Complex Numbers — JEE Mains Mathematics

Formulas (3)

• Real and imaginary parts · Dividing complex numbers
  $$\frac{a+ib}{c+id}=\frac{(ac+bd)+i(bc-ad)}{c^2+d^2}$$
• Equations in z and its conjugate · Conjugate identities
  $$z\bar z=|z|^2,\quad z+\bar z=2\,\mathrm{Re}(z),\quad z-\bar z=2i\,\mathrm{Im}(z)$$
• Modulus rules and bounds · Modulus of a sum
  $$|a\pm b|^2=|a|^2+|b|^2\pm2\,\mathrm{Re}(a\bar b)$$

Watch out for (3)

• Count every angle in the interval
  → Real and imaginary parts
• Squaring can add a root
  → Equations in z and its conjugate
• A bound must be reached
  → Modulus rules and bounds

Formulas (3)

• Polar form and the principal argument · Polar form
  $$z=r(\cos\theta+i\sin\theta)=re^{i\theta}$$
• Powers by De Moivre's theorem · De Moivre's theorem
  $$(re^{i\theta})^n=r^ne^{in\theta}$$
• Rotation and triangles · Angle at a vertex
  $$\frac{z_3-z_1}{z_2-z_1}=\frac{|z_3-z_1|}{|z_2-z_1|}\,e^{i\theta}$$

Watch out for (3)

• The inverse tangent is not the argument
  → Polar form and the principal argument
• Reduce the angle, and raise the modulus
  → Powers by De Moivre's theorem
• Turn about the right point
  → Rotation and triangles

Formulas (2)

• Cube roots of unity · The two facts
  $$\omega^3=1,\qquad1+\omega+\omega^2=0$$
• The nth roots of unity · Roots of unity
  $$z^n=1\ \Rightarrow\ z=e^{2\pi ik/n},\quad k=0,1,\dots,n-1$$

Watch out for (2)

• Count the multiples of 3
  → Cube roots of unity
• The root 1 is missing
  → The nth roots of unity

Formulas (2)

• Roots, sum and product · Sum and product of the roots
  $$\alpha+\beta=-\frac ba,\qquad\alpha\beta=\frac ca$$
• Powers of the roots · Recurrence for power sums
  $$S_{n+2}=p\,S_{n+1}+q\,S_n\qquad(\alpha^2=p\alpha+q)$$

Watch out for (2)

• Conjugate roots need real coefficients
  → Roots, sum and product
• Which root is alpha
  → Powers of the roots

Formulas (2)

• Straight lines · Line in complex form
  $$\bar az+a\bar z+c=0\quad(c\in\mathbb R)$$
• Circles · Circle in complex form
  $$z\bar z+\bar\alpha z+\alpha\bar z+d=0:\ \text{centre }-\alpha,\ r=\sqrt{|\alpha|^2-d}$$

Watch out for (2)

• Expand before trusting a sign
  → Straight lines
• Divide before reading the centre
  → Circles

Formulas (2)

• Real, imaginary or fixed-argument quotients · Radius of the arc
  $$\arg\frac{z-a}{z-b}=\theta\ \Rightarrow\ r=\frac{|a-b|}{2\sin\theta}$$
• Ellipses, hyperbolas and parabolas · Ellipse from two foci
  $$|z-z_1|+|z-z_2|=2a>|z_1-z_2|:\ b^2=a^2-c^2,\ c=\tfrac12|z_1-z_2|$$

Watch out for (2)

• A fixed argument gives one arc
  → Real, imaginary or fixed-argument quotients
• Compare the constant with the focal distance
  → Ellipses, hyperbolas and parabolas

Formulas (2)

• Greatest and least distances · Distance from a point to a circle
  $$\big||c-a|-r\big|\le|z-c|\le|c-a|+r\quad(|z-a|=r)$$
• Regions, areas and lattice points · Sector and segment
  $$\text{sector}=\tfrac12r^2\varphi,\qquad\text{segment}=\tfrac12r^2(\varphi-\sin\varphi)$$

Watch out for (2)

• Is the extreme point still in the region?
  → Greatest and least distances
• Which side of the bisector
  → Regions, areas and lattice points