Class 11 - Mathematics
Complex Numbers - Exercise 5.1
Top Block 1
Question 1:
Express the given complex number in the form a + ib: (5i)(-3i/5)
Answer:
(5i)(-3i/5) = (-5 * 3/5) * I * i
= -3 * i2
= -3 * (-1) [Since i2 = -1]
= 3
Question 2:
Express the given complex number in the form a + ib: i9 + i19
Answer:
i9 + i19 = i4*2 + 1 + i4*4 + 3
= (i4)2 * i + (i4)4 * i3
= (1)2 * i + (1)4 * i * i2 [Since i4 = 1]
= i + i * (-1) [Since i2 = -1]
= i – i
= 0
Question 3:
Express the given complex number in the form a + ib: i-39
Answer:
i-39 = i-4 * 9 – 3
= (i-4)9 * i-3
= (1/i4)9 * i-3
= 19 * i-3 [Since i4 = 1]
= i-3
= 1/ i3
= i4/ i3 [Since i4 = 1]
= i
Question 4:
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Answer:
Given, 3(7 + i7) + i(7 + i7) = 21 + 21i + 7i + 7i2
= 21 + 21i + 7i – 7 [Since i2 = -1]
= 14 + 28i
Question 5:
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Answer:
Given, (1 – i) – (–1 + i6) = 1 – i + 1 – 6i
= 2 – 7i
Question 6:
Express the given complex number in the form a + ib: (1/5 + 2i/5) – (4 + 5i/2)
Answer:
Given, (1/5 + 2i/5) – (4 + 5i/2) = 1/5 + 2i/5 – 4 – 5i/2
= (1/5 – 4) + (2i/5 – 5i/2)
= -19/5 + (4i – 25i)/10
= -19/5 – 21i/10
Question 7:
Express the given complex number in the form a + ib: [(1/3 + 7i/3) + (4 + i/3)] – (-4/3 + i)
Answer:
Given, [(1/3 + 7i/3) + (4 + i/3)] – (-4/3 + i) = 1/3 + 7i/3 + 4 + i/3 + 4/3 – i
= (1/3 + 4 + 4/3) + i(7/3 + 1/3 – 1)
= 17/3 + 5i/3
Question 8:
Express the given complex number in the form a + ib: (1 – i)4
Answer:
Given, (1 – i)4 = [(1 – i)2]2
= [1 + i2 – 2i]2
= [1 – 1 – 2i]2 [Since i2 = -1]
= (– 2i)2
= 4i2
= 4 * (-1) [Since i2 = -1]
= -4
Question 9:
Express the given complex number in the form a + ib: (1/3 + 3i)3
Answer:
Given, (1/3 + 3i)3 = (1/3)3 + (3i)3 + 3(1/3)(3i) (1/3 + 3i)
= 1/27 + 27i3 + (3i) (1/3 + 3i)
= 1/27 + 27i * i2 + i + 9i2
= 1/27 + 27i * (-1) + i + 9(-1) [Since i2 = -1]
= 1/27 – 27i + i – 9
= (1/27 – 9) – 26i
= -242/27 – 26i
Mddle block 1
Question 10:
Express the given complex number in the form a + ib: (-2 – i/3)3
Answer:
Given, (-2 – i/3)3 = (-1)3 (2 + i/3)3
= -[23 + (i/3)3 + 3(2)(i/3)(2 + i/3)]
= -[8 + i3/27+ (2i) (2 + i/3)]
= -[8 – i/27 + 4i + 2i2/3]
= -[8 – i/27 + 4i – 2/3] [Since i2 = -1]
= -[22/3 + 107i/27]
= -22/3 – 107i/27
Question 11:
Find the multiplicative inverse of the complex number 4 – 3i.
Answer:
Let z = 4 – 3i
Then,
z = 4 + 3i
|z|2 = 42 + (-3)2 = 16 + 9 = 25
Therefore, the multiplicative inverse of 4 – 3i is given by
z-1 = z/|z|2 = (4 + 3i)/25 = 4/25 + 3i/25
Question 12:
Find the multiplicative inverse of the complex number √5 + 3i.
Answer:
Let z = √5 + 3i
Then,
z = √5 – 3i
|z|2 = (√5)2 + (-3)2 = 5 + 9 = 14
Therefore, the multiplicative inverse of √5 – 3i is given by
z-1 = z/|z|2 = (√5 – 3i)/14 = √5/14 – 3i/14
Question 13:
Find the multiplicative inverse of the complex number -i.
Answer:
Let z = -i
Then,
z = i
|z|2 = 12 = 1
Therefore, the multiplicative inverse of -i is given by
z-1 = z/|z|2 = i/1 = i
Question 14:
Express the following expression in the form of a + ib.
{(3 + i√5)(3 – i√5)}/{(√3 + i√2) – (√3 – i√2)}
Answer:
Given, {(3 + i√5)(3 – i√5)}/{(√3 + i√2) – (√3 – i√2)}
= {32 – (i√5)2}/(√3 + i√2 – √3 + i√2) [(a – b)(a + b) = a2 – b2]
= (9 – 5i2)/i2√2
= {9 – 5 * (-1)}/i2√2 [Since i2 = -1]
= (9 + 5)/i2√2
= 14/i2√2
= 7/i√2
= (7/i√2) * (i/i)
= 7i/(√2 * i2)
= 7i/(-√2) [Since i2 = -1]
= -7i/√2
= (-7i/√2) * (√2/√2)
= -7√2i/2
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