Question 1

$(5i)\left(-\frac{3}{5}i\right)$

Accepted Answer

Given: (5i)\left(-\frac{3}{5}i\right)

Step 1: Multiply the constants: 5 * (-3/5) = -3
Step 2: Multiply the imaginary units: i * i = i^2 = -1
Step 3: So, (5i)(-3/5 i) = -3 * (-1) = 3

Answer: 3 — Multiply the coefficients and then multiply the imaginary units. Since i^2 = -1, the product becomes -3 * (-1) = 3.

Question 2

$i^9 + i^{19}$

Accepted Answer

Recall powers of i cycle every 4:
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1

Calculate i^9:
9 mod 4 = 1, so i^9 = i

Calculate i^{19}:
19 mod 4 = 3, so i^{19} = -i

Sum: i + (-i) = 0

Answer: 0 — Use the cyclic nature of powers of i with period 4 to simplify each term, then add.

Question 3

$i^{-39}$

Accepted Answer

Recall powers of i cycle every 4.
First, convert negative power:
i^{-39} = 1 / i^{39}

Calculate 39 mod 4:
39 mod 4 = 3
So, i^{39} = i^3 = -i

Therefore,
i^{-39} = 1 / (-i) = -1 / i

Multiply numerator and denominator by i:
= (-1 / i) * (i / i) = (-i) / (i^2) = (-i) / (-1) = i

Answer: i — Convert negative exponent to reciprocal, simplify using powers of i, and rationalize denominator.

Question 4

$3(7 + i7) + i(7 + i7)$

Accepted Answer

Given expression: 3(7 + 7i) + i(7 + 7i)

Step 1: Expand each term:
3(7 + 7i) = 21 + 21i

Step 2: i(7 + 7i) = 7i + 7i^2 = 7i + 7(-1) = 7i - 7

Step 3: Add the two results:
(21 + 21i) + (7i - 7) = (21 - 7) + (21i + 7i) = 14 + 28i

Answer: 14 + 28i — Distribute multiplication over addition, simplify i^2 = -1, then combine like terms.

Question 5

$(1 - i) - (-1 + i6)$

Accepted Answer

Given: (1 - i) - (-1 + 6i)

Step 1: Remove parentheses:
= 1 - i + 1 - 6i

Step 2: Combine like terms:
Real parts: 1 + 1 = 2
Imaginary parts: -i - 6i = -7i

Answer: 2 - 7i — Subtract the complex numbers by distributing the negative sign and combining real and imaginary parts.

Question 6

$\left(\frac{1}{5} + i\frac{2}{5}\right) - \left(4 + i\frac{5}{2}\right)$

Accepted Answer

Given: \left(\frac{1}{5} + i\frac{2}{5}\right) - \left(4 + i\frac{5}{2}\right)

Step 1: Subtract real parts:
\frac{1}{5} - 4 = \frac{1}{5} - \frac{20}{5} = -\frac{19}{5}

Step 2: Subtract imaginary parts:
\frac{2}{5} - \frac{5}{2} = \frac{2}{5} - \frac{25}{10} = \frac{4}{10} - \frac{25}{10} = -\frac{21}{10}

Answer: -\frac{19}{5} - i\frac{21}{10} — Subtract real and imaginary parts separately, converting to common denominators where necessary.

Question 7

$\left[\left(\frac{1}{3} + i\frac{7}{3}\right) + \left(4 + i\frac{1}{3}\right)\right] - \left(-\frac{4}{3} + i\right)$

Accepted Answer

Given 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)

Step 1: Add inside the bracket:
Real parts: \frac{1}{3} + 4 = \frac{1}{3} + \frac{12}{3} = \frac{13}{3}
Imaginary parts: \frac{7}{3} + \frac{1}{3} = \frac{8}{3}

So, sum inside bracket = \frac{13}{3} + i\frac{8}{3}

Step 2: Subtract the second bracket:
Real parts: \frac{13}{3} - (-\frac{4}{3}) = \frac{13}{3} + \frac{4}{3} = \frac{17}{3}
Imaginary parts: \frac{8}{3} - 1 = \frac{8}{3} - \frac{3}{3} = \frac{5}{3}

Answer: \frac{17}{3} + i\frac{5}{3} — Add the first two complex numbers by combining real and imaginary parts, then subtract the third complex number similarly.

