Mathematics · Textbook solutions

Complex Numbers and Quadratic Equations

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 39 questions

4.2-4.4 Complex Numbers, Algebra, Modulus and Conjugate

22 q

Solved Examples

Worked · 8
  1. 4.1 Eg.1
    If 4x+i(3x−y)=3+i(−6)4x + i(3x - y) = 3 + i(-6), where xx and yy are real numbers, then find the values of xx and yy.
  2. Express the following in the form of a+bia + bi:
    4.1 Eg.2(i)
    (−5i)(18i)(-5i)\left(\dfrac{1}{8}i\right)
  3. 4.1 Eg.2(ii)
    (−i)(2i)(−18i)3(-i)(2i)\left(-\dfrac{1}{8}i\right)^{3}
  4. 4.1 Eg.3
    Express (5−3i)3(5 - 3i)^{3} in the form a+iba + ib.
  5. 4.1 Eg.4
    Express (−3+−2)(23−i)\left(-\sqrt{3} + \sqrt{-2}\right)\left(2\sqrt{3} - i\right) in the form of a+iba + ib
  6. 4.1 Eg.5
    Find the multiplicative inverse of 2−3i2 - 3i.
  7. Express the following in the form a+iba + ib
    4.1 Eg.6(i)
    5+2i1−2i\dfrac{5 + \sqrt{2}i}{1 - \sqrt{2}i}
  8. 4.1 Eg.6(ii)
    i−35i^{-35}

Exercise 4.1

Practice · 14
  1. Ex 4.1 Q1
    Express the complex number (5i)(−35i)(5i)\left(-\dfrac{3}{5}i\right) in the form a+iba + ib.
  2. Ex 4.1 Q2
    Express the complex number i9+i19i^{9} + i^{19} in the form a+iba + ib.
  3. Ex 4.1 Q3
    Express the complex number i−39i^{-39} in the form a+iba + ib.
  4. Ex 4.1 Q4
    Express the complex number 3(7+i7)+i(7+i7)3(7 + i7) + i(7 + i7) in the form a+iba + ib.
  5. Ex 4.1 Q5
    Express the complex number (1−i)−(−1+i6)(1 - i) - (-1 + i6) in the form a+iba + ib.
  6. Ex 4.1 Q6
    Express the complex number (15+i25)−(4+i52)\left(\dfrac{1}{5} + i\dfrac{2}{5}\right) - \left(4 + i\dfrac{5}{2}\right) in the form a+iba + ib.
  7. Ex 4.1 Q7
    Express the complex number [(13+i73)+(4+i13)]−(−43+i)\left[\left(\dfrac{1}{3} + i\dfrac{7}{3}\right) + \left(4 + i\dfrac{1}{3}\right)\right] - \left(-\dfrac{4}{3} + i\right) in the form a+iba + ib.
  8. Ex 4.1 Q8
    Express the complex number (1−i)4(1 - i)^{4} in the form a+iba + ib.
  9. Ex 4.1 Q9
    Express the complex number (13+3i)3\left(\dfrac{1}{3} + 3i\right)^{3} in the form a+iba + ib.
  10. Ex 4.1 Q10
    Express the complex number (−2−13i)3\left(-2 - \dfrac{1}{3}i\right)^{3} in the form a+iba + ib.
  11. Ex 4.1 Q11
    Find the multiplicative inverse of the complex number 4−3i4 - 3i.
  12. Ex 4.1 Q12
    Find the multiplicative inverse of the complex number 5+3i\sqrt{5} + 3i.
  13. Ex 4.1 Q13
    Find the multiplicative inverse of the complex number −i-i.
  14. Ex 4.1 Q14
    Express the following expression in the form of a+iba + ib : (3+i5)(3−i5)(3+2 i)−(3−i2)\dfrac{\left(3 + i\sqrt{5}\right)\left(3 - i\sqrt{5}\right)}{\left(\sqrt{3} + \sqrt{2}\,i\right) - \left(\sqrt{3} - i\sqrt{2}\right)}

Miscellaneous Exercise on Chapter 4

17 q

Solved Examples

Worked · 2
  1. Misc Eg.7
    Find the conjugate of (3−2i)(2+3i)(1+2i)(2−i)\dfrac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}.
  2. Misc Eg.8
    If x+iy=a+iba−ibx + iy = \dfrac{a + ib}{a - ib}, prove that x2+y2=1x^{2} + y^{2} = 1.

