RD Chapter 13 Complex Numbers Ex 13.1 Solutions
Question - 1 : - Evaluate the following:
(i) i 457
(ii) i 528
(iii) 1/ i58
(iv) i 37 + 1/i 67
(v) [i 41 + 1/i 257]
(vi) (i 77 + i 70 + i 87 +i 414)3
(vii) i 30 + i 40 + i 60
(viii) i 49 + i 68 + i 89 +i 110
Answer - 1 : -
(i) i 457
Let us simplify weget,
i457 =i (456 + 1)
= i 4(114) ×i
= (1)114 ×i
= i [since i4 =1]
(ii) i 528
Let us simplify weget,
i 528 =i 4(132)
= (1)132
= 1 [since i4 =1]
(iii) 1/ i58
Let us simplify weget,
1/ i58 =1/ i 56+2
= 1/ i 56 ×i2
= 1/ (i4)14 ×i2
= 1/ i2 [since,i4 = 1]
= 1/-1 [since, i2 =-1]
= -1
(iv) i 37 +1/i 67
Let us simplify weget,
i 37 +1/i 67 = i36+1 + 1/ i64+3
= i + 1/i3 [since,i4 = 1]
= i + i/i4
= i + i
= 2i
(v) [i 41 +1/i 257]
Let us simplify weget,
[i 41 +1/i 257] = [i40+1 + 1/ i256+1]
= [i + 1/i]9 [since,1/i = -1]
= [i – i]
= 0
(vi) (i 77 +i 70 + i 87 + i 414)3
Let us simplify weget,
(i 77 +i 70 + i 87 + i 414)3 =(i(76 + 1) + i(68 + 2) + i(84 + 3) +i(412 + 2) ) 3
= (i + i2 +i3 + i2)3 [since i3 =– i, i2 = – 1]
= (i + (– 1) + (– i) +(– 1)) 3
= (– 2)3
= – 8
(vii) i 30 +i 40 + i 60
Let us simplify weget,
i 30 +i 40 + i 60 = i(28 + 2) + i40 +i60
= (i4)7 i2 +(i4)10 + (i4)15
= i2 +110 + 115
= – 1 + 1 + 1
= 1
(viii) i 49 +i 68 + i 89 + i 110
Let us simplify weget,
i 49 +i 68 + i 89 + i 110 =i(48 + 1) + i68 + i(88 + 1) + i(116+ 2)
= (i4)12×i+ (i4)17 + (i4)22×i + (i4)29×i2
= i + 1 + i – 1
= 2i
Question - 2 : - Show that 1 + i10 + i20 + i30 isa real number?
Answer - 2 : -
Given:
1 + i10 +i20 + i30 = 1 + i(8 + 2) + i20 +i(28 + 2)
= 1 + (i4)2 ×i2 + (i4)5 + (i4)7 ×i2
= 1 – 1 + 1 – 1[since, i4 = 1, i2 = – 1]
= 0
Hence , 1 + i10 +i20 + i30 is a real number.
Question - 3 : - Find the values of the following expressions:
(i) i49 + i68 + i89 + i110
(ii) i30 + i80 + i120
(iii) i + i2 + i3 + i4
(iv) i5 + i10 + i15
(v) [i592 + i590 + i588 +i586 + i584] / [i582 + i580 +i578 + i576 + i574]
(vi) 1 + i2 + i4 + i6 + i8 +… + i20
(vii) (1 + i)6 + (1 – i)3
Answer - 3 : -
(i) i49 +i68 + i89 + i110
Let us simplify weget,
i49 +i68 + i89 + i110 = i (48+ 1) + i68 + i(88 + 1) + i(108+ 2)
= (i4)12 ×i + (i4)17 + (i4)22 × i +(i4)27 × i2
= i + 1 + i – 1 [sincei4 = 1, i2 = – 1]
= 2i
∴ i49 +i68 + i89 + i110 = 2i
(ii) i30 +i80 + i120
Let us simplify weget,
i30 +i80 + i120 = i(28 + 2) + i80 +i120
= (i4)7 ×i2 + (i4)20 + (i4)30
= – 1 + 1 + 1 [since i4 =1, i2 = – 1]
= 1
∴ i30 +i80 + i120 = 1
(iii) i + i2 +i3 + i4
Let us simplify weget,
i + i2 +i3 + i4 = i + i2 + i2×i+ i4
= i – 1 + (– 1) × i +1 [since i4 = 1, i2 = – 1]
= i – 1 – i + 1
= 0
∴ i + i2 +i3 + i4 = 0
(iv) i5 + i10 +i15
Let us simplify weget,
i5 + i10 +i15 = i(4 + 1) + i(8 + 2) + i(12+ 3)
= (i4)1×i+ (i4)2×i2 + (i4)3×i3
= (i4)1×i+ (i4)2×i2 + (i4)3×i2×i
= 1×i + 1 × (– 1) + 1× (– 1)×i
= i – 1 – i
= – 1
∴ i5 +i10 + i15 = -1
(v) [i592 +i590 + i588 + i586 + i584]/ [i582 + i580 + i578 + i576 +i574]
Let us simplify weget,
[i592 +i590 + i588 + i586 + i584]/ [i582 + i580 + i578 + i576 +i574]
= [i10 (i582 +i580 + i578 + i576 + i574)/ (i582 + i580 + i578 + i576 +i574)]
= i10
= i8 i2
= (i4)2 i2
= (1)2 (-1) [since i4 = 1, i2 =-1]
= -1
∴ [i592 +i590 + i588 + i586 + i584]/ [i582 + i580 + i578 + i576 +i574] = -1
(vi) 1 + i2 +i4 + i6 + i8 + … + i20
Let us simplify weget,
1 + i2 +i4 + i6 + i8 + … + i20 =1 + (– 1) + 1 + (– 1) + 1 + … + 1
= 1
∴ 1 + i2 +i4 + i6 + i8 + … + i20 =1
(vii) (1 + i)6 +(1 – i)3
Let us simplify weget,
(1 + i)6 +(1 – i)3 = {(1 + i)2 }3 + (1 –i)2 (1 – i)
= {1 + i2 +2i}3 + (1 + i2 – 2i)(1 – i)
= {1 – 1 + 2i}3 +(1 – 1 – 2i)(1 – i)
= (2i)3 +(– 2i)(1 – i)
= 8i3 +(– 2i) + 2i2
= – 8i – 2i – 2 [sincei3 = – i, i2 = – 1]
= – 10 i – 2
= – 2(1 + 5i)
= – 2 – 10i
∴ (1 + i)6 +(1 – i)3 = – 2 – 10i