ID: 967 Type: Mcq Year: null

Question 1: Evaluate the value of $i^{n} + i^{n+1} + i^{n+2} + i^{n+3}$ for any integer $n$, where $i = \sqrt{-1}$.

ID: 968 Type: Mcq Year: null

Question 2: Find the multiplicative inverse of the complex number $z = 4 - 3i$.

• A. $\frac{4}{25} + \frac{3}{25}i$
• B. $\frac{4}{7} + \frac{3}{7}i$
• C. $\frac{4}{25} - \frac{3}{25}i$
• D. $4 + 3i$

ID: 969 Type: Mcq Year: null

Question 3: If $(x + iy)(2 - 3i) = 4 + i$, find the real values of $x$ and $y$.

• A. $x = \frac{5}{13}, y = \frac{14}{13}$
• B. $x = 1, y = 2$
• C. $x = \frac{14}{13}, y = \frac{5}{13}$
• D. $x = -1, y = 1$

ID: 970 Type: Mcq Year: null

Question 4: What is the modulus of the complex number $z = \frac{(1+i)(2+i)}{3+i}$?

• A. $1$
• B. $\sqrt{2}$
• C. $\frac{1}{2}$
• D. $2$

ID: 971 Type: Mcq Year: null

Question 5: Find the principal argument of the complex number $z = -1 - i\sqrt{3}$.

• A. $-\frac{2\pi}{3}$
• B. $\frac{4\pi}{3}$
• C. $\frac{\pi}{3}$
• D. $-\frac{\pi}{3}$

ID: 972 Type: Mcq Year: null

Question 6: What is the smallest positive integer $n$ for which $\left(\frac{1+i}{1-i}\right)^n = 1$?

• A. $4$
• B. $2$
• C. $8$
• D. $1$

ID: 973 Type: Mcq Year: null

Question 7: Solve the quadratic equation $x^2 + x + 1 = 0$.

• A. $\frac{-1 \pm i\sqrt{3}}{2}$
• B. $\frac{1 \pm i\sqrt{3}}{2}$
• C. $\pm i$
• D. $-1, 1$

ID: 974 Type: Mcq Year: null

Question 8: Find the conjugate of the complex number $z = \frac{1}{3+4i}$.

• A. $\frac{3}{25} + \frac{4}{25}i$
• B. $3 - 4i$
• C. $\frac{3}{25} - \frac{4}{25}i$
• D. $\frac{1}{3-4i}$

ID: 975 Type: Mcq Year: null

Question 9: If $z = r(\cos \theta + i \sin \theta)$, then the value of $\frac{z}{\bar{z}}$ is:

• A. $\cos(2\theta) + i \sin(2\theta)$
• B. $\cos \theta + i \sin \theta$
• C. $1$
• D. $\cos(2\theta) - i \sin(2\theta)$