Complex Numbers and Quadratic Equations Exercise 01
Question:
Find the multiplicative inverse of the complex number −i
Answer:
Answer: Step 1: The multiplicative inverse of any number is the reciprocal of the number.
Step 2: The reciprocal of i is 1/i.
Step 3: Therefore, the multiplicative inverse of i is 1/i.
Question:
Express the given complex number in the form a+ib:(1−i)^{4}
Answer:
(1i)^4 = (1i)(1i)(1i)(1i) = (1i)(1i)(1ii) = (1i)(12i) = (12ii^2) = (12i1) = 12i1+i^2 = 12i1+(1) = 12i2 = 12(i+1) = 12i2i = 14i = a+ib = 14i
Therefore, the complex number in the form a+ib is 14i.
Question:
Find the multiplicative inverse of the complex number √5+3i.
Answer:
Answer: Step 1: Find the conjugate of the complex number √5+3i. Conjugate = √53i
Step 2: Find the modulus of the complex number √5+3i. Modulus = √5+3i = √(5^2 + 3^2) = √34
Step 3: Find the multiplicative inverse of the complex number √5+3i. Multiplicative inverse = (√53i)/√34
Question:
(5i)(−3/5i)
Answer:

First, combine the two terms by adding them: 5i + (3/5i)

Simplify the terms by multiplying the numerator and denominator of the second term by the complex conjugate of the denominator of the second term: 5i + (3/(5i)(5i)) = 5i + (15/25)

Simplify the terms by combining the coefficients of the terms: 5i + 15/25 = (5i + 15)/25

Simplify the terms by multiplying the numerator and denominator by the reciprocal of the denominator: (5i + 15)/25 * 25/25 = (5i + 15)/25 * 1 = (5i + 15)
Question:
Solve : i^{9}+i^{19}
Answer:
Step 1: Rewrite the expression using the power of a power rule.
i^9 + i^19 = i^(9+19)
Step 2: Calculate the value of i^(9+19).
i^(9+19) = i^28
Question:
Express the given complex number 3(7+i7)+i(7+i7) in the form a+ib,
Answer:
Given: 3(7+i7)+i(7+i7)
Step 1: 3(7+i7) = 21 + 3i7
Step 2: i(7+i7) = 7 + i7
Step 3: 21 + 3i7 + (7 + i7)
Step 4: 21  7 + 4i7
Answer: 21  7 + 4i7 = 14 + 4i7
Question:
Express the given complex number in the form a+ib: (−2−1/3i)^{3}
Answer:
Given complex number, z = (2  i/3)
Step 1: Calculate z^3
z^3 = (2  i/3)^3
Step 2: Expand the expression
(2  i/3)^3 = 8  8i/3  2i^2/3  (i/3)^3
Step 3: Simplify the expression
8  8i/3  2i^2/3  (i/3)^3 = 8  8i/3  2i^2/3  i/27
Step 4: Express in the form a+ib
8  8i/3  2i^2/3  i/27 = 8  8i/3  2i^2/3  i/27 = 8  8i/3  2i  i/9 = 8  10i  i/9
Therefore, the given complex number in the form a+ib is: 8  10i  i/9
Question:
Express the following expression in the form of a+ib ; (3+i√5)(3−i√5)/(√3+√2i)−(√3−i√2)
Answer:
Given, (3+i√5)(3−i√5)/(√3+√2i)−(√3−i√2)
Step 1: Multiply the numerator and denominator by the conjugate of the denominator.
(3+i√5)(3−i√5)(√3√2i)/((√3+√2i)(√3√2i))((√3−i√2)(√3√2i))
Step 2: Simplify the numerator and denominator.
(3+i√5)(3−i√5)(3+√2i)/(9+2i√6)(3+√2i)(3+√2i)
Step 3: Simplify the numerator and denominator further.
(95√5)/(9+2i√6)9
Step 4: Simplify the expression further.
142i√6/9+2i√6
Step 5: Express the expression in the form of a+ib.
14/9+2i√6/9
Question:
Solve: [(1/3+i7/3)+(4+i1/3)]−(−4/3+i)
Answer:
Step 1: Combine like terms on the left side of the equation: [(1/3+i7/3)+(4+i1/3)]−(−4/3+i)
Step 2: [(5/3+i8/3)]−(−4/3+i)
Step 3: [5/3+i8/3 + 4/3i]
Step 4: 9/3+i7/3
Question:
Express (31+3i)3 in the form a+ib. A 343/34+23i B −343/34−23i C 242/27+26i D −242/27−26i
Answer:
Answer: C 242/27+26i
Question:
Find the modulus and amplitude of 4+3i.
Answer:
Answer: Modulus: 5 Amplitude: 5
Question:
Express (1−i)−(1+i6) as a+ib
Answer:
(1−i)−(1+i6) = (1−i) − 1  i6 = i  1  i6 = i  1  (i/6) = i  1  (1/6)i = i  1 + (1/6)i = i  1 + (1/6)i = (1  1 + (1/6))i = (2 + (1/6))i = (2 + (1/6))(1 + i) = (2 + (1/6)) + (2 + (1/6))i = (2 + (1/6)) + i(2 + (1/6)) = a + ib = (2 + (1/6)) + i(2 + (1/6)) = (2 + (1/6)) + i(2 + (1/6))
Question:
Solve the problem: (1/5+i2/5)−(4+i5/2)
Answer:
Given, (1/5 + i2/5)  (4+i5/2)
Step 1: Rewrite the given expression as,
(1/5 + i2/5)  (4+ i5/2) = (1/5  4) + (i2/5  i5/2)
Step 2: Simplify the expression,
(1/5  4) + (i2/5  i5/2) = 3.8 + (3.5i)
Hence, the solution is 3.8 + (3.5i).
Question:
Evaluate i^{−39}.
Answer:
i^−39 = 1/i^39
= 1/(i^2)^19
= 1/i^38
= 1/i^2 * 1/i^36
= 1/i^2 * (1/i^18)^2
= 1/i^2 * (1/i^2)^18
= (1/i^2)^19