \(n < 3 < \frac{1}{n}\), then n could be which of the following?
(A)5 (B)1 (C)\(\frac{1}{3}\) (D)\(\frac{1}{5}\) (E)\(-\frac{1}{5}\)
Output: D
If \(\frac { 2x } { x^2 + 1 } = \frac { 2 } { x + 2 } \), what is the value of \(x\) ?
(A)\(-\frac { 1 } { 4 } \) (B)\(\frac { 1 } { 4 } \) (C)\(\frac { 1 } { 2 } \) (D)0 (E)2
Output: C
What is 50\% of 50\% of 50?
(A)0.125 (B)0.5 (C)1.25 (D)5.0 (E)12.5
Output: E
If \(2a = 3b\), \(\frac { 1 } { 3 } c = 6b^2 + 2\), and \(b > 0\), what is \(c\) in terms of \(a\) ?
(A)\(8a^2 + 6\) (B)\(8a^2 + 12\) (C)\(12a^2 + 6\) (D)\(18a^2 + 6\) (E)\(18a^2 + 12\)
Output: A
If \(4x + 2 = 26\), then \(4x + 8\) =
(A)32 (B)34 (C)36 (D)38 (E)40
Output: A
If \(p < q\), \(r < s\), and \(r < q\), which of the following must be true? I. \(p < s\) II. \(s < q\) III. \(r < p\)
(A)None (B)I only (C)III only (D)I and II (E)II and III
Output: A
If \(b > a > 0\), which of the following is less than \(\frac{a}{b}\)?
(A)2 (B)\(\frac{b}{a}\) (C)\(\frac{1}{\frac{a}{b}}\) (D)\(\frac{a}{2b}\) (E)\(\frac{2a}{b}\)
Output: D
If exactly two of the three integers a,b, and c are odd, which of the following must be odd? I. \(a + bc\) II. \(\frac { ab } { c } \) III. \(a(b + c)\)
(A)I only (B)II only (C)III only (D)I and II only (E)II and III only
Output: A
If x, y, and z are consecutive positive integers and \(x < y < z\), which of the following could be true? I. \(y = 2x\) II. \(z = 2x\) III. \(z = 2y\)
(A)I only (B)II only (C)III only (D)I and II only (E)I, II and III
Output: D
If \(|x - 5| = 2x - 1\), which of the following is the value of \(x\)?
(A)-4 (B)-2 (C)0 (D)2 (E)4
Output: D
If \(a < -1\) and \(-1 < b < 0\), what are all of the possible values of \(ab\) ?
(A)All negative numbers (B)All negative numbers between -1 and O (C)All positive numbers less than 1 (D)All positive numbers greater than 1 (E)All positive numbers
Output: E
If t is 5 more than s, and s is 3 less than r, what is t when \(r = 3\)?
(A)\(-5\) (B)\(-2\) (C)1 (D)5 (E)8
Output: D
If \(\frac { a } { b } \) is negative, which of the following must be positive? I. \((a + b)(a - b)\) II. \(1 - ab\) III. \(\frac { b - a } { a } \)
(A)I only (B)II only (C)III only (D)I and II only (E)I,II and III
Output: B
If \(k\) is an even integer, which of the following must be an odd integer?
(A)\(k + 2\) (B)\(3k\) (C)\(3k + 2\) (D)\(k(k - 1)\) (E)\((k + 1)(k - 1)\)
Output: E
If \(f(x) = 2x^2 - 4x - 16\) and \(g(x) = x^2 - 3x - 4\), which of the following is an expression for \(\frac { f(x) } { g(x) } \) ?
(A)\(\frac { 22 } { 3 } \) (B)\(x^2 - x - 12\) (C)\(x^2 + \frac { 4 } { 3 } x + 4\) (D)\(\frac { 2x + 4 } { x + 1 } \) (E)\(\frac { 2x - 4 } { x - 1 } \)
Output: D
If \(\frac { a + b } { a - b } = \frac { 3 } { 7 } \), what is the value of \(\frac { a } { b } \) ?
(A)\(-\frac { 5 } { 2 } \) (B)-1 (C)\(-\frac { 2 } { 5 } \) (D)\(\frac { 2 } { 5 } \) (E)\(\frac { 5 } { 2 } \)
Output: A
If\( x^2 = 3\), what is the value of \(\frac { 1 } { x } + 1 - \frac { 1 } { x } - 1\)?
(A)-2 (B)-1 (C)0 (D)1 (E)2
Output: B
For what value of \(x\) does \((x-1)^3 = x^3 - 1\) ?
(A)x = 1 only (B)x = -1 only (C)x = 0 or x = 1 (D)x = 1 or x = -1 (E)x = O or x = -1
Output: C
\begin { gather* } |k^2 - 2| = 2 \end { gather* }  For how many integer values of k is the equation above true?
(A)One (B)Two (C)Three (D)Four (E)More than four
Output: C
If \(x^2 + x = 20\), which of the following is a possible value of \(x^2 - x\)?
(A)\(-30\) (B)20 (C)30 (D)40 (E)450
Output: C
If \(\frac { 3 } { 4 } \) of a number is 7 more than \(\frac { 1 } { 6 } \) of the number, what is \(\frac { 5 } { 3 } \) of the number?
(A)12 (B)15 (C)18 (D)20 (E)24
Output: D
If the difference of two numbers is greater than the sum of the numbers, which of the following must be true?
(A)Neither number is negative. (B)At least one of the numbers is negative. (C)Exactly one of the numbers is negative. (D)Both numbers are negative. (E)None of these statements must be true.
Output: B
The average (arithmetic mean) of three consecutive even integers \(a\), \(b\), and \(c\) is 8. What is the median of a, b, c, and 20 ?
(A)8 (B)9 (C)10 (D)11 (E)12
Output: B
If \(\frac{p + q}{s} = 9\), \(\frac{q}{p} = 4\), and \(\sqrt{q} = 6\), what is the value of s?
(A)\(\frac{5}{6}\) (B)5 (C)9 (D)13 (E)36
Output: B
If \(a^3 - b = a^3 - 5\), then \(b =\)
(A)\(-5\) (B)\(-\sqrt[3]{5}\) (C)\(\sqrt[3]{5}\) (D)5 (E)15
Output: D
If \(\frac { 3 } { 2x - 4 } = \frac { 3 } { x } \), what is the value of \(x\)?