Question 8

$(1 - i)^{4}$

Accepted Answer

Given: (1 - i)^4

Step 1: Find (1 - i)^2:
(1 - i)^2 = 1^2 - 2i + i^2 = 1 - 2i - 1 = -2i

Step 2: Then (1 - i)^4 = [(1 - i)^2]^2 = (-2i)^2 = (-2)^2 * i^2 = 4 * (-1) = -4

Answer: -4 — Square (1 - i) first, then square the result. Use i^2 = -1.

Question 9

$\left(\frac{1}{3} + 3i\right)^{3}$

Accepted Answer

Given: \left(\frac{1}{3} + 3i\right)^3

Step 1: Let z = \frac{1}{3} + 3i

Step 2: Compute z^2:
\left(\frac{1}{3} + 3i\right)^2 = \left(\frac{1}{3}\right)^2 + 2 * \frac{1}{3} * 3i + (3i)^2 = \frac{1}{9} + 2i + 9i^2 = \frac{1}{9} + 2i - 9 = -\frac{80}{9} + 2i

Step 3: Compute z^3 = z^2 * z:
= \left(-\frac{80}{9} + 2i\right) \left(\frac{1}{3} + 3i\right)

Multiply:
Real part: \left(-\frac{80}{9}\right) * \frac{1}{3} + 2i * 3i = -\frac{80}{27} + 6i^2 = -\frac{80}{27} - 6 = -\frac{80}{27} - \frac{162}{27} = -\frac{242}{27}
Imaginary part: \left(-\frac{80}{9}\right) * 3i + 2i * \frac{1}{3} = -\frac{240}{9}i + \frac{2}{3}i = -\frac{240}{9}i + \frac{6}{9}i = -\frac{234}{9}i = -26i

Answer: -\frac{242}{27} - 26i — Use binomial expansion or multiply stepwise, carefully simplifying powers of i and combining like terms.

Question 10

$\left(-2 - \frac{1}{3} i\right)^{3}$

Accepted Answer

Given: \left(-2 - \frac{1}{3} i\right)^3

Step 1: Let z = -2 - \frac{1}{3} i

Step 2: Compute z^2:
= (-2)^2 + 2 * (-2) * \left(-\frac{1}{3} i\right) + \left(-\frac{1}{3} i\right)^2
= 4 + \frac{4}{3} i + \frac{1}{9} i^2
= 4 + \frac{4}{3} i - \frac{1}{9}  (since i^2 = -1)
= \frac{36}{9} + \frac{12}{9} i - \frac{1}{9} = \frac{35}{9} + \frac{12}{9} i

Step 3: Compute z^3 = z^2 * z:
= \left(\frac{35}{9} + \frac{12}{9} i\right) \left(-2 - \frac{1}{3} i\right)

Multiply:
Real part: \frac{35}{9} * (-2) + \frac{12}{9} i * \left(-\frac{1}{3} i\right) = -\frac{70}{9} - \frac{12}{27} i^2 = -\frac{70}{9} + \frac{12}{27} = -\frac{210}{27} + \frac{12}{27} = -\frac{198}{27} = -\frac{22}{3}
Imaginary part: \frac{35}{9} * \left(-\frac{1}{3} i\right) + \frac{12}{9} i * (-2) = -\frac{35}{27} i - \frac{24}{9} i = -\frac{35}{27} i - \frac{72}{27} i = -\frac{107}{27} i

Answer: -\frac{22}{3} - \frac{107}{27} i — Square the complex number first, then multiply by the original number, simplifying powers of i and combining like terms carefully.

Question 11

$4 - 3i$

Accepted Answer

The question appears incomplete as it only states a complex number $4 - 3i$. Assuming the question is to find the conjugate or modulus or any other property, no explicit instruction is given. Hence, no solution can be provided. — No explicit question is provided, only a complex number is given.

Question 12

$\sqrt{5} + 3i$

Accepted Answer

The question appears incomplete as it only states a complex number $\sqrt{5} + 3i$. Assuming the question is to find the conjugate or modulus or any other property, no explicit instruction is given. Hence, no solution can be provided. — No explicit question is provided, only a complex number is given.

Question 13

$-i$

Accepted Answer

The question appears incomplete as it only states a complex number $-i$. Assuming the question is to find the conjugate or modulus or any other property, no explicit instruction is given. Hence, no solution can be provided. — No explicit question is provided, only a complex number is given.

