Q. If a^m * a^n = a^p, which of the following is true?
-
A.
m + n = p
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B.
m - n = p
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C.
m * n = p
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D.
m / n = p
Solution
The property of exponents states that when multiplying like bases, you add the exponents: m + n = p.
Correct Answer:
A
— m + n = p
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Q. If a^x = b^y and a = b, what can be inferred about x and y?
-
A.
x = y
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B.
x > y
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C.
x < y
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D.
x and y are unrelated
Solution
If a = b, then a^x = b^y implies x must equal y for the equality to hold.
Correct Answer:
A
— x = y
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Q. If f(x) = 3x^2 + 2x - 5, what is f(1)?
Solution
Substituting x = 1 into the function gives f(1) = 3(1)^2 + 2(1) - 5 = 3 + 2 - 5 = 0.
Correct Answer:
B
— 1
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Q. If log_2(8) = x, what is the value of x?
Solution
Since 8 is 2^3, log_2(8) equals 3.
Correct Answer:
C
— 3
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Q. If log_2(x) + log_2(4) = 5, what is the value of x? (2023)
Solution
log_2(x) + 2 = 5 implies log_2(x) = 3, thus x = 2^3 = 8.
Correct Answer:
A
— 16
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Q. If log_3(27) = x, what is the value of x?
Solution
Since 27 is 3^3, log_3(27) = 3.
Correct Answer:
C
— 3
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Q. If log_a(b) = c, which of the following is equivalent to this expression?
-
A.
a^c = b
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B.
b^c = a
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C.
c^a = b
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D.
a^b = c
Solution
The expression log_a(b) = c can be rewritten in exponential form as a^c = b.
Correct Answer:
A
— a^c = b
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Q. If log_a(b) = c, which of the following is equivalent?
-
A.
a^c = b
-
B.
b^c = a
-
C.
c^a = b
-
D.
b^a = c
Solution
The definition of logarithms states that if log_a(b) = c, then a raised to the power of c equals b, hence a^c = b.
Correct Answer:
A
— a^c = b
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Q. If log_a(b) = x, which of the following is equivalent to b?
-
A.
a^x
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B.
x^a
-
C.
log_a(x)
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D.
a * x
Solution
By the definition of logarithms, if log_a(b) = x, then b = a^x.
Correct Answer:
A
— a^x
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Q. If log_b(x) = y, which of the following statements is true?
-
A.
b^y = x
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B.
y^b = x
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C.
x^b = y
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D.
b^x = y
Solution
The definition of logarithms states that if log_b(x) = y, then b raised to the power of y equals x, hence b^y = x.
Correct Answer:
A
— b^y = x
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Q. If the 1st term of an arithmetic progression is 4 and the common difference is 3, what is the sum of the first 10 terms?
Solution
The sum of the first n terms S_n = n/2 * (2a + (n-1)d). Here, S_10 = 10/2 * (2*4 + 9*3) = 5 * (8 + 27) = 5 * 35 = 175.
Correct Answer:
B
— 80
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Q. If the 2nd term of a geometric progression is 8 and the 4th term is 32, what is the common ratio?
Solution
Let the first term be a and the common ratio be r. Then, 2nd term = ar = 8 and 4th term = ar^3 = 32. Dividing these gives r^2 = 4, so r = 2.
Correct Answer:
A
— 2
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Q. If the 2nd term of a GP is 12 and the 4th term is 48, what is the common ratio?
Solution
Let the first term be a and the common ratio be r. Then, 2nd term = ar = 12 and 4th term = ar^3 = 48. Dividing these gives r^2 = 4, so r = 2.
Correct Answer:
A
— 2
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Q. If the 2nd term of a GP is 8 and the 4th term is 32, what is the common ratio?
Solution
Let the first term be a and the common ratio be r. Then, 2nd term = ar = 8 and 4th term = ar^3 = 32. Dividing gives r^2 = 4, so r = 2.
Correct Answer:
A
— 2
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Q. If the 2nd term of an arithmetic progression is 10 and the 5th term is 16, what is the common difference?
Solution
Let the first term be a and the common difference be d. From the equations a + d = 10 and a + 4d = 16, we can solve for d to find it is 2.