(A)2 (B)4 (C)6 (D)8 (E)10
Output: B
For how many integers between 10 and 100 is the tens digit equal to 5, 6, or 7 and the units digit (ones digit) equal to 2, 3, or 4 ?
(A)Three (B)Four (C)Six (D)Nine (E)Ten
Output: D
If S is the set of all numbers between \(-3.5\) and 3.5, inclusive, T is the set of all prime numbers, and U is the set of all positive integers, then the intersection of S, T, and U contains how many elements?
(A)0 (B)1 (C)2 (D)3 (E)More than 3
Output: C
How many different positive two-digit integers are there such that the tens digit is greater than 5 and the units digit is odd?
(A)10 (B)12 (C)15 (D)20 (E)25
Output: D
If \(x^ { \frac { 2 } { 5 } } = (x^2)^ { k - 1 } \) for all positive values of \(x\), what is the value of \(k\)?
(A)\(\frac { 6 } { 5 } \) (B)\(\frac { 3 } { 5 } \) (C)\(\frac { 2 } { 3 } \) (D)\(\frac { 4 } { 3 } \) (E)\(\frac { 5 } { 3 } \)
Output: A
Which of the following is equal to \(\frac { 1 } { p } \) percent of \(k\), for \(p\) and \(k > 0\)?
(A)\(\frac { k } { p } \) (B)\(100pk\) (C)\(\frac { 100p } { k } \) (D)\(\frac { k } { 100p } \) (E)\(\frac { pk } { 100 } \)
Output: D
If \(f(x) = \sqrt{x^2 - 4}\), then \(f(x)\) is undefined for which of the following values of x?
(A)\(-6\) (B)\(-4\) (C)\(-2\) (D)0 (E)2
Output: D
If A is the set of integers between -50 and 50, and a number is in set B if it is the cube of a number in set A, how many elements of set B are in set A?
(A)2 (B)6 (C)7 (D)11 (E)101
Output: C
Set C consists of all of the positive integer multiples of 5 that are less than 20. Set D consists of all of the factors of 20. Which of the following is the union of Set C and Set D ?
(A)\(\lbrace 1, 2, 4, 5, 10, 15, 20\rbrace\) (B)\(\lbrace 2, 4, 5, 10, 15\rbrace\) (C)\(\lbrace 5, 10\rbrace\) (D)\(\lbrace 5, 10, 15, 20\rbrace\) (E)\(\lbrace 2, 4, 5, 10\rbrace\)
Output: A
If -10 < x < 10, which of the following must be true? I. \(x^2 < 10\) II. \(|x| < 10\) III. \((x + 10)(x - 10) < 0\)
(A)III only (B)II and III only (C)II only (D)I and II only (E)I, II, and III
Output: B
If \(3^5 = x\), which of the following expressions is equal to \(3^{11}\)?
(A)243x (B)\(3x^2\) (C)\(9x^4\) (D)\(27x^3\) (E)\(x^6\)
Output: B
If \(f(x) = \frac{x + 4}{x}\) and \(g(x) = x^2 - 6\), what is the difference between \(f(x)\) and \(g(x)\) when \(x = 2\)?
(A)1 (B)2 (C)3 (D)5 (E)6
Output: D
\begin { gather* } y = -3x + 2 \end { gather* }  According to the equation above, if the value of x increases by 10, which of the following statements is true?
(A)The value of \(y\) will increase by 30. (B)The value of \(y\) will increase by 28. (C)The value of \(y\) will decrease by 28. (D)The value of \(y\) will decrease by 30 (E)The value of \(y\) will decrease by2.
Output: D
A ``factor-rich'' integer is defined as one for which the sum of its positive factors, not including itself, is greater than itself. Which of the following is a ``factor-rich'' integer?
(A)6 (B)8 (C)9 (D)10 (E)12
Output: E
If\( a^\frac { 1 } { 2 } = b^2\) and \(a\) and \(b\) are both greater than 1, then what is the value of \(a^2\) in terms of \(b\) ?
(A)\(\frac { b^1 } { 4 } \) (B)\(\frac { b^1 } { 2 } \) (C)\(b\) (D)\(b^4\) (E)\(b^8\)
Output: E
If \(r = \frac { 2s^3 } { t } \), what happens to the value of \(t\) when both \(r\) and \(s\) are doubled?
(A)\(t\) is halved (B)\(t\) is doubled (C)\(t\) is tripled (D)\(t\) is multiplied by 4 (E)\(t\) is multiplied by 8
Output: D
What is 60 percent of 20y?
(A)1.2y (B)10y (C)12y (D)15y (E)120 y
Output: C
All of the following are equal to \(9x^2\) EXCEPT
(A)\(x^2 + 8x^2\) (B)\(4x + 5x\) (C)\((9x)(x)\) (D)\((3x)(3x)\) (E)\((-3x)(-3x)\)
Output: B
If \((x + y)^2 - (x - y)^2 = 72\), where \(x\) and \(y\) are positive integers, which of the following could be a value of \(x + y\)?
(A)10 (B)11 (C)12 (D)13 (E)14
Output: B
If \(\sqrt[3] { x + 1 } + 3 = 5\), then \(x =\)
(A)-9 (B)3 (C)7 (D)9 (E)26
Output: C
If \(d(w) = \sqrt{w^2 + 1}\) for all real values of w, which of the following is NOT a possible value of \(d(w)\)?
(A)0 (B)1 (C)4.6 (D)7.25 (E)49
Output: A
If \(n > 0\), what is the value of \(3^{\frac{n}{2}} + 3^{\frac{n}{2}} + 3^{\frac{n}{2}}\)?
(A)\(\frac{3}{2}(3^{\frac{n}{2}})\) (B)\(3^{\frac{n}{6}}\) (C)\(3^{\frac{n}{2} + 1}\) (D)\(3^{\frac{3n}{2}}\) (E)\(9^{\frac{n}{2}}\)
Output: C
If \(0 < x + y\) and \(y < 0\), which of the following statements must be true? I. \(x < 0\) II. \(x < -y\) III. \(0 < x - y\)
(A)I only (B)II only (C)III only (D)I and II (E)II and III
Output: C
If d is a positive integer, then \((d - 1)(d + 1)\) could equal which of the following?
(A)12 (B)13 (C)14 (D)15 (E)16
Output: D
If k is a positive integer, then \(\frac { 2^ { k+5 } } { 2^4 } \)\(a + b\) is an odd integer, which of the following must be an even integer?
