ACT Math Practice Test: Logarithms, Complex Numbers and Matrices
- Logarithms, Complex Numbers and Matrices
- 10 questions
- 10 minutes
These 10 questions cover the less common ACT Math topics that still appear on most tests. Remember that logb a = c means bc = a, and that i² = −1.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
What is the value of log2 32?
Correct answer: B
Since 25 = 32, log2 32 = 5. The choice 16 is 32 ÷ 2, which treats the logarithm like division.
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Question 2
If log3 x = 4, what is the value of x?
Correct answer: D
The equation means 34 = x, so x = 81. The choice 64 is 43, with the base and exponent reversed.
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Question 3
What is the value of log2 12 − log2 3?
Correct answer: A
The difference of logs with the same base is the log of the quotient: log2 (12 ÷ 3) = log2 4 = 2. The choice log2 9 subtracts the numbers inside the logs.
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Question 4
Which of the following is equal to (3 + 2i)(1 − i)? (Note: i = −1)
Correct answer: D
Expanding gives 3 − 3i + 2i − 2i² = 3 − i + 2 = 5 − i. The choice 1 − i treats i² as +1.
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Question 5
What is the value of i23? (Note: i = −1)
Correct answer: A
Powers of i repeat every 4, and 23 = 4 × 5 + 3, so i23 = i3 = −i. The choice i uses a remainder of 1 instead of 3.
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Question 6
Which of the following is equal to (4 + 3i) + (2 − 5i)?
Correct answer: B
Add real parts and imaginary parts separately: 4 + 2 = 6 and 3i − 5i = −2i. The choice 6 + 8i adds 3 and 5 instead of 3 and −5.
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Question 7
Matrix A and matrix B are shown below.
A =
2 −1 0 3 B =
1 4 −2 5 What is the entry in row 2, column 2 of the matrix A + B?
Correct answer: C
Matrices are added entry by entry, so the row 2, column 2 entry is 3 + 5 = 8. The choice 15 multiplies the entries instead of adding them.
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Question 8
The determinant of a 2 × 2 matrix with first row a, b and second row c, d is ad − bc.
3 2 1 4 What is the determinant of the matrix shown?
Correct answer: B
Using ad − bc: 3 × 4 − 2 × 1 = 12 − 2 = 10. The choice 14 adds the two products instead of subtracting them.
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Question 9
Matrix P is a 2 × 2 matrix and matrix Q is a 2 × 1 matrix.
P =
1 2 3 0 Q =
2 5 The product PQ is a 2 × 1 matrix. What is its bottom entry?
Correct answer: A
The bottom entry is row 2 of P times the column Q: 3 × 2 + 0 × 5 = 6. The choice 12 is the top entry, 1 × 2 + 2 × 5.
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Question 10
If log x = 2 and log y = 3, where both logarithms have base 10, what is the value of log (x²y)?
Correct answer: C
By the log rules, log (x²y) = 2 log x + log y = 2(2) + 3 = 7. The choice 5 ignores the exponent on x.
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