Class 11: Maths Chapter 13 solutions. Complete Class 11 Maths Chapter 13 Notes.
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RD Sharma Solutions for Class 11 Maths Chapter 13–Complex Numbers
RD Sharma 11th Maths Chapter 13, Class 11 Maths Chapter 13 solutions
EXERCISE 13.1 PAGE NO: 13.3
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
Solution:
(i) i 457
Let us simplify we get,
i457 = i (456 + 1)
= i 4(114) × i
= (1)114 × i
= i [since i4 = 1]
(ii) i 528
Let us simplify we get,
i 528 = i 4(132)
= (1)132
= 1 [since i4 = 1]
(iii) 1/ i58
Let us simplify we get,
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 we get,
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 we get,[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 we get,
(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 we get,
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 we get,
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
2. Show that 1 + i10 + i20 + i30 is a real number?
Solution:
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.
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
Solution:
(i) i49 + i68 + i89 + i110
Let us simplify we get,
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 [since i4 = 1, i2 = – 1]
= 2i
∴ i49 + i68 + i89 + i110 = 2i
(ii) i30 + i80 + i120
Let us simplify we get,
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 we get,
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 we get,
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 we get,[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 we get,
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 we get,
(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 [since i3 = – i, i2 = – 1]
= – 10 i – 2
= – 2(1 + 5i)
= – 2 – 10i
∴ (1 + i)6 + (1 – i)3 = – 2 – 10i
EXERCISE 13.2 PAGE NO: 13.31
1. Express the following complex numbers in the standard form a + ib:
(i) (1 + i) (1 + 2i)
(ii) (3 + 2i) / (-2 + i)
(iii) 1/(2 + i)2
(iv) (1 – i) / (1 + i)
(v) (2 + i)3 / (2 + 3i)
(vi) [(1 + i) (1 +√3i)] / (1 – i)
(vii) (2 + 3i) / (4 + 5i)
(viii) (1 – i)3 / (1 – i3)
(ix) (1 + 2i)-3
(x) (3 – 4i) / [(4 – 2i) (1 + i)]
(xi)
(xii) (5 +√2i) / (1-√2i)
Solution:
(i) (1 + i) (1 + 2i)
Let us simplify and express in the standard form of (a + ib),
(1 + i) (1 + 2i) = (1+i)(1+2i)
= 1(1+2i)+i(1+2i)
= 1+2i+i+2i2
= 1+3i+2(-1) [since, i2 = -1]
= 1+3i-2
= -1+3i
∴ The values of a, b are -1, 3.
(ii) (3 + 2i) / (-2 + i)
Let us simplify and express in the standard form of (a + ib),
(3 + 2i) / (-2 + i) = [(3 + 2i) / (-2 + i)] × (-2-i) / (-2-i) [multiply and divide with (-2-i)]
= [3(-2-i) + 2i (-2-i)] / [(-2)2 – (i)2]
= [-6 -3i – 4i -2i2] / (4-i2)
= [-6 -7i -2(-1)] / (4 – (-1)) [since, i2 = -1]
= [-4 -7i] / 5
∴ The values of a, b are -4/5, -7/5
(iii) 1/(2 + i)2
Let us simplify and express in the standard form of (a + ib),
1/(2 + i)2 = 1/(22 + i2 + 2(2) (i))
= 1/ (4 – 1 + 4i) [since, i2 = -1]
= 1/(3 + 4i) [multiply and divide with (3 – 4i)]
= 1/(3 + 4i) × (3 – 4i)/ (3 – 4i)]
= (3-4i)/ (32 – (4i)2)
= (3-4i)/ (9 – 16i2)
= (3-4i)/ (9 – 16(-1)) [since, i2 = -1]
= (3-4i)/25
∴ The values of a, b are 3/25, -4/25
(iv) (1 – i) / (1 + i)
Let us simplify and express in the standard form of (a + ib),
(1 – i) / (1 + i) = (1 – i) / (1 + i) × (1-i)/(1-i) [multiply and divide with (1-i)]
= (12 + i2 – 2(1)(i)) / (12 – i2)
= (1 + (-1) -2i) / (1 – (-1))
= -2i/2
= -i
∴ The values of a, b are 0, -1
(v) (2 + i)3 / (2 + 3i)
Let us simplify and express in the standard form of (a + ib),
