Q. If z = re^(iθ), what is the value of r if z = 4 + 3i?
Solution
r = |z| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.
Correct Answer: A — 5
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Q. If z = re^(iθ), what is the value of r if z = 4 + 4i?
Solution
r = |z| = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2.
Correct Answer: A — 4√2
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Q. If z = re^(iθ), what is the value of z^2?
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A.
r^2e^(i2θ)
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B.
re^(iθ)
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C.
2re^(iθ)
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D.
r^2e^(iθ)
Solution
Using the property of exponents, z^2 = (re^(iθ))^2 = r^2e^(i2θ).
Correct Answer: A — r^2e^(i2θ)
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Q. If z = re^(iθ), what is the value of z^3?
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A.
r^3 e^(i3θ)
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B.
r^3 e^(iθ)
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C.
3re^(iθ)
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D.
r^3 e^(iθ^3)
Solution
Using the properties of exponents, z^3 = (re^(iθ))^3 = r^3 e^(i3θ).
Correct Answer: A — r^3 e^(i3θ)
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Q. If z = re^(iθ), what is the value of |z|?
Solution
The modulus |z| = r, as |re^(iθ)| = r.
Correct Answer: A — r
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Q. If z = x + yi is a complex number such that |z| = 10, what is the equation of the circle in the complex plane?
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A.
x^2 + y^2 = 100
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B.
x^2 + y^2 = 10
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C.
x^2 + y^2 = 50
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D.
x^2 + y^2 = 25
Solution
The equation of the circle with radius 10 is x^2 + y^2 = 10^2 = 100.
Correct Answer: A — x^2 + y^2 = 100
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Q. If z = x + yi, find the real part of z^3.
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A.
x^3 - 3xy^2
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B.
3x^2y - y^3
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C.
x^3 + 3xy^2
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D.
3x^2 - y^3
Solution
Using the binomial expansion, z^3 = (x + yi)^3 = x^3 - 3xy^2 + (3x^2y - y^3)i.
Correct Answer: A — x^3 - 3xy^2
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Q. If z1 = 1 + i and z2 = 1 - i, find z1 * z2.
Solution
z1 * z2 = (1 + i)(1 - i) = 1 - i^2 = 1 - (-1) = 2.
Correct Answer: A — 2
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Q. If z1 = 1 + i and z2 = 2 - 3i, find z1 * z2.
-
A.
7 - i
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B.
7 + i
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C.
5 - i
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D.
5 + i
Solution
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i + 3 = 5 - i.
Correct Answer: A — 7 - i
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Q. If z1 = 1 + i and z2 = 2 - 3i, what is z1 * z2?
-
A.
5 - i
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B.
8 - i
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C.
7 + i
-
D.
1 + 5i
Solution
z1 * z2 = (1 + i)(2 - 3i) = 2 - 3i + 2i - 3i^2 = 2 - i + 3 = 5 - i.
Correct Answer: A — 5 - i
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Q. If z1 = 1 + i and z2 = 2 - i, find z1 * z2.
-
A.
3 + i
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B.
3 - i
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C.
2 + 3i
-
D.
2 - 3i
Solution
z1 * z2 = (1 + i)(2 - i) = 2 - i + 2i - 1 = 3 + i.
Correct Answer: A — 3 + i
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Q. If z1 = 2 + 3i and z2 = 4 - 5i, then z1 + z2 is?
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A.
6 - 2i
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B.
6 + 2i
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C.
2 - 2i
-
D.
2 + 2i
Solution
z1 + z2 = (2 + 3i) + (4 - 5i) = 6 - 2i.
Correct Answer: A — 6 - 2i
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Q. If z1 = 2 + 3i and z2 = 4 - i, find z1 + z2.
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A.
6 + 2i
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B.
6 + 4i
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C.
2 + 6i
-
D.
8 + 2i
Solution
z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.
Correct Answer: A — 6 + 2i
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Q. If z1 = 2 + 3i and z2 = 4 - i, what is z1 + z2?
-
A.
6 + 2i
-
B.
6 + 4i
-
C.
2 + 4i
-
D.
2 + 2i
Solution
z1 + z2 = (2 + 3i) + (4 - i) = 6 + 2i.
Correct Answer: A — 6 + 2i
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Q. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), find \( |A| \).
Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |2A| \)?
Solution
The determinant of \( 2A \) is \( 2^2 * |A| = 4 * (-2) = -8 \).
Correct Answer: B — 8
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Q. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |A| \)?
Solution
The determinant is calculated as (1*4) - (2*3) = 4 - 6 = -2.
Correct Answer: A — -2
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Q. If \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), what is the determinant of A?
-
A.
ad - bc
-
B.
bc - ad
-
C.
a + d
-
D.
b + c
Solution
The determinant of matrix A is given by the formula \( ad - bc \).
Correct Answer: A — ad - bc
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Q. If \( B = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 1 \end{pmatrix} \), what is \( |B| \)?
Solution
The determinant is calculated as 1(1*1 - 0*3) - 2(0*1 - 1*2) + 1(0*3 - 1*2) = 1 - 4 - 2 = -5.
Correct Answer: B — 2
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Q. If \( B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix} \), what is \( \det(B) \)?
Solution
Using the determinant formula, we find \( \det(B) = 1(1*0 - 4*6) - 2(0 - 4*5) + 3(0 - 1*5) = -24 \).
Correct Answer: A — -24
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Q. If \( B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \), what is \( |B| \)?
Solution
The determinant is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. If \( B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \), what is \( |B| \)?
Solution
The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).
Correct Answer: A — -2
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Q. If \( B = \begin{pmatrix} 1 & 2 \\ 3 & 5 \end{pmatrix} \), what is \( |B| \)?
Solution
The determinant is \( 1*5 - 2*3 = 5 - 6 = -1 \).
Correct Answer: B — 1
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Q. If \( B = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \), find \( |B| \).
Solution
The determinant is \( 2*4 - 3*1 = 8 - 3 = 5 \).
Correct Answer: A — 5
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Q. If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( |C| \).
Solution
The determinant is 0 because the first column is a linear combination of the others.
Correct Answer: A — 0
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Q. If \( C = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), find \( \det(C) \).
Solution
The determinant is 0 because the first column is a linear combination of the other columns.
Correct Answer: A — 0
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Q. If \( C = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \), what is the determinant of C?
-
A.
ad - bc
-
B.
bc - ad
-
C.
a + d
-
D.
b + c
Solution
The determinant of C is given by the formula \( ad - bc \).
Correct Answer: A — ad - bc
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Q. If \( D = \begin{vmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{vmatrix} \), find \( D \).
Solution
Calculating gives \( 2(0*0 - 2*1) - 3(1*0 - 2*4) + 1(1*1 - 0*4) = -4 + 24 + 1 = 21 \).
Correct Answer: A — -10
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Q. If \( F = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \), what is the value of the determinant?
Solution
The determinant is 0 because the first column is a linear combination of the other columns.
Correct Answer: A — 0
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Q. If \( J = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 1 & 3 \end{pmatrix} \), what is the value of the determinant?
Solution
The determinant is calculated as \( 1(1*3 - 0*1) - 2(0*3 - 1*2) + 1(0*1 - 1*2) = 3 + 4 - 2 = 5 \).
Correct Answer: A — 0
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