Mathematics · Textbook solutions

Complex Numbers

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 254 questions

1.2.1 Equality of Complex Numbers

1 q

Worked Example

Worked · 1
  1. 1.2.1 SolvedEx.1
    If 7a+i(3a−b)=21−3i7a + i(3a - b) = 21 - 3i then find aa and bb.

1.2.3 Addition

2 q

Worked Examples

Worked · 2
  1. 1.2.3 SolvedEx.1
    Find (2+3i)+(4+3i)(2 + 3i) + (4 + 3i).
  2. 1.2.3 SolvedEx.2
    Find (−2+5i)+(7+3i)+(6−4i)(-2 + 5i) + (7 + 3i) + (6 - 4i).

1.2.4 Scalar Multiplication

2 q

Worked Examples

Worked · 2
  1. 1.2.4 SolvedEx.1
    If z=7+3iz = 7 + 3i, find 5z5z.
  2. 1.2.4 SolvedEx.2
    If z1=3−4iz_1 = 3 - 4i and z2=10−9iz_2 = 10 - 9i, find 2z1+5z22z_1 + 5z_2.

1.2.5 Subtraction

2 q

Worked Examples

Worked · 2
  1. 1.2.5 SolvedEx.1
    If z1=4+3iz_1 = 4 + 3i and z2=2+iz_2 = 2 + i, find z1−z2z_1 - z_2.
  2. 1.2.5 SolvedEx.2
    If z1=7+iz_1 = 7 + i, z2=4iz_2 = 4i and z3=−3+2iz_3 = -3 + 2i, then find 2z1−(5z2+2z3)2z_1 - (5z_2 + 2z_3).

1.2.6 Multiplication

2 q

Worked Examples

Worked · 2
  1. 1.2.6 SolvedEx.1
    If z1=2+3iz_1 = 2 + 3i and z2=3−2iz_2 = 3 - 2i, find z1⋅z2z_1 \cdot z_2.
  2. 1.2.6 SolvedEx.2
    If z1=2−7iz_1 = 2 - 7i, z2=4−3iz_2 = 4 - 3i and z3=1+iz_3 = 1 + i, then find (2z1)⋅(z2)⋅(z3)(2z_1) \cdot (z_2) \cdot (z_3).

1.2.7 Powers of i

3 q

Worked Examples

Worked · 3
  1. 1.2.7 SolvedEx.1
    Evaluate i50i^{50}.
  2. 1.2.7 SolvedEx.2
    Evaluate i318i^{318}.
  3. 1.2.7 SolvedEx.3
    Evaluate i999i^{999}.

1.2.8 Division

1 q

Worked Example

Worked · 1
  1. 1.2.8 SolvedEx.1
    If z1=3+2iz_1 = 3 + 2i and z2=1+iz_2 = 1 + i, then find z1z2\dfrac{z_1}{z_2}.

1.2 Algebra of Complex Numbers

5 q

Solved Examples

Worked · 5
  1. 1.2 SolvedEx.1
    Write (1+2i)(1+3i)(2+i)−1(1 + 2i)(1 + 3i)(2 + i)^{-1} in the form a+iba + ib.
  2. 1.2 SolvedEx.2
    (Activity) Express 1i+2i2+3i3+5i4\dfrac{1}{i} + \dfrac{2}{i^2} + \dfrac{3}{i^3} + \dfrac{5}{i^4} in the form of (a+ib)(a + ib).
  3. 1.2 SolvedEx.3
    If aa and bb are real and (i4+3i)a+(i−1)b+5i3=0(i^4 + 3i)a + (i - 1)b + 5i^3 = 0, find aa and bb.
  4. 1.2 SolvedEx.4
    If x+2i+15i6y=7x+i3(y+4)x + 2i + 15i^6 y = 7x + i^3(y + 4) find x+yx + y, given that x,y∈Rx, y \in R.
  5. 1.2 SolvedEx.5
    Show that (32+i2)3=i\left(\dfrac{\sqrt{3}}{2} + \dfrac{i}{2}\right)^3 = i.