(2 + i)3 / (2 + 3i) = (23 + i3 + 3(2)2(i) + 3(i)2(2)) / (2 + 3i)
= (8 + (i2.i) + 3(4)(i) + 6i2) / (2 + 3i)
= (8 + (-1)i + 12i + 6(-1)) / (2 + 3i)
= (2 + 11i) / (2 + 3i)[multiply and divide with (2-3i)]
= (2 + 11i)/(2 + 3i) × (2-3i)/(2-3i)
= [2(2-3i) + 11i(2-3i)] / (22 – (3i)2)
= (4 – 6i + 22i – 33i2) / (4 – 9i2)
= (4 + 16i – 33(-1)) / (4 – 9(-1)) [since, i2 = -1]
= (37 + 16i) / 13
∴ The values of a, b are 37/13, 16/13
(vi) [(1 + i) (1 +√3i)] / (1 – i)
Let us simplify and express in the standard form of (a + ib),[(1 + i) (1 +√3i)] / (1 – i) = [1(1+√3i) + i(1+√3i)] / (1-i)
= (1 + √3i + i + √3i2) / (1 – i)
= (1 + (√3+1)i + √3(-1)) / (1-i) [since, i2 = -1]
= [(1-√3) + (1+√3)i] / (1-i)[multiply and divide with (1+i)]
= [(1-√3) + (1+√3)i] / (1-i) × (1+i)/(1+i)
= [(1-√3) (1+i) + (1+√3)i(1+i)] / (12 – i2)
= [1-√3+ (1-√3)i + (1+√3)i + (1+√3)i2] / (1-(-1)) [since, i2 = -1]
= [(1-√3)+(1-√3+1+√3)i+(1+√3)(-1)] / 2
= (-2√3 + 2i) / 2
= -√3 + i
∴ The values of a, b are -√3, 1
(vii) (2 + 3i) / (4 + 5i)
Let us simplify and express in the standard form of (a + ib),
(2 + 3i) / (4 + 5i) = [multiply and divide with (4-5i)]
= (2 + 3i) / (4 + 5i) × (4-5i)/(4-5i)
= [2(4-5i) + 3i(4-5i)] / (42 – (5i)2)
= [8 – 10i + 12i – 15i2] / (16 – 25i2)
= [8+2i-15(-1)] / (16 – 25(-1)) [since, i2 = -1]
= (23 + 2i) / 41
∴ The values of a, b are 23/41, 2/41
(viii) (1 – i)3 / (1 – i3)
Let us simplify and express in the standard form of (a + ib),
(1 – i)3 / (1 – i3) = [13 – 3(1)2i + 3(1)(i)2 – i3] / (1-i2.i)
= [1 – 3i + 3(-1)-i2.i] / (1 – (-1)i) [since, i2 = -1]
= [-2 – 3i – (-1)i] / (1+i)
= [-2-4i] / (1+i)[Multiply and divide with (1-i)]
= [-2-4i] / (1+i) × (1-i)/(1-i)
= [-2(1-i)-4i(1-i)] / (12 – i2)
= [-2+2i-4i+4i2] / (1 – (-1))
= [-2-2i+4(-1)] /2
= (-6-2i)/2
= -3 – i
∴ The values of a, b are -3, -1
(ix) (1 + 2i)-3
Let us simplify and express in the standard form of (a + ib),
(1 + 2i)-3 = 1/(1 + 2i)3
= 1/(13+3(1)2 (2i)+2(1)(2i)2 + (2i)3)
= 1/(1+6i+4i2+8i3)
= 1/(1+6i+4(-1)+8i2.i) [since, i2 = -1]
= 1/(-3+6i+8(-1)i) [since, i2 = -1]
= 1/(-3-2i)
= -1/(3+2i)[Multiply and divide with (3-2i)]
= -1/(3+2i) × (3-2i)/(3-2i)
= (-3+2i)/(32 – (2i)2)
= (-3+2i) / (9-4i2)
= (-3+2i) / (9-4(-1))
= (-3+2i) /13
∴ The values of a, b are -3/13, 2/13
(x) (3 – 4i) / [(4 – 2i) (1 + i)]
Let us simplify and express in the standard form of (a + ib),
(3 – 4i) / [(4 – 2i) (1 + i)] = (3-4i)/ [4(1+i)-2i(1+i)]
= (3-4i)/ [4+4i-2i-2i2]
= (3-4i)/ [4+2i-2(-1)] [since, i2 = -1]
= (3-4i)/ (6+2i)[Multiply and divide with (6-2i)]
= (3-4i)/ (6+2i) × (6-2i)/(6-2i)
= [3(6-2i)-4i(6-2i)] / (62 – (2i)2)
= [18 – 6i – 24i + 8i2] / (36 – 4i2)
= [18 – 30i + 8 (-1)] / (36 – 4 (-1)) [since, i2 = -1]
= [10-30i] / 40
= (1 – 3i) / 4
∴ The values of a, b are 1/4, -3/4
(xi)
(xii) (5 +√2i) / (1-√2i)
Let us simplify and express in the standard form of (a + ib),
(5 +√2i) / (1-√2i) = [Multiply and divide with (1+√2i)]
= (5 +√2i) / (1-√2i) × (1+√2i)/(1+√2i)
= [5(1+√2i) + √2i(1+√2i)] / (12 – (√2)2)
= [5+5√2i + √2i + 2i2] / (1 – 2i2)
= [5 + 6√2i + 2(-1)] / (1-2(-1)) [since, i2 = -1]
= [3+6√2i]/3
= 1+ 2√2i
∴ The values of a, b are 1, 2√2
2. Find the real values of x and y, if
(i) (x + iy) (2 – 3i) = 4 + i
(ii) (3x – 2i y) (2 + i)2 = 10(1 + i)
(iv) (1 + i) (x + iy) = 2 – 5i
Solution:
(i) (x + iy) (2 – 3i) = 4 + i
Given:
(x + iy) (2 – 3i) = 4 + i
Let us simplify the expression we get,
x(2 – 3i) + iy(2 – 3i) = 4 + i
2x – 3xi + 2yi – 3yi2 = 4 + i
2x + (-3x+2y)i – 3y (-1) = 4 + i [since, i2 = -1]
2x + (-3x+2y)i + 3y = 4 + i [since, i2 = -1]