(A)20 (B)14 (C)12 (D)8 (E)2
Output: D
If \(a = b^ { -2 } \) and \(b = \frac { m^1 } { 2 } \), what is \(a\) in terms of \(m\) ?
(A)\(\frac { 1 } { m } \) (B)\(\sqrt { \frac { 1 } { m } } \) (C)\(m^2\) (D)\(\sqrt { m } \) (E)\(-m\)
Output: A
If \(|x - 10| = -5\), for how many values of \(x\) is the equation true?
(A)None (B)One (C)Two (D)Four (E)More than four
Output: A
For all real numbers, \(f(x) = \frac { x^2 + 1 } { 2 } \). If \(f(a) = 25\) and \(f(11) = b\), which of the following could be the value of \(b - a\) ?
(A)-14 (B)36 (C)68 (D)77 (E)86
Output: C
If \(\frac { 100 - n } { 25 } \) is a positive integer, for how many possible positive values of \(n\) is the statement true?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: C
Which of the following is equivalent to \(\frac { x^2 + 4x + 3 } { -3 -x } > 0\) ?
(A)\(x < -1\) and \(x \neq -3\) (B)\(x > -1\) (C)\(x > -1\) and \(x \neq 3\) (D)\(-3 < x < -1\) (E)\(x < -1\) or \(x > 3\)
Output: A
If \(k\) represents an odd integer, which of the following also represents an odd integer?
(A)\(k + 3\) (B)\(2(k + 3) + k\) (C)\(k^2 + k - 2\) (D)\((k + 1)(k - 1)\) (E)\((k + 1) (k + 2)\)
Output: B
If \(x + y = a\), \(y + z = b\) and \(x + z = c\), what is the average (arithmetic mean) of \(x\), \(y\), and \(z\)?
(A)\(\frac { a + b + c } { 2 } \) (B)\(\frac { a + b + c } { 3 } \) (C)\(\frac { a + b + c } { 4 } \) (D)\(\frac { a + b + c } { 6 } \) (E)\(\frac { a + b + c } { 12 } \)
Output: D
The average (arithmetic mean) of \(a\), \(2a\), \(b\), and \((b + 1)\) is equal to the median of the four numbers, where \(a < 2a < b < (b + 1)\). What is the value of \(a\)?
(A)\(\frac { b + 1 } { 2 } \) (B)\(2b - 3\) (C)\(6\) (D)\(4\) (E)\(1\)
Output: E
Which of the following is divisible by 5 and 7, but is NOT divisible by 10?
(A)28 (B)50 (C)90 (D)105 (E)135
Output: D
For which of the following function does \(f(3) =-f(-3)\)?
(A)\(f(x) = x^2\) (B)\(f(x) =4\) (C)\(f(x)=x+2\) (D)\(f (x) = x^3 +4\) (E)\(f(x) = \frac { 3 } { x } \)
Output: E
If \(\frac { 4 } { p } \) percent of a positive number is equal to \(p\) percent of the same number, what is the value of \(p\)?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: B
If \(a = \frac { 4 } { 3 } \), then \(\frac { 4 } { a } - \frac { a } { a - 1 } \) =
(A)-3 (B)-1 (C)\(\frac { 3 } { 4 } \) (D)1 (E)3
Output: B
How many integers greater than 199, when doubled, will yield a 3-digit integer?
(A)200 (B)300 (C)400 (D)1000 (E)More than 1000
Output: B
The sum of 8 positive integers is 31. If no individual integer value can appear more than twice in the list of 8 integers, what is the greatest possible value one of these integers can have?
(A)16 (B)15 (C)10 (D)9 (E)7
Output: B
Which of the following numbers disproves the statement "A number that is divisible by 4 and 8 is also divisible by 12"?
(A)24 (B)36 (C)40 (D)48 (E)60
Output: C
The equation \(|10 - \sqrt { x } | = 7\) has two solutions. What is the sum of these solutions?
(A)0 (B)9 (C)18 (D)20 (E)298
Output: E
If \(pq = l\) and \(3q = 9\), what is the value of \(p\)?
(A)2 (B)\(\frac { 1 } { 2 } \) (C)3 (D)\(\frac { 1 } { 3 } \) (E)\(\frac { 1 } { 9 } \)
Output: D
Which of the following CANNOT be expressed as the sum of three consecutive integers?
(A)18 (B)24 (C)28 (D)33 (E)36
Output: C
If \(\frac { b^x } { 2 } = 8\), then \(\frac { b^x } { 2 } \) =
(A)1 (B)2 (C)4 (D)8 (E)16
Output: C
If \(12 = r^y\), then \(12r = \)
(A)\(r^{y + 1}\) (B)\(r^{y + 2}\) (C)\(r^{y + 12}\) (D)\(r^{12y}\) (E)\(r^{2y}\)
Output: A
If \(p = 3^a\), \(q = 3^b\), and \(a - b = 3\), what is \(p\) in terms of \(q\)?
(A)\(9q\) (B)\(18q\) (C)\(\frac { 1 } { 27 } q\) (D)\(27q\) (E)\(\frac { q } { 27 } \)
Output: D
\begin { gather* } Set A = \{ x|x = prime number\} \\ Set B = \{ x|x = 2n + 1, n = positive integer\} \end { gather* }  Which of the following is an element of both set \(A\) and \(B\) shown above?
(A)2 (B)9 (C)13 (D)15 (E)33
Output: C
If \(m^{-1}n = 2\), what does n equal in terms of m?
(A)\(-2m\) (B)\(\frac{1}{m}\) (C)\(\frac{2}{m}\) (D)2m (E)\(m^2\)
Output: D
If \(x > 0 > y\), which of the following inequalities must be true?
(A)\(x^3 > y^3\) (B)\(x^2 < y^2\) (C)\(x - y < 0\) (D)\(\frac { 1 } { x^2 } < \frac { 1 } { y^2 } \) (E)\(\frac { 1 } { x } < \frac { 1 } { y } \)
Output: A
For any positive odd integer k, how many even integers are greater than zero and less than k?
(A)\(k - 1\) (B)\(\frac{k}{2}\) (C)\(\frac{2}{k}\) (D)\(\frac{k - 1}{2}\) (E)\(\frac{2}{k - 1}\)
Output: D
If \(f(x) = 4x - 8\) and \(g(x) = 3x^2 + 7\), then \(f(g(3))\) =
(A)9 (B)21 (C)30 (D)42 (E)48
Output: D
The square of \(b\) is equal to the sum of \(b\) and 12. Which of the following equations could be used to find the value of \(b\) ?