Exercise 1.1

59 q
  1. Simplify :
    Ex 1.1 Q1(i)
    −16+3−25+−36−−625\sqrt{-16} + 3\sqrt{-25} + \sqrt{-36} - \sqrt{-625}
  2. Ex 1.1 Q1(ii)
    4−4+5−9−3−164\sqrt{-4} + 5\sqrt{-9} - 3\sqrt{-16}
  3. Write the conjugates of the following complex numbers
    Ex 1.1 Q2(i)
    3+i3 + i
  4. Ex 1.1 Q2(ii)
    3−i3 - i
  5. Ex 1.1 Q2(iii)
    −5−7 i-\sqrt{5} - \sqrt{7}\,i
  6. Ex 1.1 Q2(iv)
    −−5-\sqrt{-5}
  7. Ex 1.1 Q2(v)
    5i5i
  8. Ex 1.1 Q2(vi)
    5−i\sqrt{5} - i
  9. Ex 1.1 Q2(vii)
    2+3 i\sqrt{2} + \sqrt{3}\,i
  10. Ex 1.1 Q2(viii)
    cos⁡θ+isin⁡θ\cos\theta + i\sin\theta
  11. Find aa and bb if
    Ex 1.1 Q3(i)
    a+2b+2ai=4+6ia + 2b + 2ai = 4 + 6i
  12. Ex 1.1 Q3(ii)
    (a−b)+(a+b)i=a+5i(a - b) + (a + b)i = a + 5i
  13. Ex 1.1 Q3(iii)
    (a+b)(2+i)=b+1+(10+2a)i(a + b)(2 + i) = b + 1 + (10 + 2a)i
  14. Ex 1.1 Q3(iv)
    abi=3a−b+12iabi = 3a - b + 12i
  15. Ex 1.1 Q3(v)
    1a+ib=3−2i\dfrac{1}{a + i\mathrm{b}} = 3 - 2i
  16. Ex 1.1 Q3(vi)
    (a+ib)(1+i)=2+i(a + ib)(1 + i) = 2 + i
  17. Express the following in the form of a+iba + ib, a,b∈Ra, b \in R, i=−1i = \sqrt{-1}. State the values of aa and bb.
    Ex 1.1 Q4(i)
    (1+2i)(−2+i)(1 + 2i)(-2 + i)
  18. Ex 1.1 Q4(ii)
    (1+i)(1−i)−1(1 + i)(1 - i)^{-1}
  19. Ex 1.1 Q4(iii)
    i(4+3i)(1−i)\dfrac{i(4 + 3i)}{(1 - i)}
  20. Ex 1.1 Q4(iv)
    (2+i)(3−i)(1+2i)\dfrac{(2 + i)}{(3 - i)(1 + 2i)}
  21. Ex 1.1 Q4(v)
    (1+i1−i)2\left(\dfrac{1 + i}{1 - i}\right)^2
  22. Ex 1.1 Q4(vi)
    3+2i2−5i+3−2i2+5i\dfrac{3 + 2i}{2 - 5i} + \dfrac{3 - 2i}{2 + 5i}
  23. Ex 1.1 Q4(vii)
    (1+i)−3(1 + i)^{-3}
  24. Ex 1.1 Q4(viii)
    2+−34+−3\dfrac{2 + \sqrt{-3}}{4 + \sqrt{-3}}
  25. Ex 1.1 Q4(ix)
    (−5+2−4)+(1−−9)+(2+3i)(2−3i)\left(-\sqrt{5} + 2\sqrt{-4}\right) + \left(1 - \sqrt{-9}\right) + (2 + 3i)(2 - 3i)
  26. Ex 1.1 Q4(x)
    (2+3i)(2−3i)(2 + 3i)(2 - 3i)
  27. Ex 1.1 Q4(xi)
    4i8−3i9+33i11−4i10−2\dfrac{4i^8 - 3i^9 + 3}{3i^{11} - 4i^{10} - 2}
  28. Ex 1.1 Q5
    Show that (−1+3 i)3\left(-1 + \sqrt{3}\,i\right)^3 is a real number.
  29. Ex 1.1 Q6
    Find the value of (3+2i)(i6−i7)(1+i11)\left(3 + \dfrac{2}{i}\right)\left(i^6 - i^7\right)\left(1 + i^{11}\right).
  30. Evaluate the following :
    Ex 1.1 Q7(i)
    i35i^{35}
  31. Ex 1.1 Q7(ii)
    i888i^{888}
  32. Ex 1.1 Q7(iii)
    i93i^{93}
  33. Ex 1.1 Q7(iv)
    i116i^{116}
  34. Ex 1.1 Q7(v)
    i403i^{403}
  35. Ex 1.1 Q7(vi)
    1i58\dfrac{1}{i^{58}}
  36. Ex 1.1 Q7(vii)
    i−888i^{-888}
  37. Ex 1.1 Q7(viii)
    i30+i40+i50+i60i^{30} + i^{40} + i^{50} + i^{60}
  38. Ex 1.1 Q8
    Show that 1+i10+i20+i301 + i^{10} + i^{20} + i^{30} is a real number.
  39. Find the value of
    Ex 1.1 Q9(i)
    i49+i68+i89+i110i^{49} + i^{68} + i^{89} + i^{110}
  40. Ex 1.1 Q9(ii)
    i+i2+i3+i4i + i^2 + i^3 + i^4
  41. Ex 1.1 Q10
    Simplify : i592+i590+i588+i586+i584i582+i580+i578+i576+i574\dfrac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}
  42. Ex 1.1 Q11
    Find the value of 1+i2+i4+i6+i8+…+i201 + i^2 + i^4 + i^6 + i^8 + \ldots + i^{20}.
  43. Ex 1.1 Q12
    Show that 1+i10+i100−i1000=01 + i^{10} + i^{100} - i^{1000} = 0.
  44. Ex 1.1 Q13
    Is (1+i14+i18+i22)\left(1 + i^{14} + i^{18} + i^{22}\right) a real number? Justify your answer.
  45. Ex 1.1 Q14
    Evaluate : (i37+1i67)\left(i^{37} + \dfrac{1}{i^{67}}\right)
  46. Ex 1.1 Q15
    Prove that (1+i)4×(1+1i)4=16(1 + i)^4 \times \left(1 + \dfrac{1}{i}\right)^4 = 16.
  47. Ex 1.1 Q16
    Find the value of i6+i7+i8+i9i2+i3\dfrac{i^6 + i^7 + i^8 + i^9}{i^2 + i^3}
  48. Ex 1.1 Q17
    If a=−1+3i2a = \dfrac{-1 + \sqrt{3}i}{2}, b=−1−3i2b = \dfrac{-1 - \sqrt{3}i}{2} then show that a2=ba^2 = b and b2=ab^2 = a.
  49. Ex 1.1 Q18
    If x+iy=(a+ib)3x + iy = (a + ib)^3, show that xa+yb=4(a2−b2)\dfrac{x}{a} + \dfrac{y}{b} = 4(a^2 - b^2)
  50. Ex 1.1 Q19
    If a+3i2+ib=1−i\dfrac{a + 3i}{2 + ib} = 1 - i, show that (5a−7b)=0(5a - 7b) = 0.
  51. Ex 1.1 Q20
    If x+iy=a+ibc+idx + iy = \sqrt{\dfrac{a + ib}{c + id}}, prove that (x2+y2)2=a2+b2c2+d2\left(x^2 + y^2\right)^2 = \dfrac{a^2 + b^2}{c^2 + d^2}
  52. Ex 1.1 Q21
    If (a+ib)=1+i1−i(a + ib) = \dfrac{1 + i}{1 - i}, then prove that (a2+b2)=1\left(a^2 + b^2\right) = 1.
  53. Ex 1.1 Q22
    Show that (7+i37−i3+7−i37+i3)\left(\dfrac{\sqrt{7} + i\sqrt{3}}{\sqrt{7} - i\sqrt{3}} + \dfrac{\sqrt{7} - i\sqrt{3}}{\sqrt{7} + i\sqrt{3}}\right) is real.
  54. Ex 1.1 Q23
    If (x+iy)3=y+vi(x + iy)^3 = y + vi the show that (yx+vy)=4(x2−y2)\left(\dfrac{y}{x} + \dfrac{v}{y}\right) = 4\left(x^2 - y^2\right)
  55. Find the value of xx and yy which satisfy the following equations (x,y∈R)(x, y \in R)
    Ex 1.1 Q24(i)
    (x+2y)+(2x−3y)i+4i=5(x + 2y) + (2x - 3y)i + 4i = 5
  56. Ex 1.1 Q24(ii)
    x+11+i+y−11−i=i\dfrac{x + 1}{1 + i} + \dfrac{y - 1}{1 - i} = i
  57. Ex 1.1 Q24(iii)
    (x+iy)2+3i+2+i2−3i=913(1+i)\dfrac{(x + iy)}{2 + 3i} + \dfrac{2 + i}{2 - 3i} = \dfrac{9}{13}(1 + i)
  58. Ex 1.1 Q24(iv)
    If x(1+3i)+y(2−i)−5+i3=0x(1 + 3i) + y(2 - i) - 5 + i^3 = 0, find x+yx + y
  59. Ex 1.1 Q24(v)
    If x+2i+15i6y=7x+i3(y+4)x + 2i + 15i^6 y = 7x + i^3(y + 4), find x+yx + y

