niryuu/sni-each-converted huggingface dataset:
If \(m\) and \(n\) are constants, what is the value of \(n\) if the equation \((x + 9)(x + m) = x^2 + 4mx + n\) is true for all values of \(x\)?
(A)9 (B)18 (C)27 (D)36 (E)48
Output: C
If a = 5b + 3, how much greater is a when b = 9 than when b = 7?
(A)10 (B)12 (C)14 (D)16 (E)18
Output: A
If\( q = \frac { 1 } { s } \) and \(qs \neq 0\), what is the result of \(\frac { 1 + q } { 1 + s } \) ?
(A)0 (B)\(-q\) (C)1 (D)s (E)q
Output: E
When the number \(k\), where \(k > 0\), is cubed and multiplied by 4, the result is the same as when the number is divided by 9. What is the value of k?
(A)36 (B)6 (C)\(\frac { 1 } { 6 } \) (D)\(\frac { 1 } { 18 } \) (E)\(\frac { 1 } { 36 } \)
Output: C
If \( |k-5| \leq 8 \), which of the following CANNOT be the value of \(k\) ?
(A)10 (B)8 (C)5 (D)-3 (E)-4
Output: E
Set A contains the letters A, E, I, O, and U. Set B contains the letters A, B, C, D, and E. What is the difference between the number of elements in the union of the two sets and the number of elements in their intersection?
(A)4 (B)6 (C)7 (D)8 (E)10
Output: B
If \(\sqrt { x } = \frac { x } { 10 } \) and \(x > 0\), what is the value of \(x\) ?
(A)-1 (B)1 (C)10 (D)100 (E)1,000
Output: D
Each of the following is a factor of 120 EXCEPT
(A)4 (B)10 (C)18 (D)24 (E)60
Output: C
10 is what percent of A?
(A)\(10A\%\) (B)\(\frac { 1 } { 10A } \%\) (C)\(\frac { 10 } { A } \%\) (D)\(\frac { 100 } { A } \%\) (E)\(\frac { 1000 } { A } \%\)
Output: E
If \(f(x) = 3x + 8\), for what value of a is \(f(a) = a\)?
(A)-4 (B)-2 (C)-\(\frac { 3 } { 8 } \) (D)0 (E)\(\frac { 3 } { 8 } \)
Output: A
If \(a > b\) and \(k < 0\), which of the following must be true? I. \(ak > bk\) II. \(a^2 > b^2\) III. \(\frac { a + k } { k } < \frac { b + k } { k } \)
(A)I only (B)II only (C)III only (D)II and III only (E)I, II and III
Output: C
If a, b, and c are positive numbers and \(abc = b^2\), which of the following must equal b?
(A)ac (B)ab (C)bc (D)\(\frac{a}{c}\) (E)\(\frac{c}{a}\)
Output: A
if \(a^x = 4\) and \(a^c = 64\), what is the value of \(a^ { x - c } \) ?
(A)-60 (B)-16 (C)\(\frac { 1 } { 16 } \) (D)16 (E)32
Output: C
If \(a - 2b = 10\), \(b - 3c = 5\), and \(c = 1\), what is the value of a?
(A)\(-4\) (B)10 (C)18 (D)26 (E)36
Output: D
If \(x\) and \(y\) are distinct prime numbers, which of the following has exactly four distinct factors?
(A)\(x^2\) (B)\(xy\) (C)\(xy^2\) (D)\((x + y)^2\) (E)\((x - y)^2\)
Output: B
If \(10a + 8b = 4x\), then, in terms of \(x\), \(5a + 4b\) =
(A)\(8x\) (B)\(4x\) (C)\(2x\) (D)\(x\) (E)\(\frac { x } { 2 } \)
Output: C
If \(x\) is \(\frac { 2 } { 5 } \)of \(y\) and \(y\) is \(\frac { 5 } { 12 } \)of \(z\), what is the value of \(\frac { x } { z } \)?
(A)\(\frac { 1 } { 12 } \) (B)\(\frac { 1 } { 6 } \) (C)\(\frac { 1 } { 3 } \) (D)\(\frac { 1 } { 2 } \) (E)\(\frac { 2 } { 3 } \)
Output: B
Some integers in set T are negative. If the statement above is true, which of the following must also be true?
(A)If an integer is positive, it is in set T. (B)If an integer is negative, it is in set T. (C)All integers in set T are positive. (D)All integers in set T are negative. (E)Not all integers in set T are positive.
Output: E
If \(\frac{a}{b} = 4\), \(a = 8c\), and \(c = 9\), what is the value of b?
(A)2 (B)8 (C)18 (D)36 (E)72
Output: C
If the sum of all the positive even integers less than 1000 is A, what is the sum of all the positive odd integers less than 1000?
(A)\(A - 998\) (B)\(A - 499\) (C)\(A + 1\) (D)\(A + 500\) (E)\(\frac { A } { 2 } + 999\)
Output: D
\(\frac { w - x } { y + z } = \frac { 3 } { 4 } \) In the equation above, w, x, y, and z are distinct numbers. What is the value of \(\frac { 7w - 7x } { 15y + 15z } \)?
(A)\(\frac { 3 } { 4 } \) (B)\(\frac { 3 } { 5 } \) (C)\(\frac { 7 } { 15 } \) (D)\(\frac { 7 } { 20 } \) (E)\(\frac { 28 } { 45 } \)
Output: D
If \(|x - \frac { 1 } { 2 } | < \frac { 3 } { 2 } \), which of the following must be true?
(A)\(x < 0\) (B)\(x > 0\) (C)\(-3 < x < 0\) (D)\(-2 < x < 1\) (E)\(-1 < x < 2\)
Output: E
The four-digit number 7,X7X is divisible by 3 if X is replaced by which of the following digits?
(A)4 (B)5 (C)6 (D)7 (E)9
Output: B
If a ``sump'' number is defined as one in which the sum of the digits of the number is greater than the product of the digits of the same number, which of the following is a ``sump'' number?
(A)123 (B)234 (C)332 (D)411 (E)521
Output: D
If \(f(x) = |x| + 2\), which of the following is the LEAST possible value of \(f(x)\) ?
(A)-2 (B)-1 (C)0 (D)1 (E)2
Output: E
If n is an integer, which of the following must be true about \(12n + 3\) ? I. It is even II. It is odd III. It is divisible by 3
(A)I only (B)II only (C)III only (D)II and III only (E)I, II, and III
Output: D
If \(g(t) = 2t - 6\), then at what value of t does the graph of g(t) cross the x-axis?
(A)\(-6\) (B)\(-3\) (C)0 (D)2 (E)3
Output: E
Which of the following represents "the square of the sum of a and b"?
(A)\(a^2 + b^2\) (B)\((a + b)^2\) (C)\(a^2 + b\) (D)\(2a + 2b\) (E)\(a + b^2\)
Output: B
What is the absolute value of the product of all the integers from -6 to 3, inclusive?
(A)-120 (B)-15 (C)0 (D)15 (E)120
Output: C
A certain fraction is equal to \(\frac{1}{3}\). If 1 is subtracted from the numerator and the denominator is divided by 2, the resulting fraction is equivalent to \(\frac{1}{2}\). What is the denominator of the original fraction?
(A)3 (B)4 (C)6 (D)9 (E)12
Output: E
If k and s are positive integers and the ratio of 2k to 6s is the same as the ratio of \(6k + 5\) to \(18s + 10\), which of the following must be true? I. \(k = s\) II. \(k = 1.5\) III. \(k = 1.5s\)
(A)None (B)I only (C)II only (D)III only (E)I and II
Output: D
If \(a^b = 0\), \(c^b = c\), \(bc \neq 0\), and \(c \neq 1\), what must be the value of \(bc^a\) ?
(A)0 (B)1 (C)a (D)\(a^b\) (E)c
Output: B
Let the operation \(\blacktriangle\) be defined by \(a \blacktriangle b = a^b - b^a\) for all positive integers \(a\) and \(b\), where \(a \neq b\). If \(2 \blacktriangle x = x \blacktriangle 2\), where \(x\) is a positive integer, what is the value of \(x\) ?
(A)8 (B)6 (C)4 (D)2 (E)1
Output: C
If \(\frac{q}{r} = 4\), \(\frac{q}{s} = 6\), and \(s = 2\), what is the value of r?