(A)\(b^2 + 12b = 0\) (B)\(b^2 - 12b = 0\) (C)\(b^2 + b + 12 = 0\) (D)\(b^2 - b + 12 = 0\) (E)\(b^2 - b - 12 = 0\)
Output: E
What is the product of 1.1 and 1.9 rounded to the nearest tenth?
(A)1.5 (B)1.7 (C)2.0 (D)2.1 (E)3.0
Output: D
If \(a + 2 > 5\) and \(a - 4 < 1\), which of the following could be a value for a?
(A)2 (B)3 (C)4 (D)5 (E)6
Output: C
\begin { gather* } \frac { 1 } { x^2 } , \frac { 1 } { x } , x, x^2, x^3 \end { gather* } In the set of numbers above, there is no mode and the median is \(x^3\). What is one possible value of \(x\) ?
(A)2 (B)1 (C)\(\frac { 1 } { 2 } \) (D)\(-\frac { 1 } { 2 } \) (E)-1
Output: D
In how many integers from 1 to 200 does the digit 9 appear at least once?
(A)38 (B)37 (C)25 (D)20 (E)18
Output: A
If \(x - y = 5 \)and \(x^2 + y^2 = 15\), what is the value of \(xy\) ?
(A)-25 (B)-5 (C)-3 (D)3 (E)25
Output: B
For which of the following functions is \(f(-2) > f(2)\)?
(A)\(f(x) = x^2\) (B)\(f(x) = x + 2\) (C)\(f(x) = x^3 + 2\) (D)\(f(x) = \frac { 2 } { x } \) (E)\(f(x) = -x^3 + 2\)
Output: E
1. Choose a number between 0 and 9.9. 2. Multiply the number from the previous step by 100. 3. Determine the smallest integer greater than or equal to the number obtained from the previous step. 4. Add 12 to the number found in the previous step. 5. Print the resulting number. If 6.127 is the number chosen in step 1, what is the number printed in step 5?
(A)12 (B)74 (C)624 (D)624.7 (E)625
Output: E
Which of the following is equivalent to \(|k| < 10\) ? I. \(k^2 < 10\) II. \(-10 < k < 10\) III. \((k + 10)(k - 10) < 0\)
(A)I only (B)II only (C)III only (D)I and II only (E)II and III only
Output: E
If \(f = 1 + 2 + 4 + 8 + 16\) and \(g = \frac { 1 } { 2 } f + \frac { 1 } { 2 } \), then \(f\) exceeds \(g\) by
(A)2 (B)4 (C)8 (D)15 (E)16
Output: D
\begin { gather* } -5p, -3p, p, 3p \end { gather* } If \(p < 0\), which of the number above has the least value?
(A)-5p (B)-3p (C)p (D)3p (E)It cannot be determined from the information given.
Output: D
For any positive integer \(n > 1\), \(n!\) represents the product of the first \(n\) positive integers. For example, \(3! = 1 \times 2 \times 3 = 6\). Which of the following is (are) equal to \(\frac { 10! } { 8! } \) ? I. \(5! - 4! - 3!\) II. \(\frac { 5! } { 4! } \) III. \(15(3!)\)
(A)I only (B)II only (C)III only (D)I and III only (E)I, II and III
Output: D
If \(5 = m^x\), then \(5m =\)
(A)\(m^{x + 1}\) (B)\(m^{x + 2}\) (C)\(m^{x + 5}\) (D)\(m^{5x}\) (E)\(m^{2x}\)
Output: A
What is the value of \(a\) if \(a\) is positive and \(a \times a \times a = a + a + a\) ?
(A)\(\frac { 1 } { 3 } \) (B)\(\sqrt { 3 } \) (C)3 (D)\(3\sqrt { 3 } \) (E)9
Output: B
Let \(f(x) = \frac { \sqrt { x - \pi } } { x - 4 } \). What is the smallest integer for which \(f(x)\) is defined?
(A)0 (B)1 (C)3 (D)4 (E)5
Output: E
If \(x > 0\) and \(y < 0\), which of the following must be positive?
(A)\(2y\) (B)\(-2x\) (C)\(-4xy\) (D)\(x + 2y\) (E)\(4(x + y)\)
Output: C
If the average (arithmetic mean) of a, b, and c is p, which of the following is the average of a, b, c, and d?
(A)\(\frac{3p + d}{4}\) (B)\(\frac{3p + d}{3}\) (C)\(\frac{p + d}{4}\) (D)\(\frac{p + d}{2}\) (E)\(\frac{3(p + d)}{4}\)
Output: A
Which of the following represents the same value as 0.00326691?
(A)\(3.26691 \times 10^{-3}\) (B)\(3.26691 \times 10^{-2}\) (C)\(3.26691 \times 10^{-1}\) (D)\(32.6691 \times 10^{-2}\) (E)\(32.6691 \times 10^{-3}\)
Output: A
Which of the following expressions represents \(b - 3a\) subtracted from twice the sum of \(2a\) and \(5b\) ?
(A)\(-9b - 7a\) (B)\(-4b - a\) (C)\(4b + a\) (D)\(7b + 3a\) (E)\(9b + 7a\)
Output: E
If \((a^ { \frac { 1 } { 2 } } )^3 (a^3)^ { \frac { 1 } { 2 } } = a^n\), then \(n\) =
(A)2 (B)4 (C)8 (D)9 (E)19.25
Output: B
83 \(\frac { - AB } { 31 } \) In the correctly solved subtraction problem above, A and B represent digits. What is the value of \(A \times B\) ?
(A)2 (B)5 (C)7 (D)10 (E)52
Output: D
If \(x > 0\), then \((4^x)(8^x) =\)
(A)\(2^{9x}\) (B)\(2^{8x}\) (C)\(2^{6x}\) (D)\(2^{5x}\) (E)\(2^{4x}\)
Output: D
If a, b, and c are distinct positive odd integers less than 6, how many different values of \(a(b^c)\) are possible?