(2x+3y) + i(-3x+2y) = 4 + i
Equating Real and Imaginary parts on both sides, we get
2x+3y = 4… (i)
And -3x+2y = 1… (ii)
Multiply (i) by 3 and (ii) by 2 and add
On solving we get,
6x – 6x – 9y + 4y = 12 + 2
13y = 14
y = 14/13
Substitute the value of y in (i) we get,
2x+3y = 4
2x + 3(14/13) = 4
2x = 4 – (42/13)
= (52-42)/13
2x = 10/13
x = 5/13
x = 5/13, y = 14/13
∴ The real values of x and y are 5/13, 14/13
(ii) (3x – 2i y) (2 + i)2 = 10(1 + i)
Given:
(3x – 2iy) (2+i)2 = 10(1+i)
(3x – 2yi) (22+i2+2(2)(i)) = 10+10i
(3x – 2yi) (4 + (-1)+4i) = 10+10i [since, i2 = -1]
(3x – 2yi) (3+4i) = 10+10i
Let us divide with 3+4i on both sides we get,
(3x – 2yi) = (10+10i)/(3+4i)
= Now multiply and divide with (3-4i)
= [10(3-4i) + 10i(3-4i)] / (32 – (4i)2)
= [30-40i+30i-40i2] / (9 – 16i2)
= [30-10i-40(-1)] / (9-16(-1))
= [70-10i]/25
Now, equating Real and Imaginary parts on both sides we get
3x = 70/25 and -2y = -10/25
x = 70/75 and y = 1/5
x = 14/15 and y = 1/5
∴ The real values of x and y are 14/15, 1/5
(4+2i) x-3i-3 + (9-7i)y = 10i
(4x+9y-3) + i(2x-7y-3) = 10i
Now, equating Real and Imaginary parts on both sides we get,
4x+9y-3 = 0 … (i)
And 2x-7y-3 = 10
2x-7y = 13 … (ii)
Multiply (i) by 7 and (ii) by 9 and add
On solving these equations we get
28x + 18x + 63y – 63y = 117 + 21
46x = 117 + 21
46x = 138
x = 138/46
= 3
Substitute the value of x in (i) we get,
4x+9y-3 = 0
9y = -9
y = -9/9
= -1
x = 3 and y = -1
∴ The real values of x and y are 3 and -1
(iv) (1 + i) (x + iy) = 2 – 5i
Given:
(1 + i) (x + iy) = 2 – 5i
Divide with (1+i) on both the sides we get,
(x + iy) = (2 – 5i)/(1+i)
Multiply and divide by (1-i)
= (2 – 5i)/(1+i) × (1-i)/(1-i)
= [2(1-i) – 5i (1-i)] / (12 – i2)
= [2 – 7i + 5(-1)] / 2 [since, i2 = -1]
= (-3-7i)/2
Now, equating Real and Imaginary parts on both sides we get
x = -3/2 and y = -7/2
∴ Thee real values of x and y are -3/2, -7/2
3. Find the conjugates of the following complex numbers:
(i) 4 – 5i
(ii) 1 / (3 + 5i)
(iii) 1 / (1 + i)
(iv) (3 – i)2 / (2 + i)
(v) [(1 + i) (2 + i)] / (3 + i)
(vi) [(3 – 2i) (2 + 3i)] / [(1 + 2i) (2 – i)]
Solution:
(i) 4 – 5i
Given:
4 – 5i
We know the conjugate of a complex number (a + ib) is (a – ib)
So,
∴ The conjugate of (4 – 5i) is (4 + 5i)
(ii) 1 / (3 + 5i)
Given:
1 / (3 + 5i)
Since the given complex number is not in the standard form of (a + ib)
Let us convert to standard form by multiplying and dividing with (3 – 5i)
We get,
We know the conjugate of a complex number (a + ib) is (a – ib)
So,
∴ The conjugate of (3 – 5i)/34 is (3 + 5i)/34
(iii) 1 / (1 + i)
Given:
1 / (1 + i)
Since the given complex number is not in the standard form of (a + ib)
Let us convert to standard form by multiplying and dividing with (1 – i)
We get,
We know the conjugate of a complex number (a + ib) is (a – ib)
So,
∴ The conjugate of (1-i)/2 is (1+i)/2
(iv) (3 – i)2 / (2 + i)
Given:
(3 – i)2 / (2 + i)
Since the given complex number is not in the standard form of (a + ib)
Let us convert to standard form,
We know the conjugate of a complex number (a + ib) is (a – ib)
So,
∴ The conjugate of (2 – 4i) is (2 + 4i)
(v) [(1 + i) (2 + i)] / (3 + i)
Given:[(1 + i) (2 + i)] / (3 + i)
Since the given complex number is not in the standard form of (a + ib)
Let us convert to standard form,
We know the conjugate of a complex number (a + ib) is (a – ib)
So,
∴ The conjugate of (3 + 4i)/5 is (3 – 4i)/5
(vi) [(3 – 2i) (2 + 3i)] / [(1 + 2i) (2 – i)]
Given:[(3 – 2i) (2 + 3i)] / [(1 + 2i) (2 – i)]
Since the given complex number is not in the standard form of (a + ib)