(A)2 (B)3 (C)5 (D)6 (E)12
Output: B
If \(-1 < N < 1\) and \(N \neq 0\), which of the following statements is always true? I. \(N < 2N\) II. \(N^2 < N\) III. \(N^2 < \frac{1}{N^2}\)
(A)I only (B)II only (C)III only (D)I and III (E)II and III
Output: C
If \(p\), \(q\), and \(r\) are positive, which of the following represents \(p\) percent of \(q\) percent of \(r\)?
(A)\(0.0001pqr\) (B)\(0.001pqr\) (C)\(0.1pqr\) (D)\(100pqr\) (E)\(10000pqr\)
Output: A
If \(ab = k\), \(b = ka\),and \(k \neq 0\), which of the following could be equal to \(a\)?
(A)0 (B)\(\frac { 1 } { 2 } \) (C)1 (D)b (E)\(\frac { 1 } { b } \)
Output: C
If r and s are positive numbers satisfying the inequality \(\frac { r } { s } < \frac { r + 1 } { s + 1 } \), which of the following must be true?
(A)\(r < s\) (B)\(s < r\) (C)\(r > 1\) (D)\(s > 1\) (E)r and s can be any numbers as long as \(r \neq s\)
Output: A
If the product of 3 positive integers is even, which of the following must be true?
(A)All of the integers are odd. (B)All of the integers are even. (C)Two of the integers are even and one is odd. (D)At least one of the integers is odd. (E)At least one of the integers is even.
Output: E
Let the function \(f\) be defined by \(f(2k) = 2f(k)\) and \(f(5) = 10\). Which of the following functions could be the definition of \(f(x)\)?
(A)\(f(x) = x + 5\) (B)\(f(x) = x - 5\) (C)\(f(x) = 15 - x \) (D)\(f(x) = 2x\) (E)\(f(x) = x\)
Output: D
\begin { gather* } 10, 15, 5, 10 \end { gather* } In the sequence above, the first term is 10, the second term is 15 and each term after the second is the nonnegative difference between the two previous terms. If \(a_n\) is defined as the nth term, what is the value of \(a_124\)?
(A)0 (B)5 (C)10 (D)15 (E)30
Output: A
For any numbers, x and y, \(x \blacksquare y\) is defined as \(x \blacksquare y = x^2 - y\). If \(a \blacksquare b = (a + 2) \blacksquare (b + 2)\), which of the following must be true?
(A)\(a = 0\) (B)\(a = -\frac { 1 } { 2 } \) (C)\(a = -1\) (D)\(b = 0\) (E)\(b = -1\)
Output: B
If \(((k^2 \div 2) \div 3) \div 4 \) is an integer, which of the following could be a value of \(k\)?
(A)6 (B)8 (C)10 (D)12 (E)14
Output: D
If the sum of \(a\) and \(b\) is \(c\), what is the average (arithmetic mean) of \(a\), \(b\), and \(c\)?
(A)\(2c\) (B)\(\frac { 2c } { 3 } \) (C)\(\frac { a + b } { c } \) (D)\(\frac { 3c } { a + b } \) (E)\(c - (a + b)\)
Output: B
If \(2x + 4 = b\), then \(6x + 12 =\)
(A)\(b + 3\) (B)\(b + 12\) (C)3b (D)\(3b + 3\) (E)\(3b + 12\)
Output: C
If \(\frac { a + b } { x - y } = \frac { 1 } { 2 } \) and \(\frac { a - b } { x + y } = \frac { 2 } { 3 } \), then \(\frac { 2a^2 - 2b^2 } { 3x^2 - 3y^2 } = \)
(A)\(\frac { 1 } { 3 } \) (B)\(\frac { 1 } { 2 } \) (C)\(\frac { 2 } { 7 } \) (D)\(\frac { 2 } { 9 } \) (E)\(\frac { 2 } { 3 } \)
Output: D
If \(f(x) = 3 + \frac { 5 } { x } \) , which of the following CANNOT be a value of \(f(x)\)?
(A)\(-\frac { 5 } { 3 } \) (B)\(-\frac { 3 } { 5 } \) (C)0 (D)\(\frac { 5 } { 3 } \) (E)3
Output: E
If \(r^2s^3t^4u > 0\), then which of the following must be true?
(A)r > 0 (B)s < 0 (C)t < 0 (D)su > 0 (E)st > 0
Output: D
If \(3(a + 2b) = 7\), what is the value of \(9a + 18b\)?
(A)21 (B)28 (C)49 (D)63 (E)81
Output: A
If \(a > 0\) and \(b < 0\), which of the following must be true?
(A)\(a + b = 0\) (B)\(\frac { a } { b } < 0\) (C)\(a + b < 0\) (D)\(a + b > 0\) (E)\(ab > 0\)
Output: B
How much greater than 15\% of z is 30\% of \(z\) ?
(A)\(15z\) (B)\(2z\) (C)\(1.5z\) (D)\(z\) (E)\(0.15z\)
Output: E
If a linear function passes through the points \((1, a)\), \((2, b)\), and \((4,18)\), what is the value of \(\frac{3}{2}b - a\)?
(A)4 (B)7 (C)9 (D)18 (E)It cannot be determined from the information given.
Output: C
If \(7^ { x+3 } + 7^2 = 98\), what is \(x \)?
(A)-5 (B)-3 (C)-1 (D)1 (E)3
Output: C
If \(y < O\) and \(z \neq O\), each of the following could be true EXCEPT
(A)\(y^2z^2 < 0\) (B)\(yz^2 < 0\) (C)\(y^2 z < 0\) (D)\(yz < 0\) (E)\(y^3z < 0\)
Output: A
The number 0.1234 is between \(\frac { n } { 1000 } \) and \(\frac { n + 1 } { 1000 } \) for some positive integer \(n\). Which of the following is the value of \(n\) ?
(A)120 (B)121 (C)123 (D)124 (E)130
Output: C
For positive integers \(x\) and \(y\), \(x + y = 20\). Which of the following cannot be a possible value of \(x - y\) ?
(A)-8 (B)-2 (C)3 (D)4 (E)8
Output: C
Squaring the product of z and 6 gives the same result as squaring the difference of z and 6. Which of the following equations could be used to find all the possible values of z?
(A)\(6z^2 = (z + 6)^2\) (B)\((6z)^2 = z^2 + 6^2\) (C)\(6^2 z = z^2 - 6^2\) (D)\((6z)^2 = (z - 6)^2\) (E)\(6z^2 = z^2 + 6^2\)
Output: D
The value of \(x + 6\) is how much greater than the value of \(x - 2\)?
(A)4 (B)6 (C)8 (D)\(x + 4\) (E)\(x + 8\)
Output: C
If \(|12r - 2| = 58\) and \(|3r + 2| = 17\), then what is the value of r?
(A)\(-5\) (B)\(-\frac{19}{6}\) (C)\(-3\) (D)5 (E)15
Output: D
If \(2x - 1 = 9\), what is \(10x - 5\) ?
(A)35 (B)45 (C)55 (D)75 (E)95
Output: B
If \(k\) is a negative number, which of the following must be a positive number?
(A)\(k - 5 \) (B)\(k + 5\) (C)\(k^2 - 5 \) (D)\((k + 5)(k - 5)\) (E)\(k(k - 5)\)
Output: E
What is the value of \(b\) if \(1 - \frac { 4 } { b } = \frac { 5 } { 3 } \) ?
(A)-6 (B)-2 (C)\(\frac { 1 } { 4 } \) (D)\(\frac { 2 } { 3 } \) (E)6
Output: A
If \(q = \frac{5}{4}st\), what is the value of t when \(s = 2\) and \(q = 50\)?
(A)2.5 (B)20 (C)35 (D)40 (E)70
Output: B
Which of the following is equivalent to \(\frac { 2x^2 - 8 } { x^2 - 4x + 4 } \)?
(A)2 (B)\(\frac { 2(x + 2 } { x - 2 } \) (C)\(\frac { 2(x + 4) } { x - 4 } \) (D)\(\frac { 2x + 2 } { x - 2 } \) (E)\(\frac { 6 } { 4x - 4 } \)
Output: B
Between which two digits of the number 987,654 should a decimal point be placed so that the value of the resulting number is equal to \(9.87654 \times 10^4\)?
(A)9 and 8 (B)8 and 7 (C)7 and 6 (D)6 and 5 (E)5 and 4
Output: E
If \((0.008)x = 0.032\), then what is the value of \(0.08x\)?