(A)Two (B)Three (C)Four (D)Five (E)Six
Output: D
\begin { gather* } 8, 10, x, 12, 8, 10, 8, 10, 10 \end { gather* } If the set of numbers above has both a median and a single mode of 10, then each of the number below could be the value of x EXCEPT
(A)4 (B)5 (C)6 (D)7 (E)8
Output: E
If \((a + b)^2 = (a - b)^2\) , which of the following must be true?
(A)\(a = 0\) (B)\(b < 0\) (C)\(a = b\) (D)\(ab = 0\) (E)\(a + b = 0\)
Output: D
What is the sum of the prime factors of 42 ?
(A)3 (B)6 (C)12 (D)13 (E)76
Output: C
If \(a^3 = 4\), what is the value of \(a^6\) ?
(A)8 (B)10 (C)12 (D)16 (E)64
Output: D
If the sum of the consecutive integers from \(-15\) to x, inclusive, is 51, what is the value of x ?
(A)15 (B)16 (C)18 (D)53 (E)66
Output: C
For how many integers, \(x\), is the function \(f(x) = \sqrt { x^2 - 9 } \) undefined?
(A)None (B)4 (C)5 (D)7 (E)Infinitely many
Output: C
If r is a multiple of both 12 and 18, which of the following must be a factor of r ?
(A)8 (B)24 (C)27 (D)30 (E)36
Output: E
If the average (arithmetic mean) of \(p\), \(q\), and 10 is 12, and the average of \(2p\), \(q\), and 7 is 13, what is the value of p?
(A)3 (B)6 (C)9 (D)12 (E)15
Output: B
What is the hundredths digit in the number 123,456.789?
(A)1 (B)4 (C)5 (D)8 (E)9
Output: D
If y is inversely proportional to \(x\), and \(y = 25\) when \(x = 10\), what is the value of \(y\) when \(x = 20\)?
(A)50 (B)30 (C)25 (D)12.5 (E)6.25
Output: A
If a is an odd negative number and b is a positive even number, which of the following must be even and positive?
(A)\(a + b\) (B)\(-ab\) (C)ab (D)\(\frac{b}{a}\) (E)\(b - a\)
Output: B
If a, b, and c are all integers greater than 1 and \(ab = 14\) and \(bc = 21\), which of the following must be true?
(A)\(c > a > b\) (B)\(b > c > a\) (C)\(b > a > c\) (D)\(a > c > b\) (E)\(a > b > c\)
Output: B
The integer 90 can be expressed as the sum of \(z\) consecutive integers. The value of z could be any of the following EXCEPT
(A)3 (B)4 (C)5 (D)6 (E)9
Output: D
Set A consists of all integers such that each element is equal to its square. How many elements are in set A ?
(A)Zero (B)One (C)Two (D)Three (E)Four
Output: C
For the function \(f\), \(y = f(x)\) is inversely proportional to \(x\). If \(f(5) = 24\), what is the value of \(f(10)\)?
(A)6 (B)12 (C)24 (D)36 (E)48
Output: B
If \(\frac{14x}{\sqrt{2x + 2}} = 7\), what is one possible value of x?
(A)\(-7\) (B)\(-1\) (C)\(\frac{1}{2}\) (D)1 (E)7
Output: D
If \(2x + 10 = 16\), what is the value of \(2x -10\) ?
(A)-4 (B)-3 (C)3 (D)4 (E)6
Output: A
If \(\frac { a } { b } = \frac { 7 } { 4 } \), and \(\frac { b } { c } = \frac { 8 } { 6 } \), then \(\frac { a } { c } \)=
(A)\(\frac { 3 } { 7 } \) (B)\(\frac { 4 } { 7 } \) (C)\(\frac { 16 } { 21 } \) (D)\(\frac { 21 } { 16 } \) (E)\(\frac { 7 } { 3 } \)
Output: E
If \(\frac{x}{y} = \frac{5}{7}\), what is the value of \(\frac{7x}{5y}\)?
(A)\(\frac{2}{7}\) (B)\(\frac{5}{7}\) (C)1 (D)\(\frac{7}{5}\) (E)\(\frac{49}{25}\)
Output: C
What number is 18 more than 25\% of itself?
(A)32 (B)24 (C)20 (D)12 (E)8
Output: B
The smallest integer of a set of even consecutive integers is -20. If the sum of these integers is 72, how many integers are in this set?
(A)20 (B)21 (C)22 (D)23 (E)24
Output: E
If \(m = |\frac{1}{x}|\) and \(n = \frac{1}{y}\), what is the value of \(m + n\) when \(x = -2\) and \(y = -3\)?
(A)\(-\frac{5}{6}\) (B)\(-\frac{1}{6}\) (C)0 (D)\(\frac{1}{6}\) (E)\(\frac{5}{6}\)
Output: D
If \(3x = 8\), what is the value of \(3(8 - x)\) ?
(A)16 (B)20 (C)24 (D)28 (E)32
Output: A
A clever integer is defined as an integer that is greater than 20, less than 100 and such that the sum of its digits is 9. What fraction of all clever integers is divisible by 27?
(A)\(\frac { 1 } { 8 } \) (B)\(\frac { 1 } { 4 } \) (C)\(\frac { 3 } { 8 } \) (D)\(\frac { 1 } { 2 } \) (E)\(\frac { 5 } { 8 } \)
Output: C
If \(\frac{2x + 4}{x - 3} = \frac{4}{3}\), then what is the value of x?
(A)\(-12\) (B)\(-3\) (C)\(-2\) (D)3 (E)12
Output: A
1. Choose a number between 0 and 9.9. 2. Multiply the number from the previous step by 100. 3. Determine the smallest integer greater than or equal to the number obtained from the previous step. 4. Add 12 to the number found in the previous step. 5. Print the resulting number. Which of the following could be a number printed in step 5 after steps 1 through 4 are performed?
(A)\(-1\) (B)10 (C)27.3 (D)674 (E)1,050
Output: D
If \(xy^2z^3\) is negative, which of the following must be negative?
(A)\(xy\) (B)\(xz\) (C)\(yz\) (D)\(x^2z\) (E)\(xz^2\)
Output: B
If \(-3 \leq x \leq 4\), which of the following is NOT a possible value for \(f(x)\) in the function \(f(x) = x^2 - 2\) ?
(A)-2 (B)5 (C)7 (D)14 (E)16
Output: E
If the ratio of a to b is equal to the ratio of \(2a\) to \(b\), and \(b \neq 0\), which of the following must be true? I. b = 2a II. a = 0 III. b = 2b
(A)I only (B)II only (C)III only (D)I and II (E)II and III
Output: B
If \(x^4 = 10\), what is \(x^6\) ?