1.3 Square Root of a Complex Number

2 q

Solved Examples

Worked · 2
  1. 1.3 SolvedEx.1
    Find the square root of 6+8i6+8i.
  2. 1.3 SolvedEx.2
    Find the square root of 3−4i3 - 4i.

1.4 Fundamental Theorem of Algebra

4 q

Solved Examples

Worked · 4
  1. 1.4 SolvedEx.1
    Solve x2+x+1=0x^2 + x + 1 = 0.
  2. 1.4 SolvedEx.2
    Solve x2−(23+3i) x+63 i=0x^2 - (2\sqrt{3} + 3i)\,x + 6\sqrt{3}\,i = 0.
  3. 1.4 SolvedEx.3
    Find the value of x3−x2+2x+10x^3 - x^2 + 2x + 10 when x=1+3 ix = 1 + \sqrt{3}\,i.
  4. 1.4 SolvedEx.4
    If x=−5+2−4x = -5 + 2\sqrt{-4}, find the value of x4+9x3+35x2−x+64x^4 + 9x^3 + 35x^2 - x + 64.

Exercise 1.2

22 q
  1. Find the square root of the following complex numbers
    Ex 1.2 Q1(i)
    −8−6i-8-6i
  2. Ex 1.2 Q1(ii)
    7+24i7+24i
  3. Ex 1.2 Q1(iii)
    1+43 i1+4\sqrt{3}\,i
  4. Ex 1.2 Q1(iv)
    3+210 i3+2\sqrt{10}\,i
  5. Ex 1.2 Q1(v)
    2(1−3 i)2(1-\sqrt{3}\,i)
  6. Solve the following quadratic equations.
    Ex 1.2 Q2(i)
    8x2+2x+1=08x^2 + 2x + 1 = 0
  7. Ex 1.2 Q2(ii)
    2x2−3 x+1=02x^2 - \sqrt{3}\,x + 1 = 0
  8. Ex 1.2 Q2(iii)
    3x2−7x+5=03x^2 - 7x + 5 = 0
  9. Ex 1.2 Q2(iv)
    x2−4x+13=0x^2 - 4x + 13 = 0
  10. Solve the following quadratic equations.
    Ex 1.2 Q3(i)
    x2+3ix+10=0x^2 + 3ix + 10 = 0
  11. Ex 1.2 Q3(ii)
    2x2+3ix+2=02x^2 + 3ix + 2 = 0
  12. Ex 1.2 Q3(iii)
    x2+4ix−4=0x^2 + 4ix - 4 = 0
  13. Ex 1.2 Q3(iv)
    ix2−4x−4i=0ix^2 - 4x - 4i = 0
  14. Solve the following quadratic equations.
    Ex 1.2 Q4(i)
    x2−(2+i) x−(1−7i)=0x^2 - (2+i)\,x - (1-7i) = 0
  15. Ex 1.2 Q4(ii)
    x2−(32+2i) x+62 i=0x^2 - (3\sqrt{2} + 2i)\,x + 6\sqrt{2}\,i = 0
  16. Ex 1.2 Q4(iii)
    x2−(5−i) x+(18+i)=0x^2 - (5-i)\,x + (18+i) = 0
  17. Ex 1.2 Q4(iv)
    (2+i)x2−(5−i) x+2(1−i)=0(2+i)x^2 - (5-i)\,x + 2(1-i) = 0
  18. Find the value of
    Ex 1.2 Q5(i)
    x3−x2+x+46x^3 - x^2 + x + 46, if x=2+3ix = 2+3i.
  19. Ex 1.2 Q5(ii)
    2x3−11x2+44x+272x^3 - 11x^2 + 44x + 27, if x=253−4ix = \frac{25}{3-4i}.
  20. Ex 1.2 Q5(iii)
    x3+x2−x+22x^3 + x^2 - x + 22, if x=51−2ix = \frac{5}{1-2i}.
  21. Ex 1.2 Q5(iv)
    x4+9x3+35x2−x+4x^4 + 9x^3 + 35x^2 - x + 4, if x=−5+−4x = -5+\sqrt{-4}.
  22. Ex 1.2 Q5(v)
    2x4+5x3+7x2−x+412x^4 + 5x^3 + 7x^2 - x + 41, if x=−2−3 ix = -2-\sqrt{3}\,i.