(A)0.032 (B)0.32 (C)3.2 (D)32 (E)320
Output: B
Let the function \(g(x)\) be defined by \(g(x) = 2f(x) + k\) , where \(f(x)\) is a linear function. If \(g(2) = 10\) and \(g(5) = 18\), what is the slope of function \(f\)?
(A)\(\frac { 2 } { 3 } \) (B)\(\frac { 4 } { 3 } \) (C)2 (D)3 (E)\(\frac { 9 } { 2 } \)
Output: B
If x is directly proportional to \(\sqrt { y } \), then y is inversely proportional to
(A)\(x^2\) (B)\(\sqrt { x } \) (C)\(\frac { 1 } { \sqrt { x } } \) (D)\(\frac { 1 } { x } \) (E)\(\frac { 1 } { x^2 } \)
Output: E
If \(3a + 3(b + 1) = c\), what is \(b + 1\) in terms of a and c?
(A)\(\frac{c}{9a}\) (B)\(\frac{c}{3} - a\) (C)\(\frac{c}{3} + a\) (D)\(\frac{c}{3} - 3a\) (E)\(\frac{c}{3} + 3a\)
Output: B
If \(x^{\frac{1}{2}} = 4\), then what is the value of \(x^2\)?
(A)\(\frac{1}{2}\) (B)2 (C)4 (D)16 (E)256
Output: E
In the equation \(S = 4\pi r^2h\), if the value of \(S\) is doubled, and the value of \(r\) is doubled, then the value of \(h\) is multiplied by...
(A)\(\frac { 1 } { 4 } \) (B)\(\frac { 1 } { 2 } \) (C)2 (D)4 (E)8
Output: B
What is the sum of 15 consecutive integers if the middle integer is 60?
(A)45 (B)90 (C)450 (D)500 (E)900
Output: E
The various averages (arithmetic mean) of two of the three numbers a, b, and c are calculated, and are arranged as follows: \(P\) = The average of \(a\) and \(b\) \(Q\) = The average of \(b\) and \(c\) \(R\) = The average of \(c\) and \(a\) If \(P > Q > R\), then which of the following is true?
(A)\(a = b = c\) (B)\(a > b > c\) (C)\(b > c > a\) (D)\(c > b > a\) (E)\(b > a > c\)
Output: E
If \(a + b - c = d + 6\), \(c - b = 8\), and \(3a = 2 - d\), what is the value of \(a\)?
(A)2 (B)4 (C)8 (D)12 (E)16
Output: B
The number represented by \(\frac { 1 } { x } \) has a tens digit larger than its units digit. If the tens digit of \(\frac { 1 } { x } \) is odd, what is the greatest possible value of x?
(A)0.010204... (B)0.027027... (C)0.03333333... (D)0.09090909... (E)0.1
Output: E
If \(x^3 = 1\), what is the value of \(\frac { x^2 + 3 } { x } \) ?
(A)0 (B)\(\frac { 1 } { 3 } \) (C)2 (D)\(\frac { 7 } { 2 } \) (E)4
Output: E
The first term of a sequence is \(a\). Every term after the first is equal to \(d\) plus the preceding term. If the seventh term is 20 and the fourth term is 11, what is the value of \(a + d\)?
(A)5 (B)6 (C)7 (D)8 (E)9
Output: A
If \(a\) is 4 greater than \(b\) and \(a^2 + b^2 = 10\), what is the value of \(ab\) ?
(A)13 (B)6 (C)3 (D)-3 (E)-6
Output: D
What is the least positive integer a for which 25a is the cube of an integer?
(A)1 (B)3 (C)5 (D)25 (E)125
Output: C
If \(a = 3(2c^2 + 3c + 4)\) and \(b = -c + 4\), what is a in terms of b?
(A)\(6b^2 - 48b + 96\) (B)\(6b^2 - 57b + 132\) (C)\(6b^2 - 57b + 144\) (D)\(6b^2 - 9b - 132\) (E)\(6b^2 - 9b + 144\)
Output: C
If \(3^ { x^2 + y^2 } = \frac { 1 } { 9^xy } \), what is the value of \(x + y\) ?
(A)-2 (B)-1 (C)0 (D)1 (E)2
Output: C
If \(F = k \frac { v^2 } { r } \), where \(k\) is a constant, what happens to the value of F when \(v\) is tripled and \(r\) is halved?
(A)F is doubled (B)F is tripled (C)F is multiplied by 8 (D)F is multiplied by 12 (E)F is multiplied by 18
Output: E
If the average (arithmetic mean) of 6 numbers is greater than 30 and less than 60, which of the following could be the sum of the 6 numbers?
(A)100 (B)180 (C)250 (D)360 (E)420
Output: C
If \(p < q < 0\), which of the following MUST be greater than 1?
(A)\(pq\) (B)\(p + q\) (C)\(p - q\) (D)\(\frac { p } { q } \) (E)\(\frac { q } { p } \)
Output: D
If \(4^{4x + 6} = 64^{2x}\), what is the value of x?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: C
The values of \(x\) and y are such that \(f(x) = f(2y)\) for a certain function \(f\). Which of the following is equivalent to \(f(x) + f(x)\) ? I. \(2f(2y)\) II. \(f(4y)\) III. \(2 (f(y) + f(y))\)
(A)I only (B)II only (C)I and III (D)II and III (E)I, II, and III
Output: A
A ``complex prime'' is any integer greater than 1 that has exactly three positive integer factors--itself, its square root, and 1. Of the following, which is a complex prime?
(A)169 (B)144 (C)81 (D)64 (E)37
Output: A
\(a + b + c = 4\) \(d + e = 5 \) What is the value of \(ae + ce + bd + be + cd + ad\) ?
(A)6 (B)9 (C)14 (D)20 (E)34
Output: D
How many primes less than 1000 are divisible by 7?
(A)None (B)1 (C)More than 1 but less than 142 (D)142 (E)More than 142
Output: B
If a positive number a is 150 percent of \(4p\), and if \(p\) is 25 percent of \(8b\), then what is the value of \(\frac { a } { b } \)?
(A)4 (B)8 (C)12 (D)16 (E)32
Output: C
If \(2 < |x - 4| < 5\), which of the following could be the value of \(x\)?
(A)-6 (B)-7 (C)6 (D)8 (E)9
Output: D
If the positive four-digit integer JKLM contains the digits J, K, L and M, what is the decimal equivalent of \(JKLM \times 10^ { -3 } \)
(A)0.JKLM (B)J.KLM (C)JK.LM (D)JK,LM0 (E)J,KLM,000
Output: B
Let \(<a, b>\) be defined as the set of all integers between, but not including, a and b. For example, \(<-3,3> = \{-2, -1,0,1,2\}\). Which of the following does NOT have the same number of elements as the intersection of \(<-4,4>\) and \(<0,7>\)?
(A)\(<-2, 2>\) (B)\(<-1, 4>\) (C)\(<-3, 1>\) (D)\(<3, 7>\) (E)\(<1, 5>\)
Output: B
If \(\sqrt { x + a + b } = \sqrt { x - a - b } \), which of the following must be true?
(A)\(a = 0\) (B)\(b = 0\) (C)\(ab = 0\) (D)\(a + b = 0\) (E)\(a - b = 0\)
Output: D
When \(n\) is a positive integer, \(9 + 3^ { n + 2 } = m\), what is the value of \(3^n\) in terms of \(m\)?
(A)\(m + 3\) (B)\(m^2 - 3\) (C)\(\frac { m + 1 } { 9 } \) (D)\(\frac { m - 9 } { 9 } \) (E)\(\frac { 1 } { 3 } m\)
Output: D
If the average (arithmetic mean) of 7 numbers is greater than 7 and less than 12, which of the following could be the sum of the 7 numbers?
(A)84 (B)77 (C)49 (D)42 (E)35
Output: B
Which of the following groups contains three fractions that are equal?
(A)\(\frac{1}{3}, \frac{1}{6}, \frac{1}{9}\) (B)\(\frac{2}{5}, \frac{10}{25}, \frac{12}{30}\) (C)\(\frac{1}{2}, \frac{2}{4}, \frac{4}{6}\) (D)\(\frac{2}{3}, \frac{2}{6}, \frac{2}{9}\) (E)\(\frac{3}{4}, \frac{10}{12}, \frac{75}{100}\)
Output: B
If q and r are prime numbers, which of the following CANNOT be a prime number?