Let us convert to standard form,
We know the conjugate of a complex number (a + ib) is (a – ib)
So,
∴ The conjugate of (63 – 16i)/25 is (63 + 16i)/25
4. Find the multiplicative inverse of the following complex numbers:
(i) 1 – i
(ii) (1 + i √3)2
(iii) 4 – 3i
(iv) √5 + 3i
Solution:
(i) 1 – i
Given:
1 – i
We know the multiplicative inverse of a complex number (Z) is Z-1 or 1/Z
So,
∴ The multiplicative inverse of (1 – i) is (1 + i)/2
(ii) (1 + i √3)2
Given:
(1 + i √3)2
Z = (1 + i √3)2
= 12 + (i √3)2 + 2 (1) (i√3)
= 1 + 3i2 + 2 i√3
= 1 + 3(-1) + 2 i√3 [since, i2 = -1]
= 1 – 3 + 2 i√3
= -2 + 2 i√3
We know the multiplicative inverse of a complex number (Z) is Z-1 or 1/Z
So,
Z = -2 + 2 i√3
∴ The multiplicative inverse of (1 + i√3)2 is (-1-i√3)/8
(iii) 4 – 3i
Given:
4 – 3i
We know the multiplicative inverse of a complex number (Z) is Z-1 or 1/Z
So,
Z = 4 – 3i
∴ The multiplicative inverse of (4 – 3i) is (4 + 3i)/25
(iv) √5 + 3i
Given:
√5 + 3i
We know the multiplicative inverse of a complex number (Z) is Z-1 or 1/Z
So,
Z = √5 + 3i
∴ The multiplicative inverse of (√5 + 3i) is (√5 – 3i)/14
6. If z1 = (2 – i), z2 = (-2 + i), find
Solution:
Given:
z1 = (2 – i) and z2 = (-2 + i)
7. Find the modulus of [(1 + i)/(1 – i)] – [(1 – i)/(1 + i)]
Solution:
Given:[(1 + i)/(1 – i)] – [(1 – i)/(1 + i)]
So,
Z = [(1 + i)/(1 – i)] – [(1 – i)/(1 + i)]
Let us simplify, we get
= [(1+i) (1+i) – (1-i) (1-i)] / (12 – i2)
= [12 + i2 + 2(1)(i) – (12 + i2 – 2(1)(i))] / (1 – (-1)) [Since, i2 = -1]
= 4i/2
= 2i
We know that for a complex number Z = (a+ib) it’s magnitude is given by |z| = √(a2 + b2)
So,
|Z| = √(02 + 22)
= 2
∴ The modulus of [(1 + i)/(1 – i)] – [(1 – i)/(1 + i)] is 2.
8. If x + iy = (a+ib)/(a-ib), prove that x2 + y2 = 1
Solution:
Given:
x + iy = (a+ib)/(a-ib)
We know that for a complex number Z = (a+ib) it’s magnitude is given by |z| = √(a2 + b2)
So,
|a/b| is |a| / |b|
Applying Modulus on both sides we get,
9. Find the least positive integral value of n for which [(1+i)/(1-i)]n is real.
Solution:
Given:[(1+i)/(1-i)]n
Z = [(1+i)/(1-i)]n
Now let us multiply and divide by (1+i), we get
= i [which is not real]
For n = 2, we have[(1+i)/(1-i)]2 = i2
= -1 [which is real]
So, the smallest positive integral ‘n’ that can make [(1+i)/(1-i)]n real is 2.
∴ The smallest positive integral value of ‘n’ is 2.
10. Find the real values of θ for which the complex number (1 + i cos θ) / (1 – 2i cos θ) is purely real.
Solution:
Given:
(1 + i cos θ) / (1 – 2i cos θ)
Z = (1 + i cos θ) / (1 – 2i cos θ)
Let us multiply and divide by (1 + 2i cos θ)
For a complex number to be purely real, the imaginary part should be equal to zero.
So,
3cos θ = 0 (since, 1 + 4cos2θ ≥ 1)
cos θ = 0
cos θ = cos π/2
θ = [(2n+1)π] / 2, for n ∈ Z
= 2nπ ± π/2, for n ∈ Z
∴ The values of θ to get the complex number to be purely real is 2nπ ± π/2, for n ∈ Z
11. Find the smallest positive integer value of n for which (1+i) n / (1-i) n-2 is a real number.
Solution:
Given:
(1+i) n / (1-i) n-2
Z = (1+i) n / (1-i) n-2
Let us multiply and divide by (1 – i)2
For n = 1,
Z = -2i1+1
= -2i2
= 2, which is a real number.
∴ The smallest positive integer value of n is 1.
12. If [(1+i)/(1-i)]3 – [(1-i)/(1+i)]3 = x + iy, find (x, y)
Solution:
Given:[(1+i)/(1-i)]3 – [(1-i)/(1+i)]3 = x + iy
Let us rationalize the denominator, we get
i3–(-i)3 = x + iy
2i3 = x + iy
2i2.i = x + iy
2(-1)I = x + iy
-2i = x + iy
Equating Real and Imaginary parts on both sides we get
x = 0 and y = -2
∴ The values of x and y are 0 and -2.