1.5.2 Modulus and Argument

2 q

Solved Examples

Worked · 2
  1. 1.5.2 SolvedEx.1
    If z=1+3iz = 1+3i, find the modulus and amplitude of zz.
  2. 1.5.2 SolvedEx.2
    Find the modulus, argument of the complex number −7+24i-7 + 24i.

1.5.5 Exponential Form

8 q

Solved Examples

Worked · 8
  1. 1.5.5 SolvedEx.1
    Represent the complex numbers z=1+iz = 1+i, z‾=1−i\overline{z} = 1-i, −z‾=−1+i-\overline{z} = -1+i, −z=−1−i-z = -1-i in Argand's diagram and hence find their arguments from the figure. (In the book's figure the four numbers are plotted as A(1,1)≡z(1, 1) \equiv z, B(−1,1)≡−z‾(-1, 1) \equiv -\overline{z}, C(−1,−1)≡−z(-1, -1) \equiv -z and D(1,−1)≡z‾(1, -1) \equiv \overline{z} on axes marked from −4-4 to 44.)
  2. Represent the following complex numbers in the polar form and in the exponential form.
    1.5.5 SolvedEx.2(i)
    4+43 i4+4\sqrt{3}\,i
  3. 1.5.5 SolvedEx.2(ii)
    −2-2
  4. 1.5.5 SolvedEx.2(iii)
    3i3i
  5. 1.5.5 SolvedEx.2(iv)
    −3+i-\sqrt{3} + i
  6. 1.5.5 SolvedEx.3
    Express z=2⋅e3π4iz = \sqrt{2}\cdot e^{\frac{3\pi}{4}i} in the a+iba + ib form.
  7. Express the following in a+iba + ib form.
    1.5.5 SolvedEx.4(i)
    3⋅e5π12i×4⋅eπ12i3\cdot e^{\frac{5\pi}{12}i} \times 4\cdot e^{\frac{\pi}{12}i}
  8. 1.5.5 SolvedEx.4(ii)
    2(cos⁡π12+isin⁡π12)2(cos⁡5π6+isin⁡5π6)\dfrac{\sqrt{2}\left(\cos\frac{\pi}{12} + i\sin\frac{\pi}{12}\right)}{2\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right)}