(A)\(q + r\) (B)\(q - r\) (C)qr (D)\(qr + 1\) (E)\(qr + 2\)
Output: C
If \(|1 - 2k| = k + 4\), which of the following could be the value of \(k\)?
(A)-2 (B)-1 (C)0 (D)2 (E)4
Output: B
The first term of sequence I is 2, and each subsequent term is 2 more than the preceding term. The first term of sequence II is 2 and each subsequent term is 2 times the preceding term. What is the ratio of the 32nd term of sequence II to the 32nd term of sequence I?
(A)1 (B)2 (C)\(2^26\) (D)\(2^27\) (E)\(2^32\)
Output: C
The absolute value of a certain integer is greater than 3 and less than 6. Which of the following could NOT be 2 less than the integer?
(A)\(-7\) (B)\(-6\) (C)2 (D)3 (E)4
Output: E
If \(2(ax + bx) = 48\), what is the value of \(x\) when \(a + b = 8\)?
(A)3 (B)6 (C)12 (D)18 (E)24
Output: A
If \(\frac{5}{d} = \frac{3}{2}\), then \(2d =\)
(A)\(1 \frac{2}{3}\) (B)\(3 \frac{1}{3}\) (C)\(6 \frac{2}{3}\) (D)\(7 \frac{1}{2}\) (E)15
Output: C
S = 1 + 2 + 3 + 4 + ... + 49 + 50 T = 51 + 52 + 53 + ... + 99 + 100 If the sum of the positive integers from 1 to 50 is \(S\), and the sum of the positive integers from 51 to 100 is \(T\), what is \(T\) in terms of \(S\)?
(A)\(2S\) (B)\(S + 50\) (C)\(S + 100\) (D)\(S + 2500\) (E)\(S + 10000\)
Output: D
For any positive integer [n] represents the sum of the integers from 1 to n. For example, [5] = 1 + 2 + 3 + 4 + 5 = 15. Which of the following is equal to [10] - [9] ?
(A)[1] (B)[2] (C)[3] (D)[4] (E)[5]
Output: D
If \(ab = 8\), \(bc = 4\) and \(ac = 2\), and \(c > O\), then what is the value of \(abc\) ?
(A)2 (B)4 (C)8 (D)32 (E)64
Output: C
If a, b, and c are consecutive even integers, such that \(0 < a < b < c\), what is \(c^2 - a^2\) in terms of \(a\) ?
(A)\(8a + 16\) (B)\(4a + 4\) (C)\(8a\) (D)\(4a + 16\) (E)\(4a\)
Output: A
If \(a(\frac { 7 } { 11 } ) = (\frac { 7 } { 11 } )b\), then \(\frac { a } { b } \) =
(A)\(\frac { 49 } { 121 } \) (B)\(\frac { 7 } { 11 } \) (C)1 (D)\(\frac { 11 } { 7 } \) (E)\(\frac { 121 } { 49 } \)
Output: C
If the average (arithmetic mean) of 8 numbers is greater than 10 and less than 12, which of the following could be the sum of the 8 numbers?
(A)70 (B)80 (C)90 (D)100 (E)110
Output: C
If \(|5x + 15| = 10\), then x could equal
(A)1 and 5 (B)-1 and 5 (C)-1 and -5 (D)-1 (E)-5
Output: C
If \(4 \leq x \leq 12\) and \(-1 \leq y \leq 1\), which of the following gives the set of all possible values of \(xy\)?
(A)\(-4 \leq xy \leq 12\) (B)\(0 \leq xy \leq 12\) (C)\(-12 \leq xy \leq -4\) (D)\(-12 \leq xy \leq 4\) (E)\(-12 \leq xy \leq 12\)
Output: E
If X is the set of positive multiples of 2 and Y is the set of positive multiples of 3, then the intersection of X and Y is
(A)the set of all positive integers (B)the set of all positive real numbers (C)the set of positive multiples of 3 (D)the set of positive multiples of 5 (E)the set of positive multiples of 6
Output: E
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. If 0.62 is the number chosen in step 1, what is the number written in step 5?
(A)6.2 (B)8 (C)9 (D)16 (E)62
Output: B
For all positive integers p and q, let \(\Phi\) be defined by \(p \Phi q = (p + Q)^2 - (p - q)^2\). If \(a\) and \(b\) are positive integers, which of the following CANNOT be the value of \(a \Phi b\)?
(A)4 (B)8 (C)12 (D)18 (E)20
Output: D
If \(f(x) = \frac{x^3 - 6}{x^2 - 2x + 6}\), then what is \(f(6)\)?
(A)0 (B)3 (C)6 (D)7 (E)35
Output: D
\(A = \lbrace 2, 3\rbrace\) \(B = \lbrace 4, 5\rbrace\) \(C = \lbrace 6, 7\rbrace\) In how many ways is it possible to pick 1 number from each set, so that the 3 numbers could be the lengths of the three sides of a triangle?
(A)0 (B)2 (C)4 (D)6 (E)8
Output: C
If \(a^ { \frac { 2 } { 5 } } = b\), what does \(a^6\) equal in terms of \(b\)?
(A)\(b^2\) (B)\(b^ { \frac { 12 } { 5 } } \) (C)\(b^6\) (D)\(b^10\) (E)\(b^15\)
Output: E
If \(x^2 - y^2 = 30\) and \(x + y > 10\), where \(x\) and \(y\) are positive numbers, then which of the following must be true?
(A)\(x = y\) (B)\(x < y\) (C)\(x - y < 30\) (D)\(x - y > 3\) (E)\(x - y < 3\)
Output: E
If \(2^ { a + b } = 8\) and \(3^ { a - b } = 81\), what is the value of \(a\)?
(A)2 (B)3.5 (C)6 (D)8.5 (E)10
Output: C
If \(\frac{y}{y - 3} = \frac{42}{39}\), then what does y equal?
(A)39 (B)41 (C)42 (D)45 (E)81
Output: C
If m is an integer, which of the following could be true? I. \(\frac { 17 } { m } \) is an even integer II. \(\frac { m } { 17 } \) is an even integer III. \(17m\) is a prime
(A)I only (B)II only (C)III only (D)I and II only (E)II and III only
Output: E
For the inequality \(c - d > c\), which of the following must be true?
(A)\(c < 0\) (B)\(c > 0\) (C)\(c < d\) (D)\(d < 0\) (E)\(d > 0\)
Output: D
If the average (arithmetic mean) of 14 consecutive integers is 20.5, what is the average of the first seven integers?
(A)10.25 (B)14.5 (C)17 (D)18.25 (E)19
Output: C
\(XY + YX = 88\) In the addition problem above, X and Y are distinct positive integers. How many different integer values of X are possible?
(A)4 (B)5 (C)6 (D)7 (E)8
Output: C
\(x + y = 10\) \(y + z = 15\) \(x + z = 17\) What is the average (arithmetic mean) of \(x\), \(y\), and \(z\)?
(A)7 (B)14 (C)15 (D)21 (E)It cannot be determined from the information given
Output: A
If \(c\) and \(d\) are integers and \(c^2 - d^2\) is even, which of the following must be true?
(A)\(cd\) is odd (B)\(cd\) is even (C)\(c + d\) is odd (D)\(c + d\) is even (E)\(c - d\) is odd
Output: D
If \(xyz \neq 0\), then \(\frac{x^2 y^3 z^6}{x^6 y^3 z^2} =\)
(A)xyz (B)\(\frac{z}{x}\) (C)\(\frac{z^2}{x^2}\) (D)\(\frac{z^3}{x^3}\) (E)\(\frac{z^4}{x^4}\)
Output: E
If \(a < b < c < d\) and the average (arithmetic mean) of \(a\), \(b\), \(c\), and \(d\) is 10, which of the following MUST be true? I. \(a + d - b + c\) II. \(a < 10 and d > 10\) III. \(b < 10 and c > 10\)
(A)I only (B)II only (C)I and II only (D)I and III only (E)I, II, and III
Output: B
The average (arithmetic mean) of a student's grades on four history tests is 86. If the student received two grades of 83 and one of 85, what grade did the student receive on fourth test?
(A)95 (B)93 (C)90 (D)89 (E)88
Output: B
If the ratio of \(a\) to \(b\) is 2 to 5, and the ratio of \(b\) to \(c\) is 1 to 4, what is the ratio of \(a\) to \(c\) ?
