Commerce Exam > Commerce Notes > Mathematics (Maths) Class 11 > RD Sharma Class 11 Solutions Chapter - Complex Numbers
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Page 1
13. Complex Numbers
Exercise 13.1
1. Question
Evaluate the following:
(i) i
457
(ii) i
528
(iii)
(iv)
(v)
(vi) (i
77
+ i
70
+ i
87
+ i
414
)
3
(vii) (vii) i
30
+ i
40
+ i
60
(viii) i
49
+ i
68
+ i
89
+ i
118
Answer
i. i
457
= i
(456 + 1)
= i
4(114)
× i
= (1)
114
× i = i since i
4
= 1
ii. i
528
= i
4(132)
= (1)
132
=1 since i
4
= 1
iii.
since i
4
= 1
= – 1 since i
2
= – 1
iv.
[since i
4
= 1]
v.
= (i – i) = 0
[since ]
Page 2
13. Complex Numbers
Exercise 13.1
1. Question
Evaluate the following:
(i) i
457
(ii) i
528
(iii)
(iv)
(v)
(vi) (i
77
+ i
70
+ i
87
+ i
414
)
3
(vii) (vii) i
30
+ i
40
+ i
60
(viii) i
49
+ i
68
+ i
89
+ i
118
Answer
i. i
457
= i
(456 + 1)
= i
4(114)
× i
= (1)
114
× i = i since i
4
= 1
ii. i
528
= i
4(132)
= (1)
132
=1 since i
4
= 1
iii.
since i
4
= 1
= – 1 since i
2
= – 1
iv.
[since i
4
= 1]
v.
= (i – i) = 0
[since ]
vi. (i
77
+ i
70
+ i
87
+ i
414
)
3
= (i
(76 + 1)
+ i
(68 + 2)
+ i
(84 + 3)
+ i
(412 + 2)
)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= (i + i
2
+ i
3
+ i
2
)
3
[since i
3
= – i, i
2
= – 1]
= (i + (– 1) + (– i) + (– 1))
3
= (– 2)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= –8
vii. i
30
+ i
40
+ i
60
= i
(28 + 2) +
i
40
+ i
60
= (i
4
)
7
i
2
+ (i
4
)
10
+ (i
4
)
15
= i
2
+ 1
10
+ 1
15
= – 1 + 1 + 1 = 1
viii. i
49
+ i
68
+ i
89
+ i
118
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(116 + 2)
= (i
4
)
12
×i + (i
4
)
17
+ (i
4
)
11
×i + (i
4
)
29
×i
2
= i + 1 + i – 1 = 2i
2. Question
Show that 1 + i
10
+ i
20
+ i
30
is a real number ?
Answer
1 + i
10
+ i
20
+ i
30
= 1 + i
(8 + 2) +
i
20
+ i
(28 + 2)
= 1 + (i
4
)
2
× i
2
+ (i
4
)
5
+ (i
4
)
7
× i
2
= 1 – 1 + 1 – 1 = 0
[ since i
4
= 1, i
2
= – 1]
Hence , 1 + i
10
+ i
20
+ i
30
is a real number.
3 A. Question
Find the value of following expression:
i
49
+ i
68
+ i
89
+ i
110
Answer
i
49
+ i
68
+ i
89
+ i
110
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(108 + 2)
= (i
4
)
12
× i + (i
4
)
17
+ (i
4
)
11
× i + (i
4
)
27
× i
2
= i + 1 + i – 1 = 2i
[since i
4
= 1, i
2
= – 1]
i
49
+ i
68
+ i
89
+ i
110
= 2i
3 B. Question
Find the value of following expression:
i
30
+ i
80
+ i
120
Answer
i
30
+ i
80
+ i
120
= i
(28 + 2)
+ i
80
+ i
120
= (i
4
)
7
× i
2
+ (i
4
)
20
+ (i
4
)
30
= – 1 + 1 + 1 = 1
Page 3
13. Complex Numbers
Exercise 13.1
1. Question
Evaluate the following:
(i) i
457
(ii) i
528
(iii)
(iv)
(v)
(vi) (i
77
+ i
70
+ i
87
+ i
414
)
3
(vii) (vii) i
30
+ i
40
+ i
60
(viii) i
49
+ i
68
+ i
89
+ i
118
Answer
i. i
457
= i
(456 + 1)
= i
4(114)
× i
= (1)
114
× i = i since i
4
= 1
ii. i
528
= i
4(132)
= (1)
132
=1 since i
4
= 1
iii.
since i
4
= 1
= – 1 since i
2
= – 1
iv.