Exercise 1.3

34 q
  1. Find the modulus and amplitude for each of the following complex numbers.
    Ex 1.3 Q1(i)
    7−5i7 - 5i
  2. Ex 1.3 Q1(ii)
    3+2 i\sqrt{3} + \sqrt{2}\,i
  3. Ex 1.3 Q1(iii)
    −8+15i-8 + 15i
  4. Ex 1.3 Q1(iv)
    −3(1−i)-3(1-i)
  5. Ex 1.3 Q1(v)
    −4−4i-4 - 4i
  6. Ex 1.3 Q1(vi)
    3−i\sqrt{3} - i
  7. Ex 1.3 Q1(vii)
    33
  8. Ex 1.3 Q1(viii)
    1+i1 + i
  9. Ex 1.3 Q1(ix)
    1+i31 + i\sqrt{3}
  10. Ex 1.3 Q1(x)
    (1+2i)2 (1−i)(1+2i)^2\,(1-i)
  11. Ex 1.3 Q2
    Find real values of θ\theta for which (4+3isin⁡θ1−2isin⁡θ)\left(\dfrac{4 + 3i\sin\theta}{1 - 2i\sin\theta}\right) is purely real.
  12. Ex 1.3 Q3
    If z=3+5iz = 3 + 5i then represent the zz, z‾\overline{z}, −z-z, −z‾-\overline{z} in Argand's diagram.
  13. Express the following complex numbers in polar form and exponential form.
    Ex 1.3 Q4(i)
    −1+3 i-1 + \sqrt{3}\,i
  14. Ex 1.3 Q4(ii)
    −i-i
  15. Ex 1.3 Q4(iii)
    −1-1
  16. Ex 1.3 Q4(iv)
    11+i\dfrac{1}{1+i}
  17. Ex 1.3 Q4(v)
    1+2i1−3i\dfrac{1 + 2i}{1 - 3i}
  18. Ex 1.3 Q4(vi)
    1+7i(2−i)2\dfrac{1 + 7i}{(2-i)^2}
  19. Express the following numbers in the form x+iyx + iy.
    Ex 1.3 Q5(i)
    3(cos⁡π6+isin⁡π6)\sqrt{3}\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)
  20. Ex 1.3 Q5(ii)
    2(cos⁡7π4+isin⁡7π4)\sqrt{2}\left(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}\right)
  21. Ex 1.3 Q5(iii)
    7(cos⁡(−5π6)+isin⁡(−5π6))7\left(\cos\left(-\frac{5\pi}{6}\right) + i\sin\left(-\frac{5\pi}{6}\right)\right)
  22. Ex 1.3 Q5(iv)
    eπ3ie^{\frac{\pi}{3}i}
  23. Ex 1.3 Q5(v)
    e−4π3ie^{\frac{-4\pi}{3}i}
  24. Ex 1.3 Q5(vi)
    e5π6ie^{\frac{5\pi}{6}i}
  25. Ex 1.3 Q6
    Find the modulus and argument of the complex number 1+2i1−3i\dfrac{1 + 2i}{1 - 3i}.
  26. Ex 1.3 Q7
    Convert the complex number z=i−1cos⁡π3+isin⁡π3z = \dfrac{i - 1}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}} in the polar form.
  27. For z=2+3iz = 2 + 3i verify the following :
    Ex 1.3 Q8(i)
    (z‾)‾=z\overline{(\overline{z})} = z
  28. Ex 1.3 Q8(ii)
    z z‾=∣z∣2z\,\overline{z} = |z|^2
  29. Ex 1.3 Q8(iii)
    (z+z‾)(z + \overline{z}) is real
  30. Ex 1.3 Q8(iv)
    z−z‾=6iz - \overline{z} = 6i
  31. z1=1+iz_1 = 1 + i, z2=2−3iz_2 = 2 - 3i. Verify the following :
    Ex 1.3 Q9(i)
    z1+z2‾=z1‾+z2‾\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}
  32. Ex 1.3 Q9(ii)
    z1−z2‾=z1‾−z2‾\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}
  33. Ex 1.3 Q9(iii)
    z1⋅z2‾=z1‾⋅z2‾\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}
  34. Ex 1.3 Q9(iv)
    (z1z2)‾=z1‾z2‾\overline{\left(\dfrac{z_1}{z_2}\right)} = \dfrac{\overline{z_1}}{\overline{z_2}}

1.6 De Moivre's Theorem

4 q

Solved Examples

Worked · 4
  1. Use De Moivres Theorem and simplify.
    1.6 SolvedEx.1(i)
    (cos⁡π3+isin⁡π3)8\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)^{8}
  2. 1.6 SolvedEx.1(ii)
    (cos⁡π10−isin⁡π10)15\left(\cos\frac{\pi}{10}-i\sin\frac{\pi}{10}\right)^{15}
  3. 1.6 SolvedEx.1(iii)
    (cos⁡5θ+isin⁡5θ)2(cos⁡4θ−isin⁡4θ)3\dfrac{(\cos 5\theta+i\sin 5\theta)^{2}}{(\cos 4\theta-i\sin 4\theta)^{3}}
  4. 1.6 SolvedEx.2
    Express (1+i)4(1+i)^{4} in a+iba+ib form.

1.8 Sets of Points in the Complex Plane

8 q

Solved Examples

Worked · 8
  1. If ww is a complex cube root of unity, then prove that
    1.8 SolvedEx.1(i)
    1w+1w2=−1\dfrac{1}{w}+\dfrac{1}{w^{2}}=-1
  2. 1.8 SolvedEx.1(ii)
    (1+w2)3=−1(1+w^{2})^{3}=-1
  3. 1.8 SolvedEx.1(iii)
    (1−w+w2)3=−8(1-w+w^{2})^{3}=-8
  4. If ww is a complex cube root of unity, then show that
    1.8 SolvedEx.2(i)
    (1−w+w2)5+(1+w−w2)5=32(1-w+w^{2})^{5}+(1+w-w^{2})^{5}=32
  5. 1.8 SolvedEx.2(ii)
    (1−w)(1−w2)(1−w4)(1−w5)=9(1-w)(1-w^{2})(1-w^{4})(1-w^{5})=9
  6. If ww is a complex cube root of unity such that x=a+bx=a+b, y=aw+bw2y=aw+bw^{2} and z=aw2+bwz=aw^{2}+bw, a, b∈Ra,\ b\in R prove that
    1.8 SolvedEx.3(i)
    x+y+z=0x+y+z=0
  7. 1.8 SolvedEx.3(ii)
    x3+y3+z3=3(a3+b3)x^{3}+y^{3}+z^{3}=3(a^{3}+b^{3})
  8. 1.8 SolvedEx.4
    Prove that 1+wn+w2n=31+w^{n}+w^{2n}=3, if nn is multiple of 3 1+wn+w2n=01+w^{n}+w^{2n}=0, if nn is not multiple of 3, n∈Nn\in N

