Question 1

Q1: Express the complex number $(5 i)\left(-\frac{3}{5} i\right)$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $(5 i)\left(-\frac{3}{5} iight)$. To Find: Express the given complex number in the form $a+i b$. Solution: We have, $$(5 i)\left(-\frac{3}{5} iight) = 5 	imes \left(-\frac{3}{5}ight) 	imes (i 	imes i)$$ $$= -3 	imes i^2$$ We know that $i^2 = -1$. Substi...

Question 2

Q10: Express the complex number $\left(-2-\frac{1}{3} i\right)^{3}$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $\left(-2-\frac{1}{3} i\right)^{3}$. To Find: Express the given complex number in the form $a+i b$. Solution: We can factor out $-1$ from the expression: $$\left(-2-\frac{1}{3} i\right)^{3} = \left[-1\left(2+\frac{1}{3} i\right)\right]^{3} = (-1)^3 \left(2+\frac{...

Question 3

Q11: Find the multiplicative inverse of the complex number $4-3i$.

Accepted Answer

Given: The complex number $z = 4-3i$. To Find: The multiplicative inverse of $z$. Formula: The multiplicative inverse of a complex number $z = a+ib$ is given by $z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2}$, where $\bar{z} = a-ib$ is the conjugate and $|z|^2 = a^2+b^2$. Solution: For the given comp...

Question 4

Q12: Find the multiplicative inverse of the complex number $\sqrt{5}+3 i$.

Accepted Answer

Given: The complex number $z = \sqrt{5}+3i$. To Find: The multiplicative inverse of $z$. Formula: The multiplicative inverse of a complex number $z = a+ib$ is given by $z^{-1} = \frac{1}{z} = \frac{\bar{z}}{|z|^2}$, where $\bar{z} = a-ib$ is the conjugate and $|z|^2 = a^2+b^2$. Solution: For the giv...

Question 5

Q13: Find the multiplicative inverse of the complex number $-i$.

Accepted Answer

Given: The complex number $z = -i$. To Find: The multiplicative inverse of $z$. Solution: We can write $z$ in the form $a+ib$ as $z = 0 - 1i$. So, $a=0$ and $b=-1$. Method 1: Using the formula Formula: $z^{-1} = \frac{\bar{z}}{|z|^2}$ 1.  Find the conjugate ($\bar{z}$):     $$\bar{z} = 0 - (-1i) = i...

Question 6

Q14: Express the following expression in the form of $a+i b$ : $$\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$$

Accepted Answer

Given: The expression $\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+\sqrt{2} i)-(\sqrt{3}-i \sqrt{2})}$. To Find: Express the given expression in the form $a+i b$. Solution: Let's simplify the numerator and the denominator separately. Numerator: $$(3+i \sqrt{5})(3-i \sqrt{5})$$ This is of the form...

Question 7

Q2: Express the complex number $i^{9}+i^{19}$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $i^{9}+i^{19}$. To Find: Express the given complex number in the form $a+i b$. Solution: We know the properties of powers of $i$: $i^1 = i$ $i^2 = -1$ $i^3 = -i$ $i^4 = 1$ In general, $i^{4k+r} = i^r$ for any integer $k$. First, let's simplify $i^9$: $$i^9 = i^{4...

Question 8

Q3: Express the complex number $i^{-39}$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $i^{-39}$. To Find: Express the given complex number in the form $a+i b$. Solution: We have, $$i^{-39} = \frac{1}{i^{39}}$$ First, let's simplify the denominator $i^{39}$. We know $i^{4k+r} = i^r$. $$i^{39} = i^{4 	imes 9 + 3} = i^3 = -i$$ Now substitute this ba...

Question 9

Q4: Express the complex number $3(7+i 7)+i(7+i 7)$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $3(7+i 7)+i(7+i 7)$. To Find: Express the given complex number in the form $a+i b$. Solution: We have, $$3(7+i 7)+i(7+i 7)$$ First, distribute the terms: $$= (3 	imes 7 + 3 	imes i7) + (i 	imes 7 + i 	imes i7)$$ $$= (21 + 21i) + (7i + 7i^2)$$ We know that $i^...

Question 10

Q5: Express the complex number $(1-i)-(-1+i 6)$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $(1-i)-(-1+i 6)$. To Find: Express the given complex number in the form $a+i b$. Solution: We have, $$(1-i)-(-1+i 6)$$ Distribute the negative sign: $$= 1 - i - (-1) - (i6)$$ $$= 1 - i + 1 - 6i$$ Now, group the real and imaginary parts: $$= (1 + 1) + (-i - 6i)$$...

Question 11

Q6: Express the complex number $\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)$. To Find: Express the given complex number in the form $a+i b$. Solution: We have, $$\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)$$ First, remove the parentheses: $$= \fr...

Question 12

Q7: Express the complex number $\left[\left(\frac{1}{3}+i \frac{7}{3}\right)+\left(4+i \frac{1}{3}\right)\right]-\left(-\frac{4}{3}+i\right)$ in the form...

Accepted Answer

Given: The complex number expression $\left[\left(\frac{1}{3}+i \frac{7}{3}\right)+\left(4+i \frac{1}{3}\right)\right]-\left(-\frac{4}{3}+i\right)$. To Find: Express the given complex number in the form $a+i b$. Solution: First, let's solve the expression inside the square brackets: $$\left(\frac{1}...

Question 13

Q8: Express the complex number $(1-i)^{4}$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $(1-i)^{4}$. To Find: Express the given complex number in the form $a+i b$. Solution: We can write $(1-i)^4$ as $((1-i)^2)^2$. First, let's compute $(1-i)^2$: $$(1-i)^2 = 1^2 - 2(1)(i) + i^2$$ $$= 1 - 2i + (-1)$$ $$= 1 - 2i - 1$$ $$= -2i$$ Now, we need to compute...

Question 14

Q9: Express the complex number $\left(\frac{1}{3}+3 i\right)^{3}$ in the form $a+i b$.

Accepted Answer

Given: The complex number expression $\left(\frac{1}{3}+3 i\right)^{3}$. To Find: Express the given complex number in the form $a+i b$. Formula: We use the identity $(x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$. Solution: Let $x = \frac{1}{3}$ and $y = 3i$. $$\left(\frac{1}{3}+3 i\right)^{3} = \left(\frac{1...

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