(A)\(10\sqrt { 10 } \) (B)100 (C)\(100\sqrt { 10 } \) (D)1000 (E)\(1000\sqrt { 10 } \)
Output: A
If \(|x| < 2\), then which of the following is equivalent to \(|x^2 - 4|\)?
(A)0 (B)\(x^2 - 4\) (C)\(x^2 + 4\) (D)\(4 - x^2\) (E)\(-4 - x^2\)
Output: D
In the sequence 1, 2, 3, -4, 1, 2, 3, -4, ... , the numbers 1, 2, 3, -4 repeat indefinitely. What is the sum of the first 150 terms?
(A)0 (B)3 (C)37 (D)77 (E)300
Output: D
If \(x\) and \(y\) are positive integers, \((3^x)^y = 81\), and \(3^x + 3^y = 84\), which of the following could be the value of \(x\) ?
(A)2 (B)3 (C)4 (D)5 (E)6
Output: C
Which of the following is a product of 3 consecutive integers?
(A)0 (B)1 (C)2 (D)7 (E)8
Output: A
If \(8 \times 27 \times 64 = r^3\), what is the value of r?
(A)6 (B)12 (C)18 (D)24 (E)32
Output: D
If p is an integer between 1,000 and 1,030, and if the sum of the digits of p is odd, then p must be odd. Which of the following is one possible value of p that proves the above statement FALSE ?
(A)1,017 (B)1,018 (C)1,019 (D)1,020 (E)1,021
Output: D
If f(-3) = 1, which of the following CANNOT be f(x)?
(A)\(f(x) = x^2 - 8\) (B)\(f(x) = x^2 + 2x - 4\) (C)\(f(x) = \frac { x^2 - 3 } { -2x } \) (D)\(f(x) = x^2 + 3x + 1\) (E)\(f(x) = |x-3| - 5\)
Output: B
If the ratio of \(a\) to \(b\) is 7:8 and the ratio of \(c\) to \(d\) is 7:5, what is the value of \(\frac { ad } { bc } \)?
(A)\(\frac { 5 } { 8 } \) (B)\(\frac { 49 } { 40 } \) (C)\(\frac { 8 } { 5 } \) (D)\(\frac { 40 } { 49 } \) (E)\(\frac { 7 } { 8 } \)
Output: A
\begin { gather* } 1, 2, 2, 2, 3, 3, 4, 5, 8, 10 \end { gather* } For the list of 10 numbers above, which of the following must be true?
(A)mode < mean < median (B)median < mode < mean (C)median < mean < mode (D)mean < median < mean (E)mode < median < mean
Output: E
What is nine times y if 4 more than three times y is equal to 11 ?
(A)\(2 \frac { 1 } { 3 } \) (B)7 (C)\(9 \frac { 2 } { 3 } \) (D)14 (E)21
Output: E
If a is a real number, and if \(a^2 - 1 = a + 1\), which of the following could be true? I. \(a = -1\) II. \(a = 1\) III. \(a = 2\)
(A)I only (B)II only (C)III only (D)I and III only (E)I, II and III
Output: D
If \(|x - 6| = x^2\), then \(x\) could equal which of the following?
(A)-3 (B)-2 (C)3 (D)4 (E)9
Output: A
If \(n = \frac { 3m + 2 } { 2 } \) and the value of m is decreased by 4, then the value of n is decreased by how much?
(A)4 (B)6 (C)8 (D)14 (E)18
Output: B
If \(\frac{2x}{3} = \frac{4}{5}\), then \(x =\)
(A)\(\frac{2}{5}\) (B)\(\frac{5}{12}\) (C)\(\frac{5}{6}\) (D)\(\frac{6}{5}\) (E)\(\frac{12}{5}\)
Output: D
If \((0.0013)x = 0.013\), then \(x =\)
(A)0.01 (B)0.1 (C)1 (D)10 (E)100
Output: D
If \((w + 12) - 12 = 12\), \(w\) =
(A)-12 (B)0 (C)12 (D)24 (E)36
Output: C
If \(wx = y\), \(xy = z\) and \(wxyz \neq 0\), which of the following is equal to \(w\) ?
(A)\(xz\) (B)\(xz^2\) (C)\(\frac { x } { z } \) (D)\(\frac { z } { x } \) (E)\(\frac { z } { x^2 } \)
Output: E
If \(p + q - 2 = 5 \), what is the value of \(p + q - 7\)?
(A)-3 (B)0 (C)3 (D)6 (E)9
Output: B
When 20 is divided by the positive integer x, the remainder is 2. For how many different values of x is this true?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: D
According to the function \(p(n) = \frac { 2 } { 3 } n + 100\), if \(n\) increases by 300, by how much does \(p\) increase?
(A)200 (B)250 (C)300 (D)350 (E)400
Output: A
If \(x^{\frac{y}{2}} = 64\), where x and y are positive integers, \(x < 64\), \(y > 3\), and \(x > y\), what is the value of \(x + y\)?
(A)4 (B)7 (C)8 (D)10 (E)12
Output: E
For how many positive numbers \(a\) is it true that \(a \times a \times a = a + a + a\) ?
(A)0 (B)1 (C)2 (D)3 (E)more than 3
Output: B
If \(x \neq 0\), 1, which of the following is equivalent to \(\frac { x^2 - 1 } { x^2 - x } \)?
(A)\(\frac { 1 } { x } \) (B)\(\frac { x + 1 } { x - 1 } \) (C)\(\frac { x + 1 } { x } \) (D)\(\frac { x - 1 } { x + 1 } \) (E)\(\frac { x - 1 } { x } \)
Output: C
If \(3x + 1 = 19\), then \(6x - 2 \)=
(A)6 (B)12 (C)34 (D)38 (E)40
Output: C
The difference between m and the square of n is equal to the square root of the sum of m and n. Which of the following expressions is the same as the statement above?
(A)\(m - \sqrt{n} = (m + 2)^2\) (B)\(m - n^2 = \sqrt{m + n}\) (C)\(m - n^2 = \sqrt{m} + \sqrt{n}\) (D)\(m - \sqrt{n} = m^2 + n ^2\) (E)\(m^2 - n^2 = \sqrt{m + n}\)
Output: B
If \(x + 2y = a\) and \(x - 2y = b\), which of the following is an expression for \(xy\)?