Exercise 1.4

34 q
  1. Find the value of
    Ex 1.4 Q1(i)
    w18w^{18}
  2. Ex 1.4 Q1(ii)
    w21w^{21}
  3. Ex 1.4 Q1(iii)
    w−30w^{-30}
  4. Ex 1.4 Q1(iv)
    w−105w^{-105}
  5. If ww is a complex cube root of unity, show that
    Ex 1.4 Q2(i)
    (2−w)(2−w2)=7(2-w)(2-w^{2})=7
  6. Ex 1.4 Q2(ii)
    (1+w−w2)6=64(1+w-w^{2})^{6}=64
  7. Ex 1.4 Q2(iii)
    (1+w)3−(1+w2)3=0(1+w)^{3}-(1+w^{2})^{3}=0
  8. Ex 1.4 Q2(iv)
    (2+w+w2)3−(1−3w+w2)3=65(2+w+w^{2})^{3}-(1-3w+w^{2})^{3}=65
  9. Ex 1.4 Q2(v)
    (3+3w+5w2)6−(2+6w+2w2)3=0(3+3w+5w^{2})^{6}-(2+6w+2w^{2})^{3}=0
  10. Ex 1.4 Q2(vi)
    a+bw+cw2c+aw+bw2=w2\dfrac{a+bw+cw^{2}}{c+aw+bw^{2}}=w^{2}
  11. Ex 1.4 Q2(vii)
    (a+b)+(aw+bw2)+(aw2+bw)=0(a+b)+(aw+bw^{2})+(aw^{2}+bw)=0
  12. Ex 1.4 Q2(viii)
    (a−b)(a−bw)(a−bw2)=a3−b3(a-b)(a-bw)(a-bw^{2})=a^{3}-b^{3}
  13. Ex 1.4 Q2(ix)
    (a+b)2+(aw+bw2)2+(aw2+bw)2=6ab(a+b)^{2}+(aw+bw^{2})^{2}+(aw^{2}+bw)^{2}=6ab
  14. If ww is a complex cube root of unity, find the value of
    Ex 1.4 Q3(i)
    w+1ww+\dfrac{1}{w}
  15. Ex 1.4 Q3(ii)
    w2+w3+w4w^{2}+w^{3}+w^{4}
  16. Ex 1.4 Q3(iii)
    (1+w2)3(1+w^{2})^{3}
  17. Ex 1.4 Q3(iv)
    (1−w−w2)3+(1−w+w2)3(1-w-w^{2})^{3}+(1-w+w^{2})^{3}
  18. Ex 1.4 Q3(v)
    (1+w)(1+w2)(1+w4)(1+w8)(1+w)(1+w^{2})(1+w^{4})(1+w^{8})
  19. If α\alpha and β\beta are the complex cube root of unity, show that
    Ex 1.4 Q4(a)
    α2+β2+αβ=0\alpha^{2}+\beta^{2}+\alpha\beta=0
  20. Ex 1.4 Q4(b)
    α4+β4+α−1β−1=0\alpha^{4}+\beta^{4}+\alpha^{-1}\beta^{-1}=0
  21. Ex 1.4 Q5
    If x=a+bx=a+b, y=αa+βby=\alpha a+\beta b and z=aβ+bαz=a\beta+b\alpha where α\alpha and β\beta are the complex cube-roots of unity, show that xyz=a3+b3xyz=a^{3}+b^{3}
  22. Find the equation in cartesian coordinates of the locus of zz if
    Ex 1.4 Q6(i)
    ∣z∣=10|z|=10
  23. Ex 1.4 Q6(ii)
    ∣z−3∣=2|z-3|=2
  24. Ex 1.4 Q6(iii)
    ∣z−5+6i∣=5|z-5+6i|=5
  25. Ex 1.4 Q6(iv)
    ∣z+8∣=∣z−4∣|z+8|=|z-4|
  26. Ex 1.4 Q6(v)
    ∣z−2−2i∣=∣z+2+2i∣|z-2-2i|=|z+2+2i|
  27. Ex 1.4 Q6(vi)
    ∣z+3i∣∣z−6i∣=1\dfrac{|z+3i|}{|z-6i|}=1
  28. Use De Moivres theorem and simplify the following
    Ex 1.4 Q7(i)
    (cos⁡2θ+isin⁡2θ)7(cos⁡4θ+isin⁡4θ)3\dfrac{(\cos 2\theta+i\sin 2\theta)^{7}}{(\cos 4\theta+i\sin 4\theta)^{3}}
  29. Ex 1.4 Q7(ii)
    cos⁡5θ+isin⁡5θ(cos⁡3θ−isin⁡3θ)2\dfrac{\cos 5\theta+i\sin 5\theta}{(\cos 3\theta-i\sin 3\theta)^{2}}
  30. Ex 1.4 Q7(iii)
    (cos⁡7π13+isin⁡7π13)4(cos⁡4π13−isin⁡4π13)6\dfrac{\left(\cos\frac{7\pi}{13}+i\sin\frac{7\pi}{13}\right)^{4}}{\left(\cos\frac{4\pi}{13}-i\sin\frac{4\pi}{13}\right)^{6}}
  31. Express the following in the form a+iba+ib, a, b∈Ra,\ b\in R, using De Moivre's theorem.
    Ex 1.4 Q8(i)
    (1−i)5(1-i)^{5}
  32. Ex 1.4 Q8(ii)
    (1+i)6(1+i)^{6}
  33. Ex 1.4 Q8(iii)
    (1−3 i)4(1-\sqrt{3}\,i)^{4}
  34. Ex 1.4 Q8(iv)
    (−23−2i)5(-2\sqrt{3}-2i)^{5}