(A)1 : 2 (B)1 : 10 (C)5 : 8 (D)5 : 4 (E)5 : 2
Output: B
If \(2 \leq p \leq 6\) and \(1 \leq q \leq 12\), what is the smallest value of \((p + q)^2 - (p - q)^2\)?
(A)20 (B)18 (C)8 (D)6 (E)4
Output: C
If \(-3 \leq m \leq -1\) and \(0 \leq n \leq 2\), what is the greatest possible value of \(mn\)?
(A)13 (B)14 (C)15 (D)16 (E)17
Output: C
If a and b are integers and \(2a + 5b = 15\), which of the following CANNOT be a value of b?
(A)\(-1\) (B)1 (C)2 (D)3 (E)5
Output: C
If \(4(n + 6) = 44\), what is the value of n?
(A)\(\frac{25}{2}\) (B)\(\frac{19}{2}\) (C)11 (D)5 (E)50
Output: D
If \(k = 10\), what is the value of \(\frac { 2k(k + 1) - 2k(k - 1) } { 4 } \) ?
(A)10 (B)20 (C)50 (D)100 (E)200
Output: A
If \(\frac{4}{\sqrt{r + 4}} = 2\), what is the value of r?
(A)\(-4\) (B)\(-2\) (C)0 (D)2 (E)4
Output: C
For which of the following sets of numbers is the average (arithmetic mean) greater than the median?
(A)\(\{-2,-1,0,1,2\}\) (B)\(\{-2,-1, 0,1, 3\}\) (C)\(\{-2,0,0,0, 2\}\) (D)\(\{-3,-1,0,1,2\}\) (E)\(\{-3,-1, 0,1, 3\}\)
Output: B
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 change, if any, could be made in the order of the steps without changing the number printed in step 5?
(A)Only steps 2 and 3 could be switched. (B)Only steps 2 and 4 could be switched. (C)Only steps 3 and 4 could be switched. (D)Steps 2, 3, and 4 can be done in any order. (E)None of the above changes can be made.
Output: C
If \(-1 < x < 0\), which of the following is true?
(A)\(x < x^2 < x^3\) (B)\(x^2 < x < x^3\) (C)\(x < x^3 < x^2\) (D)\(x^3 < x^2 < x\) (E)\(x^3 < x < x^2\)
Output: C
If \(12x + 5 = k + (k + m)x\) for any value of \(x\), where \(k\) and \(m\) are constants, what is the value of \(m\)?
(A)5 (B)6 (C)7 (D)12 (E)It cannot be determined from the information given.
Output: C
If \(k^6 = m^ { -2 } n^4\), where \(k\), \(m\), and \(n\) are positive numbers, which of the following is the value of \(m\)?
(A)\(k^3n^2\) (B)\(\frac { k^3 } { n^2 } \) (C)\(\frac { 1 } { k^3n^2 } \) (D)\(\frac { k^6 } { n^3 } \) (E)\(\frac { n^2 } { k^3 } \)
Output: E
If a is a positive number, which of the following is equal to \(a^2 \times a^{-2}\)?
(A)0 (B)1 (C)\(a^{-4}\) (D)a (E)\(a^4\)
Output: B
Which of the following points \((x, y)\) does NOT satisfy \(|5 - x| > y - 5\)?
(A)(12, 10) (B)(11, 10) (C)(6, 5) (D)(2, 3) (E)(4, 7)
Output: E
If \(|k| < 5\), which of the following must be true? I. \(-5 < k < 5\) II. \(k^2 < 5\) III. \(k^3 < 5\)
(A)I only (B)II only (C)III only (D)I and II only (E)II and III only
Output: A
The terms in a sequence are given by \(a_n = a + (n -1)d\), where \(n\) is the number of terms, \(a\) is the first term, and each term after the first is found by adding 8 to the term immediately preceding it. If the first term is 10, which term in the sequence is equal to 210?
(A)The 15th (B)The 18th (C)The 25th (D)The 26th (E)The 27th
Output: D
If \(k\) is an integer and 4 is the remainder when \(k^2 + 3\) is divided by 7, which of the following could be the value of \(k\) ?
(A)3 (B)4 (C)5 (D)6 (E)7
Output: D
If \(rst < 0\) and \(r^3 \times s^2 \times t > 0\), which of the following must be true?
(A)\(r > 0\) (B)\(r < 0\) (C)\(t > 0\) (D)\(s < 0\) (E)\(s > 0\)
Output: D
If \(\frac{\sqrt{x} + n}{\sqrt{x} + 9} = 1\), then \(n =\)
(A)1 (B)3 (C)5 (D)8 (E)9
Output: E
If \(w + x = 5 \) and \(y + z = 6\), what is the value of \(wy + xz + wz + xy\) ?
(A)11 (B)22 (C)30 (D)41 (E)60
Output: C
If \(\sqrt { ab } = 6\), where a and b are positive integers, which of the following could NOT be a value of \(a - b\)?
(A)16 (B)9 (C)5 (D)4 (E)0
Output: D
If \(g(t) = t^2 - 4\), then the graph of \(g(t)\) crosses the x-axis when t equals
(A)\(-2\) only (B)0 only (C)2 only (D)\(-2\) and 2 only (E)\(-2\), 0, and 2
Output: D
If \(p > 0\), then \(p^3(\frac { 1 } { p } -\frac { 1 } { p } )\) =
(A)0 (B)1 (C)\(p\) (D)\(p^2\) (E)\(p^3\)
Output: A
In a certain sequence the difference between any two consecutive terms is 5. If the 20th term is 63, what is the 2nd term?
(A)-32 (B)-27 (C)-22 (D)32 (E)37
Output: B
The nth term of a sequence is defined to be \(5n + 2\). The 35th term is how much greater than the 30th term?
(A)5 (B)18 (C)25 (D)36 (E)40
Output: C
If \(|x + 3| > 5\), which of the following is a possible value of \(x\) ?
(A)-9 (B)-8 (C)-3 (D)1 (E)2
Output: A
Which of the following expressions must be negative if \(x < 0\)?
(A)\(x^4 + x^2 + 4\) (B)\(x^5 -1\) (C)\(x^6 - 1\) (D)\(x^6 + x^2 + 1\) (E)\(x^2 + 10\)
Output: B
The lowest integer of a set of consecutive integers is \(-33\). If the sum of these integers is 34, how many integers are in the set?
(A)33 (B)34 (C)66 (D)67 (E)68
Output: E
Which of the following is NOT a factor of \(9^2 - 9 \)?
(A)6 (B)8 (C)12 (D)18 (E)32
Output: E
If \(s = 3q\) and \(q = 4\), what is the value of 3s?
(A)3 (B)4 (C)12 (D)24 (E)36
Output: E
Which of the following expressions must be positive for all values of a and b?
(A)\(a + b\) (B)\(a^2 - b^2 + 10\) (C)\(a^2 + b^2 + 1\) (D)\(a^3 + b^3 + 16\) (E)\(a^4 + b^2 + a^2\)
Output: C
If \(x\) is an even number, then each of the following must be true EXCEPT
(A)\(2x + 7\) is odd (B)\(3x^2 + 5\) is odd (C)\(x^3 - x^2 + x - 1\) is odd (D)\(3x + 4\) is even (E)\((3x - 5)(5x - 3) is even\)
Output: E
Which of the following CANNOT be written as the sum of three consecutive integers?
(A)6 (B)15 (C)21 (D)39 (E)56
Output: E
If \(x^ { -1 } h = x^ { 3 } h^2\), what does \(h\) equal in terms of \(x\)?
(A)\(\frac { 1 } { x } \) (B)\(\frac { 1 } { x^2 } \) (C)\(\frac { 1 } { x^3 } \) (D)\(\frac { 1 } { x^4 } \) (E)\(x^4\)
Output: D
If \(x = -3\), then \(||x - 2| - |2 + x||\) =
(A)-3 (B)0 (C)1 (D)2 (E)4
Output: E
If \(abcd = 3\) and \(abce = 0\), which of the following must be true?
(A)\(e = 0\) (B)\(c = 0\) (C)\(a > 0\) (D)\(b < 0\) (E)\(b > 3\)
Output: A
If A is the set of odd positive integers less than 10, and B is the set of prime numbers less than 10, how many positive integers less than 10 are in neither set A nor set B?