Miscellaneous Exercise

Practice · 15
  1. Misc Q1
    Evaluate: [i18+(1i)25]3\left[i^{18} + \left(\dfrac{1}{i}\right)^{25}\right]^{3}.
  2. Misc Q2
    For any two complex numbers z1z_{1} and z2z_{2}, prove that Re (z1z2)=Re z1 Re z2−Im z1 Im z2\mathrm{Re}\,(z_{1} z_{2}) = \mathrm{Re}\,z_{1}\,\mathrm{Re}\,z_{2} - \mathrm{Im}\,z_{1}\,\mathrm{Im}\,z_{2}.
  3. Misc Q3
    Reduce (11−4i−21+i)(3−4i5+i)\left(\dfrac{1}{1 - 4i} - \dfrac{2}{1 + i}\right)\left(\dfrac{3 - 4i}{5 + i}\right) to the standard form .
  4. Misc Q4
    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}}.
  5. Misc Q5
    If z1=2−iz_{1} = 2 - i, z2=1+iz_{2} = 1 + i, find ∣z1+z2+1z1−z2+1∣\left|\dfrac{z_{1} + z_{2} + 1}{z_{1} - z_{2} + 1}\right|.
  6. Misc Q6
    If a+ib=(x+i)22x2+1a + ib = \dfrac{(x + i)^{2}}{2x^{2} + 1}, prove that a2+b2=(x2+1)2(2x2+1)2a^{2} + b^{2} = \dfrac{(x^{2} + 1)^{2}}{\left(2x^{2} + 1\right)^{2}}.
  7. Let z1=2−iz_{1} = 2 - i, z2=−2+iz_{2} = -2 + i. Find
    Misc Q7(i)
    Re(z1z2z‾1)\mathrm{Re}\left(\dfrac{z_{1} z_{2}}{\overline{z}_{1}}\right),
  8. Misc Q7(ii)
    Im(1z1z‾1)\mathrm{Im}\left(\dfrac{1}{z_{1}\overline{z}_{1}}\right).
  9. Misc Q8
    Find the real numbers xx and yy if (x−iy)(3+5i)(x - iy)(3 + 5i) is the conjugate of −6−24i-6 - 24i.
  10. Misc Q9
    Find the modulus of 1+i1−i−1−i1+i\dfrac{1 + i}{1 - i} - \dfrac{1 - i}{1 + i}.
  11. Misc Q10
    If (x+iy)3=u+iv(x + iy)^{3} = u + iv, then show that ux+vy=4(x2−y2)\dfrac{u}{x} + \dfrac{v}{y} = 4(x^{2} - y^{2}).
  12. Misc Q11
    If α\alpha and β\beta are different complex numbers with ∣β∣=1|\beta| = 1, then find ∣β−α1−α‾β∣\left|\dfrac{\beta - \alpha}{1 - \overline{\alpha}\beta}\right|.
  13. Misc Q12
    Find the number of non-zero integral solutions of the equation ∣1−i∣x=2x|1 - i|^{x} = 2^{x}.
  14. Misc Q13
    If (a+ib)(c+id)(e+if)(g+ih)=A+iB(a + ib)(c + id)(e + if)(g + ih) = \mathrm{A} + i\mathrm{B}, then show that (a2+b2)(c2+d2)(e2+f2)(g2+h2)=A2+B2\left(a^{2} + b^{2}\right)\left(c^{2} + d^{2}\right)\left(e^{2} + f^{2}\right)\left(g^{2} + h^{2}\right) = \mathrm{A}^{2} + \mathrm{B}^{2}
  15. Misc Q14
    If (1+i1−i)m=1\left(\dfrac{1 + i}{1 - i}\right)^{m} = 1, then find the least positive integral value of mm.