Miscellaneous Exercise 1

59 q

(I) Select the correct answer

Practice · 10
  1. Misc I Q1
    If nn is an odd positive integer then the value of 1+(i)2n+(i)4n+(i)6n1+(i)^{2n}+(i)^{4n}+(i)^{6n} is :
    1. A.
      −4i-4i
    2. B.
      00
    3. C.
      4i4i
    4. D.
      44
  2. Misc I Q2
    The value of is i592+i590+i588+i586+i584i582+i580+i578+i576+i574\dfrac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}} is equal to :
    1. A.
      −2-2
    2. B.
      11
    3. C.
      00
    4. D.
      −1-1
  3. Misc I Q3
    −3 −6\sqrt{-3}\ \sqrt{-6} is equal to
    1. A.
      −32-3\sqrt{2}
    2. B.
      323\sqrt{2}
    3. C.
      32 i3\sqrt{2}\,i
    4. D.
      −32 i-3\sqrt{2}\,i
  4. Misc I Q4
    If ww is a complex cube root of unity, then the value of w99+w100+w101w^{99}+w^{100}+w^{101} is :
    1. A.
      −1-1
    2. B.
      11
    3. C.
      00
    4. D.
      33
  5. Misc I Q5
    If z=r(cos⁡θ+isin⁡θ)z=r(\cos\theta+i\sin\theta), then the value of zzˉ+zˉz\dfrac{z}{\bar{z}}+\dfrac{\bar{z}}{z}
    1. A.
      cos⁡2θ\cos 2\theta
    2. B.
      2cos⁡2θ2\cos 2\theta
    3. C.
      2cos⁡θ2\cos\theta
    4. D.
      2sin⁡θ2\sin\theta
  6. Misc I Q6
    If w (≠1)w\ (\ne 1) is a cube root of unity and (1+w)7=A+Bw(1+w)^{7}=A+Bw, then AA and BB are respectively the numbers
    1. A.
      0, 10,\ 1
    2. B.
      1, 11,\ 1
    3. C.
      1, 01,\ 0
    4. D.
      −1, 1-1,\ 1
  7. Misc I Q7
    The modulus and argument of (1+i3)8(1+i\sqrt{3})^{8} are respectively
    1. A.
      22 and 2π3\dfrac{2\pi}{3}
    2. B.
      256256 and 8π3\dfrac{8\pi}{3}
    3. C.
      256256 and 2π3\dfrac{2\pi}{3}
    4. D.
      6464 and 4π3\dfrac{4\pi}{3}
  8. Misc I Q8
    If arg⁡(z)=θ\arg(z)=\theta, then arg⁡(z‾)=\arg(\overline{z})=
    1. A.
      −θ-\theta
    2. B.
      θ\theta
    3. C.
      π−θ\pi-\theta
    4. D.
      π+θ\pi+\theta
  9. Misc I Q9
    If −1+3 i=reiθ-1+\sqrt{3}\,i=re^{i\theta}, then θ=\theta= ................. .
    1. A.
      −2π3-\dfrac{2\pi}{3}
    2. B.
      π3\dfrac{\pi}{3}
    3. C.
      −π3-\dfrac{\pi}{3}
    4. D.
      2π3\dfrac{2\pi}{3}
  10. Misc I Q10
    If z=x+iyz=x+iy and ∣z−zi∣=1|z-zi|=1 then
    1. A.
      zz lies on X-axis
    2. B.
      zz lies on Y-axis
    3. C.
      zz lies on a circle
    4. D.
      zz lies on a rectangle