(A)1 (B)2 (C)3 (D)4 (E)5
Output: C
Let the ``tricate'' of a number x be defined as one-third of the smallest multiple of 3 greater than x. If the tricate of z is 3, which of the following could be the value of z?
(A)2 (B)5 (C)7 (D)9 (E)11
Output: C
\(3a - b = 11\) \(4b + 2c = 27\) Based on the equation above, what is the value of \( 6a + 6b + 4c\) ?
(A)18 (B)28 (C)38 (D)66 (E)76
Output: E
40 percent of 210 is the same as \(33 \frac{1}{3}\) percent of what number?
(A)840 (B)280 (C)252 (D)175 (E)84
Output: C
If \(f(x) = x^2 + \sqrt { x } \) and \(g(x) = f(4x)\), what is the value of \(g(4)\)?
(A)18 (B)36 (C)72 (D)144 (E)260
Output: E
\begin{gather*} r = n^3 - 0.61 \\ s = n^2 - 0.61 \\ t = (n - 0.61)^2 \end{gather*} If n is a negative integer, what is the'ordering of r, s, and t from least to greatest?
(A)\(t < r < s\) (B)\(r < t < s\) (C)\(s < r < t\) (D)\(r < s < t\) (E)\(t < s < r\)
Output: D
The value of \(3x + 9\) is how much more than the value of \(3x - 2\)?
(A)7 (B)11 (C)\(3x + 7\) (D)\(3x + 11\) (E)\(6x + 7\)
Output: B
If y is inversely proportional to \(x^2\) and if \(y = 6\) and \(x = 5\), what is the value of \(y\) when \(x = 10\)?
(A)12 (B)5 (C)3 (D)2.5 (E)1.5
Output: E
In a sequence of numbers, the first term is x and the ratio of each subsequent term to the previous term is 3 to 1. Which of the following expressions represents the difference between the fourth term and the second term?
(A)\(3x\) (B)\(9x\) (C)\(12x\) (D)\(24x\) (E)\(27x\)
Output: D
If \(8 - (8 - m) = 8\), then \(m\) =
(A)-16 (B)-8 (C)0 (D)8 (E)16
Output: D
If \(f(x) = x^2 - 1\) and \(g(x) = 1 - x^2\), then for which of the following values of a does \(f(2a) - 7 = g(2a) + 7\)?
(A)0 (B)1 (C)\(\sqrt { 2 } \) (D)\(\sqrt { 7 } \) (E)7
Output: C
How many three-digit integers less than 400 are there such that the sum of the hundreds and units digits is equal to the square of the tens digit?
(A)4 (B)5 (C)6 (D)7 (E)8
Output: D
If \(f(x) = x + 5\), which of the following is a solution of \(f(3a) + 2 = f(2a) + 3\) ?
(A)1 (B)2 (C)5 (D)6 (E)There are no solutions
Output: A
If 0.05 percent of n is 5, what is 5 percent of n?
(A)900 (B)700 (C)500 (D)0.007 (E)0.005
Output: C
In the sequence 7, 14, 28, x, 112.. what is the value of x?
(A)35 (B)56 (C)63 (D)75 (E)91
Output: B
If \(\frac { 4 } { 5 } p\) is two times the value of \(\frac { q } { 10 } \), where \(p\) and \(q\) are both positive integers, then \(p\) is what percent of \(q\) ?
(A)625\% (B)400\% (C)80\% (D)25\% (E)4\%
Output: D
If \(3x + 2 = 14\), what is the value of \(5x - 6\)?
(A)\(-1\) (B)14 (C)19 (D)26 (E)32
Output: B
If \(4 < \sqrt { 6k } < 5\), then \(k\) =
(A)1 (B)3 (C)5 (D)7 (E)9
Output: B
If \(9\sqrt{x} + 7 = 16\), \(x =\)
(A)1 (B)3 (C)5 (D)8 (E)9
Output: A
\begin { gather* } 3, 9, 27, 81 ....\end { gather* } In the sequence above, the first term is 3 and each term after the first is 3 times the preceding term. Which of the following is an expression for the 200th term of the sequence?
(A)3(199) (B)3(200) (C)\(3(3^200)\) (D)\(3^200\) (E)\(200^3\)
Output: D
If \(\frac { x } { 2 } < 12\) and \(y = x - 4\), then \((x, y)\) could equal
(A)(5, 9) (B)(19, 23) (C)(20, 17) (D)(22, 18) (E)(30, 26)
Output: D
Which of the following conditions would make \(2a - 2b < 0\)?
(A)\(a = b\) (B)\(a > 0\) (C)\(b > 0\) (D)\(a < b\) (E)\(a > b\)
Output: D
If \(\frac { a - b - 5 } { 2 } = 10\), then what is the value of \(a - b\) ?
(A)20 (B)25 (C)30 (D)35 (E)40
Output: B
For any positive integer [n] represents the sum of the integers from 1 to n. For example, [5] = 1 + 2 + 3 + 4 + 5 = 15. If \([1000] = a\) and \([10] = b\), what is the value of [1010]?
(A)\(a + b\) (B)\(a + l0b\) (C)\(a + b + 1000\) (D)\(a + b + 10,000\) (E)\(a + 1000b\)
Output: D
If \(\frac{1}{(a + b)^{-\frac{1}{2}}} = (a + b)^{-\frac{1}{2}}\), which of the following must be true?
(A)\(a = 0\) (B)\(\sqrt{a + b} = -1\) (C)\(\sqrt{a + b} = 0\) (D)\(a + b = 1\) (E)\((a + b)^2 = 0\)
Output: D
Given the five pairs of numbers bellow, for which pair is the ratio of the smaller number to the larger number 2 to 3 ?
(A)3, 6 (B)4, 9 (C)5, 15 (D)6, 10 (E)8, 12
Output: E
Three times a number is the same as that number subtracted from 12. What is the number?
(A)-6 (B)-2 (C)2 (D)3 (E)4
Output: D
The average of \(p\) and \(q\) is 5, and the average of \(p^2\) and \(q^2\) is 35. What is the value of \(pq\)?
(A)10 (B)15 (C)30 (D)45 (E)70
Output: B
For all positive numbers \(a\) and \(b\), let \(a \square b = \sqrt { ab } \). If \(n > 1\), what does \(n \square \frac { 1 } { n } \) equal?
(A)\(\sqrt { n } \) (B)\(\sqrt { n^2 } \) (C)\(\frac { 1 } { \sqrt { n } } \) (D)\(\frac { 1 } { \sqrt { n^2 } } \) (E)1
Output: E
If \(\frac { a } { b } = \frac { 1 } { 3 } \) and \(\frac { b } { c } = \frac { 3 } { 2 } \), what is the value of \(\frac { a } { c } \)?
(A)2 (B)1 (C)\(\frac { 1 } { 2 } \) (D)\(\frac { 2 } { 9 } \) (E)\(\frac { 1 } { 3 } \)
Output: C
If \(x = \frac { w^2 } { y } \) and \(y = \frac { 1 } { w^2 } \), what is the value of \(\frac { x } { y } \) in terms of \(w\) ?
(A)\(w^-4\) (B)\(w^-2\) (C)\(w^2\) (D)\(w^4\) (E)\(w^6\)
Output: E
Let \(\_g\_\) be defined as \(\_g\_ = 2g^2 - 2\) for all values of g. If \(\_x\_ = x^2\), what is the value of x?
(A)\(-2\) (B)1 (C)\(\sqrt{2}\) (D)2 (E)\(2\sqrt{2}\)
Output: C
For any positive odd integer n, how many positive even integers are less than n?
(A)\(\frac{n - 1}{2}\) (B)\(\frac{n}{2}\) (C)\(\frac{n + 1}{2}\) (D)\(n - 1\) (E)\(2n - 1\)
Output: A
When 28 is divided by 8, the remainder is the same as that obtained when 53 is divided by which of the following numbers?
(A)3 (B)4 (C)5 (D)6 (E)7
Output: E
If \(x\), \(y\), and \(z\) are positive and \(x^4 y^3 z^3 > x^5 y^3 z^2\). Which of the following must be true ? I. \(x > y\) II. \(y > z\) III. \(z > x\)
(A)I only (B)II only (C)III only (D)I and III only (E)II and III only
Output: C
If \(5x + 3 = 21\), then \(5x + 10 =\)
(A)28 (B)30 (C)32 (D)34 (E)36
Output: A
What is the value of \(m\) if \(\frac { 4 } { 3m - 2 } = \frac { 4 } { m + 6 } \) ?