(A)\(ab\) (B)\(\frac { a + b } { 2 } \) (C)\(\frac { a - b } { 2 } \) (D)\(\frac { a^2 - b^2 } { 4 } \) (E)\(\frac { a^2 - b^2 } { 8 } \)
Output: E
If \(\frac { 5^a } { 5^4 } = 5^16\)
(A)4 (B)12 (C)20 (D)64 (E)68
Output: C
If \(a = \frac{12b^4}{c}\), what happens to the value of a when both b and c are tripled?
(A)a is multiplied by 27 (B)a is multiplied by 12 (C)a is multiplied by 9 (D)a is tripled (E)a is unchanged
Output: A
Let the function \(f\) be defined as \(f(x) = \sqrt { (x - 5)(x + 5) } \). Which of the following could represent the domain of the function \(f\)?
(A)\(x < 5\) (B)\(x > 5\) (C)\(-5 < x < 5\) (D)\(x \geq 5\) or \(x \leq -5\) (E)\(-5 \leq x \leq 5\)
Output: D
If \(y = 2^{\frac{3}{4}}\), which of the following expressions is equal to \(2^3\)?
(A)4 (B)4y (C)\(4y^2\) (D)\(y^4\) (E)\(y^5\)
Output: D
If \(m^a \times m^b = m^5\) and \((m^2a)^b = m^12\), which of the following could be a value of \(a\) ?
(A)3 (B)4 (C)5 (D)6 (E)8
Output: A
For a linear function \(f\), \(f(1) = 8\) and \(f(7) = -10\). If \(f(k) = 5\), what is the value of \(k\)?
(A)0 (B)2 (C)3 (D)5 (E)6
Output: B
The average (arithmetic mean) of the numbers in set A is 8. If a second set of numbers, B, is created by dividing each of the numbers in set A by two, then what is the average of the numbers in set B?
(A)\(\frac { 1 } { 2 } \) (B)4 (C)8 (D)12 (E)16
Output: B
The percent increase from 6 to 16 is equal to the percent increase from 12 to what number?
(A)16 (B)22 (C)23 (D)32 (E)36
Output: D
What is the fewest number of 3s that can be multiplied together to yield a number greater than 100?
(A)4 (B)5 (C)11 (D)33 (E)34
Output: B
If \(\frac { m + m } { m \times m } = \frac { 1 } { 5 } \), which of the following must be true? I. m = 0 II. m = 5 III. m = 10
(A)I only (B)II only (C)III only (D)I and II only (E)I and III only
Output: C
A sequence of numbers begins 1, 1, 1, 2, 2, 3 and then repeats this pattern of six numbers forever. What is the sum of the 135th, 136th, and 137th numbers in the sequence?
(A)3 (B)4 (C)5 (D)6 (E)7
Output: C
Which of the following represents the statement ``When the sum of the squares of 2a and 3b are added to the difference between 8c and 7d, the result is 3 more than e''?
(A)\((2a)^2 + (3b)^2 + (8c - 7d) = e + 3\) (B)\((2a)^2 + (3b)^2 + (8c - 7d) + 3 = e\) (C)\((2a + 3b)^2 + (8c - 7d) + 3 = e\) (D)\((2a + 3b)^2 + (8c - 7d) = e + 3\) (E)\(2a^2 + 3b^2 + (8c - 7d) + 3 = e\)
Output: A
1. Choose a number between 0.5 and 0.99. 2. Multiply the number from the previous step by 20. 3. Determine the largest integer less than or equal to the number obtained from the previous step. 4. Subtract 4 from the number obtained from the previous step. 5. Write down the resulting number. Which, if any, of the following changes could be made to the sequence of steps without changing the number written in step 5?
(A)Remove step 2. (B)Remove step 3. (C)Replace step 3 with step 4 and vice versa. (D)Replace step 2 with step 4 and vice versa. (E)None of the above
Output: C
Each of the following is equal to \(\frac { 1 } { 2 } \%\) EXCEPT
(A)\(\frac { \frac { 1 } { 2 } } { 100 } \) (B)\(\frac { 1 } { 200 } \) (C)0.005 (D)\(\frac { 5 } { 1000 } \) (E)\(\frac { 1\% } { 2\% } \)
Output: E
\(\frac { 5 } { \sqrt { x - 3 } = 3 } \) For \(x > 3\), which of the following equations is equivalent to the equation above?
(A)\(5 = 3(x-3)\) (B)\(3 = 9(x-3)\) (C)\(25 = 3(x - 9)\) (D)\(25 = 9(x-3)\) (E)\(25 = 3(x -3)\)
Output: D
If the sum of three numbers is 27, what is the average (arithmetic mean) of the three numbers?
(A)0 (B)9 (C)12 (D)16 (E)18
Output: B
\(h(x) = \frac { 3 } { 2 } x + 1\) \(f(x) = \frac { 4 } { 3 } x\) The functions h and f are defined above. What is the value of \(h(5) - f(3)\)?
(A)8.5 (B)5 (C)4.5 (D)4 (E)3
Output: C
\(a, b, c, d\), and \(e\) are all integers, where \(a < b < c < d < e\). If \(c\), \(d\), and \(e\) are each greater than the product of the next two smaller values (for example, \(d > bc\)), and \(e = 6\), which of the following could equal zero?
(A)\(a\) only (B)\(b\) only (C)\(b\) or \(c\) only (D)\(a\), \(b\), or \(c\) only (E)\(a\), \(b\), \(c\), or \(d\) only
Output: B
If you add some number j to 50, and then divide this sum by j, the result is 3. What is the value of j?
(A)5 (B)10 (C)15 (D)20 (E)25
Output: E
If every digit of a whole number is either a 1 or a 7, the number must be
(A)prime (B)odd (C)even (D)divisible by 7 (E)divisible by 11
Output: B
If \(a^2 \neq b^2\), then \(\frac { a^2 - b^2 } { b^2 - a^2 } + \frac { a - b } { b - a } \) =
(A)-2 (B)0 (C)2 (D)\(\frac { a + b } { a - b } \) (E)\(\frac { a - b } { b - a } \)
Output: A
If for \(x \geq 1\), \(\star x \star\) is defined as \(\star x \star = \sqrt{(x - 1)}\), which of the following is equal to \(\star 37 \star - \star 5 \star\)?