13. If (1+i)2 / (2-i) = x + iy, find x + y
Solution:
Given:
(1+i)2 / (2-i) = x + iy
Upon expansion we get,
Let us equate real and imaginary parts on both sides we get,
x = -2/5 and y = 4/5
so,
x + y = -2/5 + 4/5
= (-2+4)/5
= 2/5
∴ The value of (x + y) is 2/5
EXERCISE 13.3 PAGE NO: 13.39
1. Find the square root of the following complex numbers.
(i) – 5 + 12i
(ii) -7 – 24i
(iii) 1 – i
(iv) – 8 – 6i
(v) 8 – 15i
(vi) -11 – 60√-1
(vii) 1 + 4√-3
(viii) 4i
(ix) -i
Solution:
(i) – 5 + 12i
Given:
– 5 + 12i
We know, Z = a + ib
So, √(a + ib) = √(-5+12i)
Here, b > 0
Let us simplify now,
∴ Square root of (– 5 + 12i) is ±[2 + 3i]
(ii) -7 – 24i
Given:
-7 – 24i
We know, Z = -7 – 24i
So, √(a + ib) = √(-7 – 24i)
Here, b < 0
Let us simplify now,
∴ Square root of (-7 – 24i) is ± [3 – 4i]
(iii) 1 – i
Given:
1 – i
We know, Z = (1 – i)
So, √(a + ib) = √(1 – i)
Here, b < 0
Let us simplify now,
∴ Square root of (1 – i) is ± [(√(√2+1)/2) – i (√(√2-1)/2)]
(iv) -8 -6i
Given:
-8 -6i
We know, Z = -8 -6i
So, √(a + ib) = -8 -6i
Here, b < 0
Let us simplify now,
= [11/2 – i 91/2]
= ± [1 – 3i]
∴ Square root of (-8 -6i) is ± [1 – 3i]
(v) 8 – 15i
Given:
8 – 15i
We know, Z = 8 – 15i
So, √(a + ib) = 8 – 15i
Here, b < 0
Let us simplify now,
∴ Square root of (8 – 15i) is ± 1/√2 (5 – 3i)
(vi) -11 – 60√-1
Given:
-11 – 60√-1
We know, Z = -11 – 60√-1
So, √(a + ib) = -11 – 60√-1
= -11 – 60i
Here, b < 0
Let us simplify now,
∴ Square root of (-11 – 60√-1) is ± (5 – 6i)
(vii) 1 + 4√-3
Given:
1 + 4√-3
We know, Z = 1 + 4√-3
So, √(a + ib) = 1 + 4√-3
= 1 + 4(√3) (√-1)
= 1 + 4√3i
Here, b > 0
Let us simplify now,
∴ Square root of (1 + 4√-3) is ± (2 + √3i)
(viii) 4i
Given:
4i
We know, Z = 4i
So, √(a + ib) = 4i
Here, b > 0
Let us simplify now,
∴ Square root of 4i is ± √2 (1 + i)
(ix) –i
Given:
-i
We know, Z = -i
So, √(a + ib) = -i
Here, b < 0
Let us simplify now,
∴ Square root of –i is ± 1/√2 (1 – i)
EXERCISE 13.4 PAGE NO: 13.57
1. Find the modulus and arguments of the following complex numbers and hence express each of them in the polar form:
(i) 1 + i
(ii) √3 + i
(iii) 1 – i
(iv) (1 – i) / (1 + i)
(v) 1/(1 + i)
(vi) (1 + 2i) / (1 – 3i)
(vii) sin 120o – i cos 120o
(viii) -16 / (1 + i√3)
Solution:
We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos θ + i sin θ)
Where,
|Z| = modulus of complex number = √(x2 + y2)
θ = arg (z) = argument of complex number = tan-1 (|y| / |x|)
(i) 1 + i
Given: Z = 1 + i
So now,
|Z| = √(x2 + y2)
= √(12 + 12)
= √(1 + 1)
= √2
θ = tan-1 (|y| / |x|)
= tan-1 (1 / 1)
= tan-1 1
Since x > 0, y > 0 complex number lies in 1st quadrant and the value of θ is 00≤θ≤900.
θ = π/4
Z = √2 (cos (π/4) + i sin (π/4))
∴ Polar form of (1 + i) is √2 (cos (π/4) + i sin (π/4))
(ii) √3 + i
Given: Z = √3 + i
So now,
|Z| = √(x2 + y2)
= √((√3)2 + 12)
= √(3 + 1)
= √4
= 2
θ = tan-1 (|y| / |x|)
= tan-1 (1 / √3)
Since x > 0, y > 0 complex number lies in 1st quadrant and the value of θ is 00≤θ≤900.
θ = π/6
Z = 2 (cos (π/6) + i sin (π/6))
∴ Polar form of (√3 + i) is 2 (cos (π/6) + i sin (π/6))
(iii) 1 – i
Given: Z = 1 – i
So now,
|Z| = √(x2 + y2)
= √(12 + (-1)2)
= √(1 + 1)
= √2
θ = tan-1 (|y| / |x|)
= tan-1 (1 / 1)
= tan-1 1
Since x > 0, y < 0 complex number lies in 4th quadrant and the value of θ is -900≤θ≤00.