(A)-4 (B)-2 (C)0 (D)2 (E)4
Output: E
If \(6a = 48\) and \(ab = 1\), what is the value of b?
(A)\(\frac{1}{48}\) (B)\(\frac{1}{8}\) (C)\(\frac{1}{6}\) (D)6 (E)8
Output: B
If 70 percent of x is 2,100, then what is 40 percent of x?
(A)3,000 (B)2,020 (C)1,800 (D)1,470 (E)1,200
Output: E
\begin{gather*} A + A = CB \\ C + B = 5 \end{gather*} Each correctly worked addition problem above shows the sum of two one-digit numbers. If the digits A, B, and C are nonzero, then \(A =\)
(A)2 (B)4 (C)5 (D)7 (E)9
Output: D
The function f is defined by \(f(x) = a (x - b)2 + k\) for all real \(x\), where \(a\), \(b\), and \(k\) are constants. If \(a\) and \(k\) are negative, which of the following CANNOT be true?
(A)\(f(5) = -3\) (B)\(f(0) = -5\) (C)\(f(2) = 4\) (D)\(f(1) = k\) (E)\(f(-2) = b\)
Output: C
Subtract 4 from y. Divide this sum by 2. Add 4 to this quotient. Which of the following is the result obtained by performing the operations described above?
(A)\(\frac{y - 2}{4}\) (B)\(\frac{y + 2}{4}\) (C)\(\frac{y - 4}{2}\) (D)\(\frac{y + 4}{2}\) (E)\(\frac{4y + 4}{2}\)
Output: D
If \(3\sqrt { k^2 } + 8 = 20\), which of the following could be the value of \(k\) ?
(A)-4 (B)-1 (C)6 (D)8 (E)36
Output: A
If \(w = 3^3\), which of the following expressions is equal to \(3^8\)?
(A)5w (B)\(w^5\) (C)8w (D)\(8w^2\) (E)\(9w^2\)
Output: E
\(r = 4c - d\) \(t = (c +1)r\) If r and t are defined by the equations above, what is the value of \(t\) when \(c = 3\) and \(d = 2\) ?
(A)10 (B)12 (C)30 (D)40 (E)80
Output: D
If \(a\) and \(b\) are positive integers, and \(3^a \times 3^b = 27\), what is the value of \(3^a + 3^b\)?
(A)9 (B)12 (C)18 (D)27 (E)36
Output: B
If \(a = -2\), what is the value of \(a^4 - a^3 + a^2 - a\)?
(A)-30 (B)-10 (C)0 (D)10 (E)30
Output: E
If \(x = 1\) and \(y = -1\), then \(x^2 + 2xy + y^2 =\)
(A)\(-1\) (B)0 (C)1 (D)2 (E)4
Output: B
If \(x \wedge y\) is defined by the expression \((x - y)^x + (x + y)^y\), what is the value of \(4 \wedge 2\)?
(A)52 (B)44 (C)28 (D)20 (E)16
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 of the following could be a number written in step 5 after steps 1 through 4 have been performed?
(A)0 (B)5 (C)7.4 (D)14 (E)17
Output: D
If \(100 - 2xy - y^2 = x^2\), which of the following must be true?
(A)\(x - y = 10\) (B)\(x^2 + y^2 = 100\) (C)\((x + y)(x - y) = 100\) (D)\((x + y)^2 = 100\) (E)\((x - y)^2 = 100\)
Output: D
If \(f(x) = x^{-3} + x^3\), at which of the following values of x is \(f(x)\) undefined?
(A)\(-3\) (B)0 (C)1 (D)2 (E)3
Output: B
If \(p\) is a multiple of 3 and \(q\) is a multiple of 5, which of the following must be true? I. p+q is even. II. pq is odd. III. 5p + 3q is a multiple of 15.
(A)I only (B)II only (C)III only (D)I and II only (E)II and III only
Output: C
If \(abc = 1\), which of the following could be the number of integers among \(a\), \(b\) and \(c\) ? I. 1 II. 2 III. 3
(A)None (B)I only (C)I and II only (D)I and III only (E)I, II and III
Output: E
\begin{gather*} A = \{\frac{3}{5}, 2, \frac{7}{2}, 4, \frac{15}{3}, 8\} \\ B = \{\frac{3}{5}, \frac{5}{3}, 4, 8\} \end{gather*} If n is a member of both set A and set B above, which of the following must be true? I. n is an integer II. 5n is an integer III. n = 5
(A)II only (B)III only (C)I and II only (D)I and III only (E)I, II, and III
Output: A
If \(X = \frac { ab^2 } { c } \), what is the result on X of doubling a, tripling b, and quadrupling c?
(A)X is multiplied by 1.5. (B)X is multiplied by 3. (C)X is multiplied 4.5. (D)X is multiplied by 6. (E)X is multiplied by 9.
Output: C
If \(a = 9c^2 + 4\) and \(b = 7 + 3c\), what is a in terms of b?
(A)\(2b^2 - 7b + 49\) (B)\(b^2 - 14b + 49\) (C)\(b^2 + 14b + 49\) (D)\(b^2 - 14b + 53\) (E)\(b^2 + 14b + 53\)
Output: D
If the product of two positive integers is greater than 20, and the sum of the two integers is 10, which of the following cannot be one of the integers?
(A)3 (B)5 (C)6 (D)7 (E)8
Output: E
Let the \(n\)th term of a sequence \(a_n\) be defined by \(a_n = 4n + 2\). What is the value of \((a_{50} - a_{40})\)?
(A)40 (B)36 (C)32 (D)28 (E)24
Output: A
The expression \(\frac { 12a^2b { \frac { 1 } { 2 } } c^6 } { 4a^ { -2 } b^2c^2 } \) is equivalent to which of the following?
(A)\(\frac { 3a^4c^4 } { b } \) (B)\(\frac { 3c^3 } { ab } \) (C)\(\frac { 3bc^4 } { a^4 } \) (D)\(\frac { a^4c^3 } { 3b } \) (E)\(3c^4\)
Output: A
If a number is chosen at random from the set \(\lbrace -10, -8, -7, -2, 0, 3, 5, 7, 8, 10 \rbrace\), what is the probability that it is a member of the solution set of \(|x - 2| > 6\) ?
(A)\(\frac { 2 } { 5 } \) (B)\(\frac { 3 } { 5 } \) (C)\(\frac { 1 } { 2 } \) (D)\(\frac { 2 } { 3 } \) (E)\(\frac { 3 } { 4 } \)
Output: A
For which of the following lists of numbers is the mode equal to the average (arithmetic mean) ?
(A)10, 10, 10, 50, 70 (B)10, 20, 25, 30, 40 (C)15, 35, 35, 35, 55 (D)10, 20, 25, 35, 45 (E)5, 15, 25, 25, 50
Output: C
If the sum of two negative integers is -15 and the product of two integers is less than 40, then which of the following could be one of the two integers?
(A)-8 (B)-7 (C)-5 (D)-4 (E)-3
Output: E
When 2x is subtracted from 48 and the difference is divided by \(x + 3\), the result is 4. What is the value of x?
(A)2 (B)5 (C)6 (D)8 (E)12
Output: C
For all positive values of \(a\) and \(b\), let \(a \odot b\) be defined by \(a \odot b = \frac { ab } { a + b } \)? If \((10 \odot 10)^2 - (8 \odot 8)^2 = k\), where \(k\) is a constant, what is the value of \(k\)?
(A)1 (B)2 (C)4 (D)6 (E)9
Output: E
If \(x \neq -1\), 1, or 3, which of the following is equivalent to \(\frac { x^2 - x } { 2x - 6 } \times \frac { x^2 - 2x - 3 } { x^2 - 1 } \) ?
(A)\(\frac { x } { 2 } \) (B)\(\frac { x + 3 } { 2(x - 3) } \) (C)\(\frac { x - 1 } { 2x - 6 } \) (D)\(\frac { x^2 - 3x - 3 } { 2x - 7 } \) (E)\(\frac { x^2 - x - 3 } { -7 } \)
Output: A
Which of the following is an expression for "the product of 5 and the average (arithmetic mean) of x and y"?