[since i
4
= 1]
v.
= (i – i) = 0
[since ]
vi. (i
77
+ i
70
+ i
87
+ i
414
)
3
= (i
(76 + 1)
+ i
(68 + 2)
+ i
(84 + 3)
+ i
(412 + 2)
)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= (i + i
2
+ i
3
+ i
2
)
3
[since i
3
= – i, i
2
= – 1]
= (i + (– 1) + (– i) + (– 1))
3
= (– 2)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= –8
vii. i
30
+ i
40
+ i
60
= i
(28 + 2) +
i
40
+ i
60
= (i
4
)
7
i
2
+ (i
4
)
10
+ (i
4
)
15
= i
2
+ 1
10
+ 1
15
= – 1 + 1 + 1 = 1
viii. i
49
+ i
68
+ i
89
+ i
118
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(116 + 2)
= (i
4
)
12
×i + (i
4
)
17
+ (i
4
)
11
×i + (i
4
)
29
×i
2
= i + 1 + i – 1 = 2i
2. Question
Show that 1 + i
10
+ i
20
+ i
30
is a real number ?
Answer
1 + i
10
+ i
20
+ i
30
= 1 + i
(8 + 2) +
i
20
+ i
(28 + 2)
= 1 + (i
4
)
2
× i
2
+ (i
4
)
5
+ (i
4
)
7
× i
2
= 1 – 1 + 1 – 1 = 0
[ since i
4
= 1, i
2
= – 1]
Hence , 1 + i
10
+ i
20
+ i
30
is a real number.
3 A. Question
Find the value of following expression:
i
49
+ i
68
+ i
89
+ i
110
Answer
i
49
+ i
68
+ i
89
+ i
110
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(108 + 2)
= (i
4
)
12
× i + (i
4
)
17
+ (i
4
)
11
× i + (i
4
)
27
× i
2
= i + 1 + i – 1 = 2i
[since i
4
= 1, i
2
= – 1]
i
49
+ i
68
+ i
89
+ i
110
= 2i
3 B. Question
Find the value of following expression:
i
30
+ i
80
+ i
120
Answer
i
30
+ i
80
+ i
120
= i
(28 + 2)
+ i
80
+ i
120
= (i
4
)
7
× i
2
+ (i
4
)
20
+ (i
4
)
30
= – 1 + 1 + 1 = 1
[since i
4
= 1, i
2
= – 1]
i
30
+ i
80
+ i
120
= 1
3 C. Question
Find the value of following expression:
i + i
2
+ i
3
+ i
4
Answer
i + i
2
+ i
3
+ i
4
= i + i
2
+ i
2
×i + i
4
= i – 1 + (– 1)×i + 1
since i
4
= 1, i
2
= – 1
= i – 1 – i + 1 = 0
3 D. Question
Find the value of following expression:
i
5
+ i
10
+ i
15
Answer
i
5
+ i
10
+ i
15
= i
(4 + 1)
+ i
(8 + 2)
+ i
(12 + 3)
= (i
4
)
1
×i + (i
4
)
2
×i
2
+ (i
4
)
3
×i
3
= (i
4
)
1
×i + (i
4
)
2
×i
2
+ (i
4
)
3
×i
2
×i
= 1×i + 1×(– 1) + 1×(– 1)×i
= i – 1 – i = – 1
3 E. Question
Find the value of following expression:
Answer
= i
10
= i
8
i
2
= (i
4
)
2
i
2
Since i
4
= 1, i
2
= -1
= (1)
2
(-1)
= -1
3 F. Question
Find the value of following expression:
1 + i
2
+ i
4
+ i
6
+ i
8
+ ... + i
20
Answer
1 + i
2
+ i
4
+ i
6
+ i
8
+ ... + i
20
= 1 + (– 1) + 1 + (– 1) + 1 + ... + 1
= 1
3 G. Question
Find the value of following expression:
(1 + i)
6
+ (1 – i)
3
Page 4
13. Complex Numbers
Exercise 13.1
1. Question
Evaluate the following:
(i) i
457
(ii) i
528
(iii)
(iv)
(v)
(vi) (i
77
+ i
70
+ i
87
+ i
414
)
3
(vii) (vii) i
30
+ i
40
+ i
60
(viii) i
49
+ i
68
+ i
89
+ i
118
Answer
i. i
457
= i
(456 + 1)
= i
4(114)
× i
= (1)
114
× i = i since i
4
= 1
ii. i
528
= i
4(132)
= (1)
132
=1 since i
4
= 1
iii.