(A)0 (B)\(2\sqrt{2}\) (C)4 (D)\(4\sqrt{2}\) (E)\(\sqrt{34}\)
Output: C
If \(a = f - 2\) and \(b = 4f + 5 \), which of the following is an expression for \(b\) in terms of \(a\) ?
(A)\(4a - 4\) (B)\(4a\) (C)\(4a + 2\) (D)\(4a + 7\) (E)\(4a + 13\)
Output: E
If the variables \(a\), \(b\), \(c\), \(d\), and \(e\) have distinct values and are listed in order from least to greatest, which of the following could be true? I. \(b\) is arithmetic mean II. \(e\) is one of the modes III. \(d\) is the median
(A)I only (B)II only (C)I and II (D)I and III (E)I, II, and III
Output: A
Which of the following equations represents the statement ``When the square of q is subtracted from the square root of the sum of r and s, the result is equal to the square of the sum of t and r''?
(A)\((r + s)^2 - q^2 = (t + r)^2\) (B)\((r + s)^2 + q^2 = (t - r)^2\) (C)\(\sqrt{r + s} - q^2 = (t + r)^2\) (D)\(q^2 - \sqrt{r + s} = (t - r)^2\) (E)\(q^2 - \sqrt{(r + s)^2} = (t + r)^2\)
Output: C
For the function \(f\), \(f(x)\) is inversely proportional to \(x\). If \(f(k) = a\), what is the value of \(f(\frac { k } { 3 } )\) ?
(A)\(\frac { a } { 3 } \) (B)\(\frac { 2a } { 3 } \) (C)\(3a\) (D)\(6a\) (E)\(9a\)
Output: C
If \(m\) and \(n\) are integers and \((x + m)(x + n) = x^2 + 11x + c\), what is the greatest possible value of \(c\) ?
(A)22 (B)24 (C)28 (D)30 (E)33
Output: D
If \(x = \frac { 2 } { 3 } (x + y)\), which of the following is an expression for \(x\) in terms of \(y\) ?
(A)\(\frac { 2 } { 3 } y\) (B)\(y\) (C)\(\frac { 3 } { 2 } y\) (D)\(2y\) (E)\(3y\)
Output: D
If \(f(c) = \frac { 9 } { 5 } c + 32\), then what is the value of \(f(25)\)?
(A)13 (B)46 (C)77 (D)103 (E)113
Output: C
The function \(h\) is defined by \(h(x) = 3x - 2\) and the function \(g\) is defined by \(g(x) = 3h(x )- 1\). If \(g(k) = 11\), what is the value of \(k\)?
(A)2 (B)3 (C)4 (D)5 (E)6
Output: A
A certain fraction is equivalent to n\%, where n is an integer. Which of the following could be the value of that fraction?
(A)\(\frac { 1 } { 25 } \) (B)\(\frac { 1 } { 12 } \) (C)\(\frac { 1 } { 8 } \) (D)\(\frac { 1 } { 6 } \) (E)\(\frac { 1 } { 3 } \)
Output: A
If \(p = 2\), what is \(2q(7 - 6p)\) in terms of q?
(A)\(-10q\) (B)\(10q\) (C)\(14q - 12\) (D)\(28 - 6q\) (E)\(36q\)
Output: A
If \(25 - 3\sqrt { x } = 7\), what is the value of \(x\) ?
(A)-36 (B)-6 (C)6 (D)36 (E)There is no value of x that satisfies the equation
Output: D
Which of the following numbers is between 1 and 2 ?
(A)\(\frac{7}{9}\) (B)\(\frac{7}{3}\) (C)\(\frac{9}{4}\) (D)\(\frac{10}{7}\) (E)\(\frac{12}{5}\)
Output: D
Given a sequence of integers where, starting from the first term, every third term is a multiple of 3 and every second term is a multiple of 2, which of the following could be the value of the 144th term in the sequence?
(A)15 (B)44 (C)81 (D)114 (E)206
Output: D
What is the greatest value of \(x\) that is a solution of the following equation? \(|x - 5| + 10 = 15\)
(A)0 (B)5 (C)10 (D)20 (E)30
Output: C
\begin { gather* } 5, 5k, 5k^2 , 5k^3 , ...\end { gather* } The first term in the sequence above is 5, and each term after the first is \(k\) times the preceding term. If the sum of the first three terms is 15, which of the following could be the value of \(k\) ?
(A)-2 (B)-1 (C)0 (D)2 (E)3
Output: A
When a certain number is increased by 4, the result is equal to that number divided by 7. What is the number?
(A)\(-\frac { 14 } { 3 } \) (B)\(-\frac { 2 } { 3 } \) (C)\(\frac { 4 } { 7 } \) (D)\(\frac { 2 } { 3 } \) (E)\(\frac { 14 } { 3 } \)
Output: A
\begin { gather* } a^2 - b^2 = 12 \\ a + b > 4 \end { gather* } If \(a\) and \(b\) are positive integers in the inequalities above and \(a > b\), what is the value of \(a\)?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: D
If \(a < 0\) and \(b > 0\), which of the following could not be true?
(A)\(a + b = 0\) (B)\(a - b = 0\) (C)\(ab < 0\) (D)\(\frac { a } { b } < 0\) (E)\(b + a > 0\)
Output: B
If \((m - 1)(k - 1) = k - 1\), which of the following can be true? I. \(m = 1\) II. \(k = 1\) III. \(m = 2\)
(A)I only (B)II only (C)III only (D)I an II only (E)II and III only
Output: E
\((60 \times 10^4) + (60 \times 10^2)\) =
(A)\(606 \times 10^3\) (B)\(606 \times 10^2\) (C)\(120 \times 10^6\) (D)\(66 \times 10^4\) (E)\(60 \times 10^6\)
Output: A
If \((2x^2 + 5x - 3)(4x - 1) = ax^3 + bx^2 + cx + d\) for all values of x, what is the value of b?
(A)\(-22\) (B)\(-18\) (C)\(-17\) (D)18 (E)22
Output: D
If \(x^6 + 4 = x^6 + w\), then \(w =\)
(A)\(-4\) (B)\(-\sqrt[6]{4}\) (C)\(\sqrt[6]{4}\) (D)4 (E)\(4^6\)
Output: D
If set A is \{3,5,7,11,19\} and set B consists of all the even numbers between 1 and 11, how many elements are in the union of the two sets?
(A)0 (B)10 (C)11 (D)19 (E)30
Output: B