(A)\(\frac { 5x + y } { 2 } \) (B)\(\frac { 5x + 5y } { 2 } \) (C)\(\frac { 5 + x + y } { 3 } \) (D)\(5 + \frac { x + y } { 2 } \) (E)\(\frac { 5 + 5x + 5y } { 3 } \)
Output: B
Which of the following numbers has the same digit in the hundreds and hundredths places?
(A)2200.0022 (B)2224.2442 (C)2242.4242 (D)2246.2462 (E)2246.6422
Output: C
\begin { gather* } -4,-3,-2,-1,0,1,2,3 \end { gather* } If a number is randomly selected from the list above, what is the probability that it will be greater than -2 ?
(A)\(\frac { 1 } { 4 } \) (B)\(\frac { 3 } { 8 } \) (C)\(\frac { 1 } { 2 } \) (D)\(\frac { 5 } { 8 } \) (E)\(\frac { 3 } { 4 } \)
Output: D
If \(x < y < z\) and the average (arithmetic mean) of \(x\), \(y\), and \(z\) is \(y\), which of the following must be true?
(A)\(z - y = 2\) (B)\(x + y + z = 3y\) (C)\(x + y < z\) (D)\(x + y = z\) (E)\(2z = 3y\)
Output: B
If \(ab = k\), \(b = ka\),and \(k\neq 0\), which of the following could be equal to \(a\)?
(A)0 (B)\(\frac { 1 } { 2 } \) (C)1 (D)b (E)\(\frac { 1 } { b } \)
Output: C
If x and y are integers such that \(x^2 = 81\) and \(2y^3 = 54\), which of the following could be true? I. \(x = 9\) II. \(y = -3\) III. \(x + y = -6\)
(A)I only (B)II only (C)I and III only (D)II and III only (E)I, II, and III
Output: C
If \(-1 < y < 0\), which of the following is the greatest?
(A)\(y^2\) (B)\(1 - y\) (C)\(1 + y\) (D)\(2y\) (E)\(\frac{1}{y + 2}\)
Output: B
If \(3d - 2q = 17\) and \(2d + 2q = -32\), what is the value of \(10d\)?
(A)3 (B)-3 (C)-13 (D)-15 (E)-30
Output: E
For all u and v, let \(\triangledown\) be defined by \(u \triangledown v = u - v + 2\). What is the value of \((2 \triangledown 3) \triangledown 1\)?
(A)0 (B)1 (C)2 (D)3 (E)4
Output: C
The members of set R are the integer solutions of |y| < 10, and the members of set S are the even integer solutions of |x| >5. Which of the following includes all the members common to both set R and set S ?
(A)\(\lbrace -8, -6, 6, 8\rbrace\) (B)\(\lbrace -9, -8, -7, -6, -5, 5, 6, 7, 8, 9\rbrace\) (C)\(\lbrace -9, -8, -7, -6, 6, 7, 8, 9\rbrace\) (D)\(\lbrace 6, 8\rbrace\) (E)\(\lbrace 6, 7, 8, 9\rbrace\)
Output: A
All numbers that are divisible by both 4 and 6 are also divisible by 8. Which of the following numbers can be used to show that the above statement is FALSE?
(A)12 (B)24 (C)32 (D)40 (E)48
Output: A
If \(k\) is an integer and \(\frac { k + 5 } { 30 } < \frac { 7 } { 20 } < \frac { k } { 10 } \), then \(k\) =
(A)5 (B)6 (C)7 (D)8 (E)9
Output: A
If \(f(x) = x^ { \frac { 1 } { 2 } } + x^ { \frac { 1 } { 4 } } \), what is the value of \(f(16)\) ?
(A)6 (B)8 (C)12 (D)32 (E)64
Output: A
If \(4^x = (\frac { 1 } { 2 } )^ { y - 3 } \), what is the value of \(y\) in terms of \(x\)?
(A)\(2x\) (B)\(2x - 3\) (C)\(2x + 3\) (D)\(3 - 2x\) (E)\(3 + 2x\)
Output: D
The first term of a sequence is 1 and every term after the first one is 1 more than the square of the preceding term. What is the fifth term?
(A)25 (B)26 (C)256 (D)676 (E)677
Output: E
If \(w\) is a positive odd integer, which of the following gives a possible value of the product of one more than \(w\) and 2 less than \(w\) ?
(A)0 (B)10 (C)18 (D)28 (E)54
Output: C
If \(x^2 + y^2 = 153\) and \(xy = 36\), what is the value of \((x - y)^2\)?
(A)81 (B)117 (C)153 (D)189 (E)225
Output: A
If, for all integers \(x\) and \(y\), \(x \ddagger y\) is defined by \(x \ddagger y = x + y \times x\), what is the value of \(3 \ddagger 4\) ?
(A)10 (B)12 (C)15 (D)21 (E)36
Output: C
If \(\frac{x + y}{x} = 0\), what is the value of x?
(A)\(-y\) (B)0 (C)y (D)\(y^2\) (E)\(\frac{1}{4}\)
Output: A
If \(m^2 + 7 = 29\), then \(m^2 - 7 =\)
(A)15 (B)22 (C)71 (D)78 (E)484
Output: A
\begin{gather*} 4, -4, -1, ... \end{gather*} The first term in the sequence above is 4. Each even-numbered term is found by multiplying the previous term by \(-1\); each odd-numbered term is found by adding 3 to the previous term. For example, the second term is \(4 \times - 1 = -4\), and the third term is \(-4 + 3 = -1\). What is the 37th term of the sequence?
(A)\(-4\) (B)\(-1\) (C)0 (D)1 (E)4
Output: E
If \(xy > 0\), \(\frac{1}{x} + \frac{1}{y} = 5\), and \(\frac{1}{xy} = 6\), then \(\frac{x + y}{5} =\)
(A)\(\frac{1}{25}\) (B)\(\frac{1}{6}\) (C)\(\frac{1}{5}\) (D)5 (E)6
Output: B
If \(x > 0\) and 20 percent of \(x\) is 30 percent of\( y\) , which of the following represents 80 percent of \(x\) ?
(A)\(0.5y\) (B)\(0.8y\) (C)\(0.9y\) (D)\(y\) (E)\(1.2y\)
Output: E
\begin{gather*} 3x - 4y = 10 \\ 6x + wy = 16 \end{gather*} For which of the following values of w will the system of equations above have no solutions?
(A)\(-8\) (B)\(-4\) (C)0 (D)4 (E)8
Output: A
\((-3a^2b^5)^3\) =
(A)\(-3a^5b^8\) (B)\(-3a^6b^15\) (C)\(-27a^5b^8\) (D)\(-27a^6b^15\) (E)\(27a^6b^15\)
Output: D
If \(n = 4p^2q\), where \(p\) and \(q\) are distinct prime numbers, and not equal to 2, how many factors does the number \(n\) have?
(A)20 (B)18 (C)16 (D)14 (E)12
Output: B
What is the value of x if \(2x + 3y = 11\) and \(2x - y = 7\)?
(A)4 (B)6 (C)9 (D)11 (E)32
Output: A
Let \(\otimes\) be defined by \(a \otimes b = a + 2b + 4ab\) for all numbers \(a\) and \(b\). For what value of \(x\) could \(x \otimes y = x \) be true?
(A)-1 (B)\(-\frac { 1 } { 2 } \) (C)0 (D)\(\frac { 1 } { 2 } \) (E)1
Output: B
If \(3x = 10 - 2x\), then \((5x)(5x)\) =
(A)2 (B)4 (C)20 (D)25 (E)100
Output: E
\begin { gather* } -7, -5, -3, -1, 0, 1, 3, 5, 7 \end { gather* } How many distinct products can be obtained by multiplying pairs of numbers from the list above?
(A)9 (B)17 (C)19 (D)21 (E)31
Output: B
If \(k(a^2 - b^2) = 0\), which of the following could be true? I. \(k = 0\) II. \(a = 5\) and \(b = 5\) III. \(a = -b\)
(A)I only (B)II only (C)III only (D)I and III only (E)I, II, and III
Output: E
If \(2x + 4x + 6x = -12\) ,then \(x\) =
(A)-1 (B)\(-\frac { 1 } { 2 } \) (C)0 (D)\(\frac { 1 } { 2 } \) (E)1
Output: A
If \(|t| = 2\), what is the value of \((t + 3)(t - 3)\)?
(A)-5 (B)0 (C)5 (D)7 (E)9
Output: A