since i
4
= 1
= – 1 since i
2
= – 1
iv.
[since i
4
= 1]
v.
= (i – i) = 0
[since ]
vi. (i
77
+ i
70
+ i
87
+ i
414
)
3
= (i
(76 + 1)
+ i
(68 + 2)
+ i
(84 + 3)
+ i
(412 + 2)
)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= (i + i
2
+ i
3
+ i
2
)
3
[since i
3
= – i, i
2
= – 1]
= (i + (– 1) + (– i) + (– 1))
3
= (– 2)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= –8
vii. i
30
+ i
40
+ i
60
= i
(28 + 2) +
i
40
+ i
60
= (i
4
)
7
i
2
+ (i
4
)
10
+ (i
4
)
15
= i
2
+ 1
10
+ 1
15
= – 1 + 1 + 1 = 1
viii. i
49
+ i
68
+ i
89
+ i
118
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(116 + 2)
= (i
4
)
12
×i + (i
4
)
17
+ (i
4
)
11
×i + (i
4
)
29
×i
2
= i + 1 + i – 1 = 2i
2. Question
Show that 1 + i
10
+ i
20
+ i
30
is a real number ?
Answer
1 + i
10
+ i
20
+ i
30
= 1 + i
(8 + 2) +
i
20
+ i
(28 + 2)
= 1 + (i
4
)
2
× i
2
+ (i
4
)
5
+ (i
4
)
7
× i
2
= 1 – 1 + 1 – 1 = 0
[ since i
4
= 1, i
2
= – 1]
Hence , 1 + i
10
+ i
20
+ i
30
is a real number.
3 A. Question
Find the value of following expression:
i
49
+ i
68
+ i
89
+ i
110
Answer
i
49
+ i
68
+ i
89
+ i
110
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(108 + 2)
= (i
4
)
12
× i + (i
4
)
17
+ (i
4
)
11
× i + (i
4
)
27
× i
2
= i + 1 + i – 1 = 2i
[since i
4
= 1, i
2
= – 1]
i
49
+ i
68
+ i
89
+ i
110
= 2i
3 B. Question
Find the value of following expression:
i
30
+ i
80
+ i
120
Answer
i
30
+ i
80
+ i
120
= i
(28 + 2)
+ i
80
+ i
120
= (i
4
)
7
× i
2
+ (i
4
)
20
+ (i
4
)
30
= – 1 + 1 + 1 = 1
[since i
4
= 1, i
2
= – 1]
i
30
+ i
80
+ i
120
= 1
3 C. Question
Find the value of following expression:
i + i
2
+ i
3
+ i
4
Answer
i + i
2
+ i
3
+ i
4
= i + i
2
+ i
2
×i + i
4
= i – 1 + (– 1)×i + 1
since i
4
= 1, i
2
= – 1
= i – 1 – i + 1 = 0
3 D. Question
Find the value of following expression:
i
5
+ i
10
+ i
15
Answer
i
5
+ i
10
+ i
15
= i
(4 + 1)
+ i
(8 + 2)
+ i
(12 + 3)
= (i
4
)
1
×i + (i
4
)
2
×i
2
+ (i
4
)
3
×i
3
= (i
4
)
1
×i + (i
4
)
2
×i
2
+ (i
4
)
3
×i
2
×i
= 1×i + 1×(– 1) + 1×(– 1)×i
= i – 1 – i = – 1
3 E. Question
Find the value of following expression:
Answer
= i
10
= i
8
i
2
= (i
4
)
2
i
2
Since i
4
= 1, i
2
= -1
= (1)
2
(-1)
= -1
3 F. Question
Find the value of following expression:
1 + i
2
+ i
4
+ i