θ = -π/4
Z = √2 (cos (-π/4) + i sin (-π/4))
= √2 (cos (π/4) – i sin (π/4))
∴ Polar form of (1 – i) is √2 (cos (π/4) – i sin (π/4))
(iv) (1 – i) / (1 + i)
Given: Z = (1 – i) / (1 + i)
Let us multiply and divide by (1 – i), we get
= 0 – i
So now,
|Z| = √(x2 + y2)
= √(02 + (-1)2)
= √(0 + 1)
= √1
θ = tan-1 (|y| / |x|)
= tan-1 (1 / 0)
= tan-1 ∞
Since x ≥ 0, y < 0 complex number lies in 4th quadrant and the value of θ is -900≤θ≤00.
θ = -π/2
Z = 1 (cos (-π/2) + i sin (-π/2))
= 1 (cos (π/2) – i sin (π/2))
∴ Polar form of (1 – i) / (1 + i) is 1 (cos (π/2) – i sin (π/2))
(v) 1/(1 + i)
Given: Z = 1 / (1 + i)
Let us multiply and divide by (1 – i), we get
So now,
|Z| = √(x2 + y2)
= √((1/2)2 + (-1/2)2)
= √(1/4 + 1/4)
= √(2/4)
= 1/√2
θ = tan-1 (|y| / |x|)
= tan-1 ((1/2) / (1/2))
= tan-1 1
Since x > 0, y < 0 complex number lies in 4th quadrant and the value of θ is -900≤θ≤00.
θ = -π/4
Z = 1/√2 (cos (-π/4) + i sin (-π/4))
= 1/√2 (cos (π/4) – i sin (π/4))
∴ Polar form of 1/(1 + i) is 1/√2 (cos (π/4) – i sin (π/4))
(vi) (1 + 2i) / (1 – 3i)
Given: Z = (1 + 2i) / (1 – 3i)
Let us multiply and divide by (1 + 3i), we get
So now,
|Z| = √(x2 + y2)
= √((-1/2)2 + (1/2)2)
= √(1/4 + 1/4)
= √(2/4)
= 1/√2
θ = tan-1 (|y| / |x|)
= tan-1 ((1/2) / (1/2))
= tan-1 1
Since x < 0, y > 0 complex number lies in 2nd quadrant and the value of θ is 900≤θ≤1800.
θ = 3π/4
Z = 1/√2 (cos (3π/4) + i sin (3π/4))
∴ Polar form of (1 + 2i) / (1 – 3i) is 1/√2 (cos (3π/4) + i sin (3π/4))
(vii) sin 120o – i cos 120o
Given: Z = sin 120o – i cos 120o
= √3/2 – i (-1/2)
= √3/2 + i (1/2)
So now,
|Z| = √(x2 + y2)
= √((√3/2)2 + (1/2)2)
= √(3/4 + 1/4)
= √(4/4)
= √1
= 1
θ = tan-1 (|y| / |x|)
= tan-1 ((1/2) / (√3/2))
= tan-1 (1/√3)
Since x > 0, y > 0 complex number lies in 1st quadrant and the value of θ is 00≤θ≤900.
θ = π/6
Z = 1 (cos (π/6) + i sin (π/6))
∴ Polar form of √3/2 + i (1/2) is 1 (cos (π/6) + i sin (π/6))
(viii) -16 / (1 + i√3)
Given: Z = -16 / (1 + i√3)
Let us multiply and divide by (1 – i√3), we get
So now,
|Z| = √(x2 + y2)
= √((-4)2 + (4√3)2)
= √(16 + 48)
= √(64)
= 8
θ = tan-1 (|y| / |x|)
= tan-1 ((4√3) / 4)
= tan-1 (√3)
Since x < 0, y > 0 complex number lies in 2nd quadrant and the value of θ is 900≤θ≤1800.
θ = 2π/3
Z = 8 (cos (2π/3) + i sin (2π/3))
∴ Polar form of -16 / (1 + i√3) is 8 (cos (2π/3) + i sin (2π/3))
2. Write (i25)3 in polar form.
Solution:
Given: Z = (i25)3
= i75
= i74. i
= (i2)37. i
= (-1)37. i
= (-1). i
= – i
= 0 – i
So now,
|Z| = √(x2 + y2)
= √(02 + (-1)2)
= √(0 + 1)
= √1
θ = tan-1 (|y| / |x|)
= tan-1 (1 / 0)
= tan-1 ∞
Since x ≥ 0, y < 0 complex number lies in 4th quadrant and the value of θ is -900≤θ≤00.
θ = -π/2
Z = 1 (cos (-π/2) + i sin (-π/2))
= 1 (cos (π/2) – i sin (π/2))
∴ Polar form of (i25)3 is 1 (cos (π/2) – i sin (π/2))
3. Express the following complex numbers in the form r (cos θ + i sin θ):
(i) 1 + i tan α
(ii) tan α – i
(iii) 1 – sin α + i cos α
(iv) (1 – i) / (cos π/3 + i sin π/3)
Solution:
(i) 1 + i tan α
Given: Z = 1 + i tan α
We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos θ + i sin θ)
Where,
|Z| = modulus of complex number = √(x2 + y2)
θ = arg (z) = argument of complex number = tan-1 (|y| / |x|)
We also know that tan α is a periodic function with period π.