(II) Answer the following

Practice · 49
  1. Simplify the following and express in the form a+iba+ib.
    Misc II Q1(i)
    3+−643+\sqrt{-64}
  2. Misc II Q1(ii)
    (2i3)2(2i^{3})^{2}
  3. Misc II Q1(iii)
    (2+3i)(1−4i)(2+3i)(1-4i)
  4. Misc II Q1(iv)
    52 i(−4−3i)\dfrac{5}{2}\,i(-4-3i)
  5. Misc II Q1(v)
    (1+3i)2(3+i)(1+3i)^{2}(3+i)
  6. Misc II Q1(vi)
    4+3i1−i\dfrac{4+3i}{1-i}
  7. Misc II Q1(vii)
    (1+2i)(3+4i)(5+i)−1\left(1+\dfrac{2}{i}\right)\left(3+\dfrac{4}{i}\right)(5+i)^{-1}
  8. Misc II Q1(viii)
    5+3 i5−3 i\dfrac{\sqrt{5}+\sqrt{3}\,i}{\sqrt{5}-\sqrt{3}\,i}
  9. Misc II Q1(ix)
    3i5+2i7+i9i6+2i8+3i18\dfrac{3i^{5}+2i^{7}+i^{9}}{i^{6}+2i^{8}+3i^{18}}
  10. Misc II Q1(x)
    5+7i4+3i+5+7i4−3i\dfrac{5+7i}{4+3i}+\dfrac{5+7i}{4-3i}
  11. Solve the following equations for x, y∈Rx,\ y\in R
    Misc II Q2(i)
    (4−5i)x+(2+3i)y=10−7i(4-5i)x+(2+3i)y=10-7i
  12. Misc II Q2(ii)
    x+iy2+3i=7−i\dfrac{x+iy}{2+3i}=7-i
  13. Misc II Q2(iii)
    (x+iy)(5+6i)=2+3i(x+iy)(5+6i)=2+3i
  14. Misc II Q2(iv)
    2x+i9y (2+i)=xi7+10i162x+i^{9}y\,(2+i)=xi^{7}+10i^{16}
  15. Evaluate
    Misc II Q3(i)
    (1−i+i2)−15(1-i+i^{2})^{-15}
  16. Misc II Q3(ii)
    (i131+i49)(i^{131}+i^{49})
  17. Find the value of
    Misc II Q4(i)
    x3+2x2−3x+21x^{3}+2x^{2}-3x+21, if x=1+2ix=1+2i.
  18. Misc II Q4(ii)
    x4+9x3+35x2−x+164x^{4}+9x^{3}+35x^{2}-x+164, if x=−5+4ix=-5+4i.
  19. Find the square roots of
    Misc II Q5(i)
    −16+30i-16+30i
  20. Misc II Q5(ii)
    15−8i15-8i
  21. Misc II Q5(iii)
    2+23 i2+2\sqrt{3}\,i
  22. Misc II Q5(iv)
    18i18i
  23. Misc II Q5(v)
    3−4i3-4i
  24. Misc II Q5(vi)
    6+8i6+8i
  25. Find the modulus and amplitude of each complex number and express it in the polar form.
    Misc II Q6(i)
    8+15i8+15i
  26. Misc II Q6(ii)
    6−i6-i
  27. Misc II Q6(iii)
    1+3 i2\dfrac{1+\sqrt{3}\,i}{2}
  28. Misc II Q6(iv)
    −1−i2\dfrac{-1-i}{\sqrt{2}}
  29. Misc II Q6(v)
    2i2i
  30. Misc II Q6(vi)
    −3i-3i
  31. Misc II Q6(vii)
    12+12i\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}i
  32. Misc II Q7
    Represent 1+2i1+2i, 2−i2-i, −3−2i-3-2i, −2+3i-2+3i by points in Argand's diagram.
  33. Misc II Q8
    Show that z=5(1−i)(2−i)(3−i)z=\dfrac{5}{(1-i)(2-i)(3-i)} is purely imaginary number.
  34. Misc II Q9
    Find the real numbers xx and yy such that x1+2i+y3+2i=5+6i−1+8i\dfrac{x}{1+2i}+\dfrac{y}{3+2i}=\dfrac{5+6i}{-1+8i}
  35. Misc II Q10
    Show that (12+i2)10+(12−i2)10=0\left(\dfrac{1}{\sqrt{2}}+\dfrac{i}{\sqrt{2}}\right)^{10}+\left(\dfrac{1}{\sqrt{2}}-\dfrac{i}{\sqrt{2}}\right)^{10}=0
  36. Misc II Q11
    Show that (1+i2)8+(1−i2)8=2\left(\dfrac{1+i}{\sqrt{2}}\right)^{8}+\left(\dfrac{1-i}{\sqrt{2}}\right)^{8}=2.
  37. Convert the complex numbers in polar form and also in exponential form.
    Misc II Q12(i)
    z=2+63 i5+3 iz=\dfrac{2+6\sqrt{3}\,i}{5+\sqrt{3}\,i}
  38. Misc II Q12(ii)
    z=−6+2 iz=-6+\sqrt{2}\,i
  39. Misc II Q12(iii)
    −32+33 i2\dfrac{-3}{2}+\dfrac{3\sqrt{3}\,i}{2}
  40. Misc II Q13
    If x+iy=a+iba−ibx+iy=\dfrac{a+ib}{a-ib}, prove that x2+y2=1x^{2}+y^{2}=1.
  41. Misc II Q14
    Show that z=(−1+−32)3z=\left(\dfrac{-1+\sqrt{-3}}{2}\right)^{3} is a rational number.
  42. Misc II Q15
    Show that 1−2i3−4i+1+2i3+4i\dfrac{1-2i}{3-4i}+\dfrac{1+2i}{3+4i} is real.
  43. Simplify
    Misc II Q16(i)
    i29+i39+i49i30+i40+i50\dfrac{i^{29}+i^{39}+i^{49}}{i^{30}+i^{40}+i^{50}}
  44. Misc II Q16(ii)
    (i65+1i145)\left(i^{65}+\dfrac{1}{i^{145}}\right)
  45. Misc II Q16(iii)
    i238+i236+i234+i232+i230i228+i226+i224+i222+i220\dfrac{i^{238}+i^{236}+i^{234}+i^{232}+i^{230}}{i^{228}+i^{226}+i^{224}+i^{222}+i^{220}}
  46. Misc II Q17
    Simplify [11−2i+31+i][3+4i2−4i]\left[\dfrac{1}{1-2i}+\dfrac{3}{1+i}\right]\left[\dfrac{3+4i}{2-4i}\right]
  47. Misc II Q18
    If α\alpha and β\beta are complex cube roots of unity, prove that (1−α)(1−β)(1−α2)(1−β2)=9(1-\alpha)(1-\beta)(1-\alpha^{2})(1-\beta^{2})=9
  48. Misc II Q19
    If ww is a complex cube root of unity, prove that (1−w+w2)6+(1+w−w2)6=128(1-w+w^{2})^{6}+(1+w-w^{2})^{6}=128
  49. Misc II Q20
    If ww is the cube root of unity then find the value of (−1+i32)18+(−1−i32)18\left(\dfrac{-1+i\sqrt{3}}{2}\right)^{18}+\left(\dfrac{-1-i\sqrt{3}}{2}\right)^{18}