Q1: What is the value of \(i^2\), where \(i\) is the imaginary unit? (a) 1 (b) -1 (c) \(i\) (d) 0
Solution:
Ans: (b) Explanation: By definition, the imaginary unit \(i\) satisfies \(i^2 = -1\). This is the fundamental property that defines complex numbers.
Q2: Which of the following represents the complex conjugate of \(3 + 5i\)? (a) \(3 - 5i\) (b) \(-3 + 5i\) (c) \(-3 - 5i\) (d) \(5 + 3i\)
Solution:
Ans: (a) Explanation: The complex conjugate of a complex number \(a + bi\) is \(a - bi\). We keep the real part the same and change the sign of the imaginary part. Therefore, the conjugate of \(3 + 5i\) is \(3 - 5i\).
Q3: What is the sum \((4 + 2i) + (6 - 7i)\)? (a) \(10 - 5i\) (b) \(10 + 5i\) (c) \(2 - 5i\) (d) \(10 - 9i\)
Solution:
Ans: (a) Explanation: To add complex numbers, we add the real parts and imaginary parts separately: \((4 + 6) + (2i - 7i) = 10 - 5i\)
Q4: What is the product \((2 + 3i)(4 - i)\)? (a) \(8 - 3i\) (b) \(11 + 10i\) (c) \(5 + 10i\) (d) \(11 - 2i\)
Q8: If \(z = 2 - 5i\), what is \(z \cdot \overline{z}\), where \(\overline{z}\) is the complex conjugate of \(z\)? (a) 29 (b) -21 (c) \(4 + 25i\) (d) -29
Solution:
Ans: (a) Explanation: The conjugate of \(z = 2 - 5i\) is \(\overline{z} = 2 + 5i\). \(z \cdot \overline{z} = (2 - 5i)(2 + 5i) = 4 - 25i^2 = 4 - 25(-1) = 4 + 25 = 29\) This equals \(|z|^2\), which is always a real number.
Section B: Fill in the Blanks
Q9: The standard form of a complex number is written as __________, where \(a\) is the real part and \(b\) is the imaginary part.
Solution:
Ans: \(a + bi\) Explanation: The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit.
Q10: The value of \(i^3\) equals __________.
Solution:
Ans: \(-i\) Explanation: Since \(i^2 = -1\), we have \(i^3 = i^2 \cdot i = (-1) \cdot i = -i\).
Q11: The difference \((8 + 6i) - (3 + 2i)\) simplifies to __________.
Solution:
Ans: \(5 + 4i\) Explanation: Subtract the real parts and imaginary parts separately: \((8 - 3) + (6i - 2i) = 5 + 4i\).
Q12: If \(z_1 = 7 + 2i\) and \(z_2 = 7 - 2i\), then \(z_1\) and \(z_2\) are called __________ of each other.
Solution:
Ans: complex conjugates Explanation: Two complex numbers are complex conjugates if they have the same real part but opposite imaginary parts.
Q13: The product \(i \cdot i \cdot i \cdot i\) equals __________.
Solution:
Ans: 1 Explanation: \(i \cdot i \cdot i \cdot i = i^4 = (i^2)^2 = (-1)^2 = 1\). The powers of \(i\) cycle with period 4.
Q14: The modulus of the complex number \(5 + 12i\) is __________.
Solution:
Ans: 13 Explanation: The modulus is calculated as \(|5 + 12i| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13\).
Section C: Word Problems
Q15: An electrical engineer is analyzing an AC circuit and encounters the impedance \(Z_1 = 4 + 3i\) ohms and \(Z_2 = 2 - 5i\) ohms connected in series. The total impedance in a series circuit is the sum of individual impedances. Find the total impedance \(Z_1 + Z_2\).
Solution:
Ans: To find the total impedance, add the two complex numbers: \(Z_1 + Z_2 = (4 + 3i) + (2 - 5i)\) \(= (4 + 2) + (3i - 5i)\) \(= 6 - 2i\) Final Answer: \(6 - 2i\) ohms
Q16: A physics student is studying wave functions and needs to compute the product of two complex amplitudes: \(A_1 = 1 + 2i\) and \(A_2 = 3 - i\). Find the product \(A_1 \cdot A_2\) and express it in standard form.
Solution:
Ans: Use the distributive property to multiply: \((1 + 2i)(3 - i) = 1(3) + 1(-i) + 2i(3) + 2i(-i)\) \(= 3 - i + 6i - 2i^2\) \(= 3 + 5i - 2(-1)\) \(= 3 + 5i + 2\) \(= 5 + 5i\) Final Answer: \(5 + 5i\)
Q17: A mathematician needs to simplify the expression \(\frac{6 + 8i}{2}\). Find the result in standard form.
Solution:
Ans: Divide both the real and imaginary parts by 2: \(\frac{6 + 8i}{2} = \frac{6}{2} + \frac{8i}{2}\) \(= 3 + 4i\) Final Answer: \(3 + 4i\)
Q18: In a control systems problem, an engineer needs to find the reciprocal of the complex number \(z = 2 + i\). Express \(\frac{1}{z}\) in standard form \(a + bi\).
Solution:
Ans: To find the reciprocal, multiply numerator and denominator by the conjugate of \(z\): \(\frac{1}{2 + i} \cdot \frac{2 - i}{2 - i} = \frac{2 - i}{(2 + i)(2 - i)}\) Denominator: \((2 + i)(2 - i) = 4 - i^2 = 4 - (-1) = 5\) Result: \(\frac{2 - i}{5} = \frac{2}{5} - \frac{1}{5}i\) Final Answer: \(\frac{2}{5} - \frac{1}{5}i\)
Q19: Two complex numbers are given: \(w = 5 - 2i\) and \(v = -3 + 4i\). Find the difference \(w - v\) and express your answer in standard form.
Q20: A researcher needs to find the distance between two points in the complex plane: \(z_1 = 1 + 2i\) and \(z_2 = 4 + 6i\). The distance between two complex numbers is the modulus of their difference. Find this distance.
Solution:
Ans: First, find the difference: \(z_2 - z_1 = (4 + 6i) - (1 + 2i) = 3 + 4i\) Now find the modulus of this difference: \(|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\) Final Answer: 5 units
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