6
+ i
8
+ ... + i
20
Answer
1 + i
2
+ i
4
+ i
6
+ i
8
+ ... + i
20
= 1 + (– 1) + 1 + (– 1) + 1 + ... + 1
= 1
3 G. Question
Find the value of following expression:
(1 + i)
6
+ (1 – i)
3
Answer
(1 + i)
6
+ (1 – i)
3
= {(1 + i)
2
}
3
+ (1 – i)
2
(1 – i)
= {1 + i
2
+ 2i}
3
+ (1 + i
2 –
2i)(1 – i)
= {1 – 1 + 2i}
3
+ (1 – 1 – 2i)(1 – i)
= (2i)
3
+ (– 2i)(1 – i)
= 8i
3
+ (– 2i) + 2i
2
[since i
3
= – i, i
2
= – 1]
= – 8i – 2i – 2
= – 10 i – 2
= – 2(1 + 5i)
Exercise 13.2
1 A. Question
Express the following complex numbers in the standard form a + i b :
(1 + i) (1 + 2i)
Answer
Given:
? a+ib = (1+i)(1+2i)
? a+ib = 1(1+2i)+i(1+2i)
? a+ib = 1+2i+i+2i
2
We know that i
2
=-1
? a+ib = 1+3i+2(-1)
? a+ib = 1+3i-2
? a+ib=-1+3i
? The values of a, b are -1, 3.
1 B. Question
Express the following complex numbers in the standard form a + i b :
Answer
Given:
?
Multiplying and dividing with -2-i
?
?
We know that i
2
=-1
Page 5
13. Complex Numbers
Exercise 13.1
1. Question
Evaluate the following:
(i) i
457
(ii) i
528
(iii)
(iv)
(v)
(vi) (i
77
+ i
70
+ i
87
+ i
414
)
3
(vii) (vii) i
30
+ i
40
+ i
60
(viii) i
49
+ i
68
+ i
89
+ i
118
Answer
i. i
457
= i
(456 + 1)
= i
4(114)
× i
= (1)
114
× i = i since i
4
= 1
ii. i
528
= i
4(132)
= (1)
132
=1 since i
4
= 1
iii.
since i
4
= 1
= – 1 since i
2
= – 1
iv.
[since i
4
= 1]
v.
= (i – i) = 0
[since ]
vi. (i
77
+ i
70
+ i
87
+ i
414
)
3
= (i
(76 + 1)
+ i
(68 + 2)
+ i
(84 + 3)
+ i
(412 + 2)
)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= (i + i
2
+ i
3
+ i
2
)
3
[since i
3
= – i, i
2
= – 1]
= (i + (– 1) + (– i) + (– 1))
3
= (– 2)
3
(i
77
+ i
70
+ i
87
+ i
414
)
3
= –8
vii. i
30
+ i
40
+ i
60
= i
(28 + 2) +
i
40
+ i
60
= (i
4
)
7
i
2
+ (i
4
)
10
+ (i
4
)
15
= i
2
+ 1
10
+ 1
15
= – 1 + 1 + 1 = 1
viii. i
49
+ i
68
+ i
89
+ i
118
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(116 + 2)
= (i
4
)
12
×i + (i
4
)
17
+ (i
4
)
11
×i + (i
4
)
29
×i
2
= i + 1 + i – 1 = 2i
2. Question
Show that 1 + i
10
+ i
20
+ i
30
is a real number ?
Answer
1 + i
10
+ i
20
+ i
30
= 1 + i
(8 + 2) +
i
20
+ i
(28 + 2)
= 1 + (i
4
)
2
× i
2
+ (i
4
)
5
+ (i
4
)
7
× i
2
= 1 – 1 + 1 – 1 = 0
[ since i
4
= 1, i
2
= – 1]
Hence , 1 + i
10
+ i
20
+ i
30
is a real number.