So α is lying in the interval [0, π/2) ∪ (π/2, π].
Let us consider case 1:
α ∈ [0, π/2)
So now,
|Z| = r = √(x2 + y2)
= √(12 + tan2 α)
= √( sec2 α)
= |sec α| since, sec α is positive in the interval [0, π/2)
θ = tan-1 (|y| / |x|)
= tan-1 (tan α / 1)
= tan-1 (tan α)
= α since, tan α is positive in the interval [0, π/2)
∴ Polar form is Z = sec α (cos α + i sin α)
Let us consider case 2:
α ∈ (π/2, π]
So now,
|Z| = r = √(x2 + y2)
= √(12 + tan2 α)
= √( sec2 α)
= |sec α|
= – sec α since, sec α is negative in the interval (π/2, π]
θ = tan-1 (|y| / |x|)
= tan-1 (tan α / 1)
= tan-1 (tan α)
= -π + α since, tan α is negative in the interval (π/2, π]
θ = -π + α [since, θ lies in 4th quadrant]
Z = -sec α (cos (α – π) + i sin (α – π))
∴ Polar form is Z = -sec α (cos (α – π) + i sin (α – π))
(ii) tan α – i
Given: Z = tan α – i
We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos θ + i sin θ)
Where,
|Z| = modulus of complex number = √(x2 + y2)
θ = arg (z) = argument of complex number = tan-1 (|y| / |x|)
We also know that tan α is a periodic function with period π.
So α is lying in the interval [0, π/2) ∪ (π/2, π].
Let us consider case 1:
α ∈ [0, π/2)
So now,
|Z| = r = √(x2 + y2)
= √(tan2 α + 12)
= √( sec2 α)
= |sec α| since, sec α is positive in the interval [0, π/2)
= sec α
θ = tan-1 (|y| / |x|)
= tan-1 (1/tan α)
= tan-1 (cot α) since, cot α is positive in the interval [0, π/2)
= α – π/2 [since, θ lies in 4th quadrant]
Z = sec α (cos (α – π/2) + i sin (α – π/2))
∴ Polar form is Z = sec α (cos (α – π/2) + i sin (α – π/2))
Let us consider case 2:
α ∈ (π/2, π]
So now,
|Z| = r = √(x2 + y2)
= √(tan2 α + 12)
= √( sec2 α)
= |sec α|
= – sec α since, sec α is negative in the interval (π/2, π]
θ = tan-1 (|y| / |x|)
= tan-1 (1/tan α)
= tan-1 (cot α)
= π/2 + α since, cot α is negative in the interval (π/2, π]
θ = π/2 + α [since, θ lies in 3th quadrant]
Z = -sec α (cos (π/2 + α) + i sin (π/2 + α))
∴ Polar form is Z = -sec α (cos (π/2 + α) + i sin (π/2 + α))
(iii) 1 – sin α + i cos α
Given: Z = 1 – sin α + i cos α
By using the formulas,
Sin2 θ + cos2 θ = 1
Sin 2θ = 2 sin θ cos θ
Cos 2θ = cos2 θ – sin2 θ
So,
z= (sin2(α/2) + cos2(α/2) – 2 sin(α/2) cos(α/2)) + i (cos2(α/2) – sin2(α/2))
= (cos(α/2) – sin(α/2))2 + i (cos2(α/2) – sin2(α/2))
We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos θ + i sin θ)
Where,
|Z| = modulus of complex number = √(x2 + y2)
θ = arg (z) = argument of complex number = tan-1 (|y| / |x|)
Now,
We know that sine and cosine functions are periodic with period 2π
Here we have 3 intervals:
0 ≤ α ≤ π/2
π/2 ≤ α ≤ 3π/2
3π/2 ≤ α ≤ 2π
Let us consider case 1:
In the interval 0 ≤ α ≤ π/2
Cos (α/2) > sin (α/2) and also 0 < π/4 + α/2 < π/2
So,
∴ Polar form is Z = √2 (cos (α/2) – sin (α/2)) (cos (π/4 + α/2) + i sin (π/4 + α/2))
Let us consider case 2:
In the interval π/2 ≤ α ≤ 3π/2
Cos (α/2) < sin (α/2) and also π/2 < π/4 + α/2 < π
So,
Since, (1 – sin α) > 0 and cos α < 0 [Z lies in 4th quadrant]
= α/2 – 3π/4
∴ Polar form is Z = –√2 (cos (α/2) – sin (α/2)) (cos (α/2 – 3π/4) + i sin (α/2 – 3π/4))
Let us consider case 3:
In the interval 3π/2 ≤ α ≤ 2π
Cos (α/2) < sin (α/2) and also π < π/4 + α/2 < 5π/4
So,
θ = tan-1 (tan (π/4 + α/2))
= π – (π/4 + α/2) [since, θ lies in 1st quadrant and tan’s period is π]
= α/2 – 3π/4
∴ Polar form is Z = –√2 (cos (α/2) – sin (α/2)) (cos (α/2 – 3π/4) + i sin (α/2 – 3π/4))
(iv) (1 – i) / (cos π/3 + i sin π/3)
Given: Z = (1 – i) / (cos π/3 + i sin π/3)
Let us multiply and divide by (1 – i√3), we get
We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos θ + i sin θ)
Where,
|Z| = modulus of complex number = √(x2 + y2)
θ = arg (z) = argument of complex number = tan-1 (|y| / |x|)
Now,
Since x < 0, y < 0 complex number lies in 3rd quadrant and the value of θ is 1800≤θ≤-900.