Correct Answer:
A
— 2
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Q. If the 2nd term of an arithmetic progression is 10 and the 5th term is 16, what is the 3rd term?
Solution
Let the first term be a and the common difference be d. From a + d = 10 and a + 4d = 16, we can find a + 2d = 12, which is the 3rd term.
Correct Answer:
A
— 12
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Q. If the 2nd term of an arithmetic progression is 15 and the 4th term is 25, what is the common difference?
Solution
Let the first term be a and the common difference be d. From the equations a + d = 15 and a + 3d = 25, we can find d = 5.
Correct Answer:
A
— 5
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Q. If the 2nd term of an arithmetic progression is 8 and the 4th term is 14, what is the 1st term?
Solution
Let the first term be a and the common difference be d. From the equations a + d = 8 and a + 3d = 14, we can find a = 6.
Correct Answer:
A
— 6
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Q. If the 2nd term of an arithmetic progression is 8 and the 5th term is 14, what is the 3rd term?
Solution
Let the first term be a and the common difference be d. From the equations a + d = 8 and a + 4d = 14, we can find the 3rd term a + 2d = 10.
Correct Answer:
A
— 10
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Q. If the 2nd term of an arithmetic progression is 8 and the 5th term is 20, what is the first term?
Solution
Let the first term be a and the common difference be d. From the equations a + d = 8 and a + 4d = 20, we can solve for a to find it equals 4.
Correct Answer:
A
— 4
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Q. If the 3rd term of a GP is 27 and the common ratio is 3, what is the first term?
Solution
Let the first term be a. Then, the 3rd term is ar^2 = 27. Thus, a * 3^2 = 27, giving a = 3.
Correct Answer:
B
— 9
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Q. If the 3rd term of an arithmetic progression is 12 and the 7th term is 24, what is the common difference?
Solution
Let the first term be a and the common difference be d. We have a + 2d = 12 and a + 6d = 24. Subtracting these gives 4d = 12, so d = 3.
Correct Answer:
B
— 4
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Q. If the 3rd term of an arithmetic progression is 15 and the 6th term is 24, what is the common difference?
Solution
Let the first term be a and the common difference be d. From the equations a + 2d = 15 and a + 5d = 24, solving gives d = 3.
Correct Answer:
B
— 4
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Q. If the 3rd term of an arithmetic progression is 15 and the 7th term is 27, what is the common difference?
Solution
Let the first term be a and the common difference be d. From the equations a + 2d = 15 and a + 6d = 27, solving gives d = 3.
Correct Answer:
B
— 4
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Q. If the 5th term of an arithmetic progression is 15 and the 10th term is 30, what is the common difference?
Solution
Let the first term be a and the common difference be d. From the equations a + 4d = 15 and a + 9d = 30, we can find d = 3.
Correct Answer:
A
— 3
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Q. If the 5th term of an arithmetic progression is 20 and the 10th term is 35, what is the first term?
Solution
Let the first term be a and the common difference be d. From the equations a + 4d = 20 and a + 9d = 35, we can solve for a to find it is 10.
Correct Answer:
B
— 10
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Q. If the 6th term of an arithmetic progression is 30 and the 9th term is 45, what is the common difference?
Solution
Let the first term be a and the common difference be d. From the equations a + 5d = 30 and a + 8d = 45, we can find d = 5.
Correct Answer:
A
— 5
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Q. If the 7th term of an arithmetic progression is 25 and the common difference is 3, what is the 1st term?
Solution
Using the formula for the nth term, a + 6d = 25. Substituting d = 3 gives a + 18 = 25, thus a = 7.
Correct Answer:
A
— 10
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Q. If the 7th term of an arithmetic progression is 50 and the common difference is 5, what is the first term?
Solution
Using the formula for the nth term, a + 6d = 50. Substituting d = 5 gives a + 30 = 50, hence a = 20.
Correct Answer:
B
— 30
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Q. If the common ratio of a GP is negative, which of the following statements is true?
-
A.
The terms will always be positive.
-
B.
The terms will alternate in sign.
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C.
The sum of the terms will be negative.
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D.
The first term must be negative.
Solution
In a GP with a negative common ratio, the terms will alternate in sign, starting from the sign of the first term.
Correct Answer:
B
— The terms will alternate in sign.
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