3 A. Question
Find the value of following expression:
i
49
+ i
68
+ i
89
+ i
110
Answer
i
49
+ i
68
+ i
89
+ i
110
= i
(48 + 1)
+ i
68
+ i
(88 + 1)
+ i
(108 + 2)
= (i
4
)
12
× i + (i
4
)
17
+ (i
4
)
11
× i + (i
4
)
27
× i
2
= i + 1 + i – 1 = 2i
[since i
4
= 1, i
2
= – 1]
i
49
+ i
68
+ i
89
+ i
110
= 2i
3 B. Question
Find the value of following expression:
i
30
+ i
80
+ i
120
Answer
i
30
+ i
80
+ i
120
= i
(28 + 2)
+ i
80
+ i
120
= (i
4
)
7
× i
2
+ (i
4
)
20
+ (i
4
)
30
= – 1 + 1 + 1 = 1
[since i
4
= 1, i
2
= – 1]
i
30
+ i
80
+ i
120
= 1
3 C. Question
Find the value of following expression:
i + i
2
+ i
3
+ i
4
Answer
i + i
2
+ i
3
+ i
4
= i + i
2
+ i
2
×i + i
4
= i – 1 + (– 1)×i + 1
since i
4
= 1, i
2
= – 1
= i – 1 – i + 1 = 0
3 D. Question
Find the value of following expression:
i
5
+ i
10
+ i
15
Answer
i
5
+ i
10
+ i
15
= i
(4 + 1)
+ i
(8 + 2)
+ i
(12 + 3)
= (i
4
)
1
×i + (i
4
)
2
×i
2
+ (i
4
)
3
×i
3
= (i
4
)
1
×i + (i
4
)
2
×i
2
+ (i
4
)
3
×i
2
×i
= 1×i + 1×(– 1) + 1×(– 1)×i
= i – 1 – i = – 1
3 E. Question
Find the value of following expression:
Answer
= i
10
= i
8
i
2
= (i
4
)
2
i
2
Since i
4
= 1, i
2
= -1
= (1)
2
(-1)
= -1
3 F. Question
Find the value of following expression:
1 + i
2
+ i
4
+ i
6
+ i
8
+ ... + i
20
Answer
1 + i
2
+ i
4
+ i
6
+ i
8
+ ... + i
20
= 1 + (– 1) + 1 + (– 1) + 1 + ... + 1
= 1
3 G. Question
Find the value of following expression:
(1 + i)
6
+ (1 – i)
3
Answer
(1 + i)
6
+ (1 – i)
3
= {(1 + i)
2
}
3
+ (1 – i)
2
(1 – i)
= {1 + i
2
+ 2i}
3
+ (1 + i
2 –
2i)(1 – i)
= {1 – 1 + 2i}
3
+ (1 – 1 – 2i)(1 – i)
= (2i)
3
+ (– 2i)(1 – i)
= 8i
3
+ (– 2i) + 2i
2
[since i
3
= – i, i
2
= – 1]
= – 8i – 2i – 2
= – 10 i – 2
= – 2(1 + 5i)
Exercise 13.2
1 A. Question
Express the following complex numbers in the standard form a + i b :
(1 + i) (1 + 2i)
Answer
Given:
? a+ib = (1+i)(1+2i)
? a+ib = 1(1+2i)+i(1+2i)
? a+ib = 1+2i+i+2i
2
We know that i
2
=-1
? a+ib = 1+3i+2(-1)
? a+ib = 1+3i-2
? a+ib=-1+3i
? The values of a, b are -1, 3.
1 B. Question
Express the following complex numbers in the standard form a + i b :
Answer
Given:
?
Multiplying and dividing with -2-i
?
?
We know that i
2
=-1
?
?
?
? The values of a, b are .
1 C. Question
Express the following complex numbers in the standard form a + i b :
Answer
Given:
?
?
We know that i
2
=-1
?
?
Multiplying and diving with 3-4i
?
?
?
?
?
? The values of a, b is .
1 D. Question
Express the following complex numbers in the standard form a + i b :
Answer
Given:
?
Multiplying and dividing by 1-i
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