Revision Notes: Complex Numbers | Mathematics (Maths) for JEE Main & Advanced PDF Download

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Important Formulas

(a) Complex numberz = x + iy,    where x, y ∈ R and i = √-l.

(b) If z = x + iy then its conjugate z = x - iy.
(c) Modulus of z, i.e. | z | = (d) Argument of z, i.e.

(e) If y=0, then argument of z, i.e.
(f) If x=0, then argument of z, i.e.

(g) In Polar form x = rcosθ and y = rsinθ , therefore z = r ( cosθ+ i sinθ )
(h) In exponential form complex number z = re^iθ , where e^iθ = cosθ+ isinθ .

(i)

( j) Important properties of conjugate

(k) Important properties of modulus
If z is a complex number, then
If z₁ ,z₂ are two complex numbers, then

(l) Important properties of argument

If z₁ = r₁ (cos θ₁+ i sin θ₁ ) and z₂ = r₂ (cos θ₂ + i sin θ₂ ) , then

(m) Triangle on the complex plane

(n) ( cos θ+ i sin θ )ⁿ = cos nθ+ i sin nθ
(o) (p) Distance between A (z₁ ) and B(z₂ ) is given by |z₂ − z₁ |
(q) Section formula: The point P (z) which divides the join of the segment AB in the ratio m : n 
is given by z = 

(r) Midpoint formula: 

(s) Equation of a straight line

(i) Non-parametric form(ii) Parametric form
(iii) General equation of straight line​

(t) Complex slope of a line, Two lines with complex slopes µ₁ and µ₂ are
(i) Parallel, if µ₁= µ₂
(ii) Perpendicular, if µ₁ +µ₂ = 0
(u) Equation of a circle: |z − z₀ |= r

Solved Examples

Que 1: Find the value of smallest positive integer n, for which

Ans:

⇒iⁿ = 1 ⇒ n = 4, 8, 12
Minimum value of n is 4

Que 2: If z = 1 + i tan α (1, where π < α < 3π/2. find the value of |z| cos α.

Ans: z = 1 + itan α
sec α < 0 ⇒ |secα| = –sec α|z| = –secα|z|cosα = –1

Que 3:  Find the common roots of the equation z³  + 2z²  + 2z + 1 = 0 and z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0.

Ans: z³ + 2z² + 2z + 1 = 0
⇒ z = –1, ω, ω²
z¹⁹⁸⁵ + z¹⁰⁰ + 1 = 0
⇒ z = ω, ω²
Common roots are ω, ω²

^​Que 4: If z₁ , z₂ , z₃ , z₄ are the vertices of a square in that order, then which of the following does not hold good?

(a)

(b)

(c)(d) None of these

Ans: (c)
z1, z2, z3, z4 vertices of square
z₁ – z₂ = i2Im(z₁)
z₃ – z₂ = –2Re(z₁)
z₂ – z₄ = [–x + iy – (x – iy)]
= 2Re(z) – i2lm(z)
z₁ – z₃ = 2x + 2iy = 2Re(z) + i2Im(z)
(x = y as it is aqueous)

Que 5: If q₁ , q₂ , q₃ are the roots of the equation, x³ + 64 = 0, then the value of the determinant is:(a) 1
(b) 4
(c) 10
(d) none of these

Ans: (d)
x³ = (4)³ (–1)^1/3
x -4, -4ω, -4ω²

Que 6: The points of intersection of the two curves |z – 3| = 2 and |z| = 2 in an argand plane are:
(a)(b)(c)

(d)

Ans: (b)

|z - 3| = 2: (x - 3)² + y² = 4
|z|=2: x²+y² = 4
Points of intersection lie on the radical axis S₁ - S₂ = 0
Radical axis is x = 3/2x² + y² = 4

Que 7: If z is a complex number of unit modulus and argument θ , then arg equals(a)

(b) θ
(c) π−θ
(d)  −θ

Ans: (b)
Let θ = arg ⇒ 0 = arg (z)

Que 8: If z₁, z₂ and z₃ and are complex numbers such that then |z₁ + z₂ + z₃| is 
(a) Equal to 1
(b) Less than 1
(c) Greater than 3
(d) Equal to 3

Ans: (a)
Given, I Zl z2 z3 1=1
Now,IZI 12=1
⇒ IZ₁I²=1
⇒ Again now,

⇒⇒

⇒ Iz₁ + z₂+ z₃I = 1

Que 9: A value of θ for which is purely imaginary, is:(a) π/6(b) (c)(d) π/3

Ans: (c)

Que 10: Let a, b, c be distinct complex numberssuch that Find the value of k.

Ans: a = k – kb
b = k – kc
c = k – ka
a = k – k² + k² (k – ka)

a = k – k² + k³ –k³ a)

but a ≠ b ≠ c
i.e. k3 = –1 ⇒ k = –w, –w²