Question 14

Express the following expression in the form of  $a + ib$ :

$$
\frac {\left(3 + i \sqrt {5}\right) \left(3 - i \sqrt {5}\right)}{\left(\sqrt {3} + \sqrt {2} i\right) - \left(\sqrt {3} - i \sqrt {2}\right)}
$$

Accepted Answer

Given expression:

$$
\frac {(3 + i \sqrt{5})(3 - i \sqrt{5})}{(\sqrt{3} + \sqrt{2} i) - (\sqrt{3} - i \sqrt{2})}
$$

Step 1: Simplify numerator:

$$
(3 + i \sqrt{5})(3 - i \sqrt{5}) = 3^2 - (i \sqrt{5})^2 = 9 - (i^2)(5) = 9 - (-1)(5) = 9 + 5 = 14
$$

Step 2: Simplify denominator:

$$
(\sqrt{3} + \sqrt{2} i) - (\sqrt{3} - i \sqrt{2}) = \sqrt{3} + \sqrt{2} i - \sqrt{3} + i \sqrt{2} = \sqrt{2} i + i \sqrt{2} = 2 i \sqrt{2}
$$

Step 3: So expression becomes:

$$
\frac{14}{2 i \sqrt{2}} = \frac{14}{2 i \sqrt{2}} = \frac{7}{i \sqrt{2}}
$$

Step 4: Multiply numerator and denominator by $-i$ to rationalize denominator:

$$
\frac{7}{i \sqrt{2}} \times \frac{-i}{-i} = \frac{-7 i}{i^2 \sqrt{2}} = \frac{-7 i}{-1 \times \sqrt{2}} = \frac{-7 i}{-\sqrt{2}} = \frac{7 i}{\sqrt{2}}
$$

Step 5: Express in $a + ib$ form:

$$
0 + i \frac{7}{\sqrt{2}} = 0 + i \frac{7 \sqrt{2}}{2} \quad \text{(rationalizing denominator)}
$$

Final answer:

$$
0 + i \frac{7 \sqrt{2}}{2}
$$

Hence, the expression in the form $a + ib$ is:

$$
a = 0, \quad b = \frac{7 \sqrt{2}}{2}
$$ — Step 1: Multiply numerator terms using difference of squares.
Step 2: Simplify denominator by subtracting terms.
Step 3: Simplify fraction.
Step 4: Rationalize denominator by multiplying numerator and denominator by $-i$.
Step 5: Express final answer in $a + ib$ form.

Question 15

Evaluate: $\left[i^{18} + \left(\frac{1}{i}\right)^{25}\right]^3$.

Accepted Answer

First, evaluate each term inside the bracket:

Recall that powers of i cycle every 4:

$i^{18} = i^{4 \times 4 + 2} = i^2 = -1$.

Next, $\left(\frac{1}{i}\right)^{25} = (i^{-1})^{25} = i^{-25} = i^{-24 -1} = i^{-24} \cdot i^{-1} = (i^4)^{-6} \cdot i^{-1} = 1^{-6} \cdot i^{-1} = i^{-1} = \frac{1}{i} = -i$.

So inside the bracket we have:

$i^{18} + \left(\frac{1}{i}\right)^{25} = -1 + (-i) = -1 - i$.

Now cube this:

$(-1 - i)^3$.

Use binomial expansion or multiply stepwise:

$(-1 - i)^2 = (-1)^2 + 2(-1)(-i) + (-i)^2 = 1 + 2i + (-1)(i^2) = 1 + 2i + (-1)(-1) = 1 + 2i + 1 = 2 + 2i$.

Then multiply by $-1 - i$:

$(2 + 2i)(-1 - i) = 2(-1) + 2(-i) + 2i(-1) + 2i(-i) = -2 - 2i - 2i - 2i^2 = -2 - 4i - 2(-1) = -2 - 4i + 2 = 0 - 4i = -4i$.

Therefore, the value is $-4i$. — Step 1: Simplify powers of i using cyclicity.
Step 2: Simplify $i^{18} = -1$.
Step 3: Simplify $\left(\frac{1}{i}\right)^{25} = -i$.
Step 4: Add inside bracket: $-1 - i$.
Step 5: Cube $-1 - i$ by first squaring then multiplying.
Step 6: Final result is $-4i$.

Mathematics

Mathematics — Study Notes

4.1 Introduction

4.2 Complex Numbers

4.3 Algebra of Complex Numbers

Practice Questions — Mathematics

All 14 Chapters in Mathematics