= tan-1 (2 + √3)
= -7π/12
Z = √2 (cos (-7π/12) + i sin (-7π/12))
= √2 (cos (7π/12) – i sin (7π/12))
∴ Polar form of (1 – i) / (cos π/3 + i sin π/3) is √2 (cos (7π/12) – i sin (7π/12))
4. If z1 and z2 are two complex number such that |z1| = |z2| and arg (z1) + arg (z2) = π, then show that
Solution:
Given:
|z1| = |z2| and arg (z1) + arg (z2) = π
Let us assume arg (z1) = θ
arg (z2) = π – θ
We know that in the polar form, z = |z| (cos θ + i sin θ)
z1 = |z1| (cos θ + i sin θ) …………. (i)
z2 = |z2| (cos (π – θ) + i sin (π – θ))
= |z2| (-cos θ + i sin θ)
= – |z2| (cos θ – i sin θ)
Now let us find the conjugate of
= – |z2| (cos θ + i sin θ) …… (ii) (since, |¯¯¯¯¯¯Z2|=|Z2||Z2¯|=|Z2|)
Now,
z1 /
= [|z1| (cos θ + i sin θ)] / [-|z2| (cos θ + i sin θ)]
= – |z1| / |z2| [since, |z1| = |z2|]
= -1
When we cross multiply we get,
z1 = –
Hence proved.
5. If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that arg (z1/z4) + arg (z2/z3) = 0
Solution:
Given:
Hence proved.
6. Express sin π/5 + i (1 – cos π/5) in polar form.
Solution:
Given:
Z = sin π/5 + i (1 – cos π/5)
By using the formula,
sin 2θ = 2 sin θ cos θ
1- cos 2θ = 2 sin2 θ
So,
Z = 2 sin π/10 cos π/10 + i (2 sin2 π/10)
= 2 sin π/10 (cos π/10 + i sin π/10)
∴ The polar form of sin π/5 + i (1 – cos π/5) is 2 sin π/10 (cos π/10 + i sin π/10)
RD Sharma Solutions for Class 11 Maths Chapter 13: Download PDF
RD Sharma Solutions for Class 11 Maths Chapter 13–Complex Numbers
Download PDF: RD Sharma Solutions for Class 11 Maths Chapter 13–Complex Numbers PDF
Chapterwise RD Sharma Solutions for Class 11 Maths :
- Chapter 1–Sets
- Chapter 2–Relations
- Chapter 3–Functions
- Chapter 4–Measurement of Angles
- Chapter 5–Trigonometric Functions
- Chapter 6–Graphs of Trigonometric Functions
- Chapter 7–Values of Trigonometric Functions at Sum or Difference of Angles
- Chapter 8–Transformation Formulae
- Chapter 9–Values of Trigonometric Functions at Multiples and Submultiples of an Angle
- Chapter 10–Sine and Cosine Formulae and their Applications
- Chapter 11–Trigonometric Equations
- Chapter 12–Mathematical Induction
- Chapter 13–Complex Numbers
- Chapter 14–Quadratic Equations
- Chapter 15–Linear Inequations
- Chapter 16–Permutations
- Chapter 17–Combinations
- Chapter 18–Binomial Theorem
- Chapter 19–Arithmetic Progressions
- Chapter 20–Geometric Progressions
- Chapter 21–Some Special Series
- Chapter 22–Brief review of Cartesian System of Rectangular Coordinates
- Chapter 23–The Straight Lines
- Chapter 24–The Circle
- Chapter 25–Parabola
- Chapter 26–Ellipse
- Chapter 27–Hyperbola
- Chapter 28–Introduction to Three Dimensional Coordinate Geometry
- Chapter 29–Limits
- Chapter 30–Derivatives
- Chapter 31–Mathematical Reasoning
- Chapter 32–Statistics
- Chapter 33–Probability
About RD Sharma
RD Sharma isn’t the kind of author you’d bump into at lit fests. But his bestselling books have helped many CBSE students lose their dread of maths. Sunday Times profiles the tutor turned internet star
He dreams of algorithms that would give most people nightmares. And, spends every waking hour thinking of ways to explain concepts like ‘series solution of linear differential equations’. Meet Dr Ravi Dutt Sharma — mathematics teacher and author of 25 reference books — whose name evokes as much awe as the subject he teaches. And though students have used his thick tomes for the last 31 years to ace the dreaded maths exam, it’s only recently that a spoof video turned the tutor into a YouTube star.
R D Sharma had a good laugh but said he shared little with his on-screen persona except for the love for maths. “I like to spend all my time thinking and writing about maths problems. I find it relaxing,” he says. When he is not writing books explaining mathematical concepts for classes 6 to 12 and engineering students, Sharma is busy dispensing his duty as vice-principal and head of department of science and humanities at Delhi government’s Guru Nanak Dev Institute of Technology.