Given that x , y ∈ R , solve : 4 x 2 + 3 x y + 2 x y - 3 x 2 i = 4 y 2 - x 2 2 + 3 x y - 2 y 2 i

We know that if Z 1 = Z 2 , then Re Z 1 = Re Z 2 & Im Z 1 = Im Z 2 . Given, 4 x 2 + 3 x y + 2 x y - 3 x 2 i = 4 y 2 - x 2 2 + 3 x y - 2 y 2 i ⇒ 4 x 2 + 3 x y = 4 y 2 - x 2 2 & 2 x y - 3 x 2 = 3 x y - 2 y 2 ⇒ 9 x 2 + 6 x y = 8 y 2 & 2 y 2 = x y + 3 x 2 ⇒ 9 x y 2 + 6 x y - 8 = 0 & 3 x y 2 + x y - 2 = 0 ⇒ 9 x y 2 + 12 x y - 6 x y - 8 = 0 & 3 x y 2 + 3 x y - 2 x y - 2 = 0 ⇒ 3 x y 3 x y + 4 - 2 3 x y + 4 = 0 & 3 x y x y + 1 - 2 x y + 1 = 0 ⇒ 3 x y - 2 3 x y + 4 = 0 & 3 x y - 2 x y + 1 = 0 Hence, the given expression is possible only when 3 x y - 2 = 0 . ⇒ 3 x = 2 y ⇒ x = K & y = 3 2 K , K ∈ R .

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Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-3>Q 16

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Complex number $a, b$ and $c$ zeros of the polynomial $P (z) = z^{3} + q z + r ($ where $q, r \in R)$ and $| a |^{2} + | b |^{2} + | c |^{2} = 250$ If the points corresponding to $a, b$ and $c$ in the complex plane are the vertices of a right angled triangle with hypotenuse $h$ , then $(\frac{h^{2}}{125})$ is

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Important Questions on Complex Numbers

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Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-3>Q 17

If

\frac{z - (1 + i)}{2 + i}

is purely real, then value of

2 lm (z) - Re (z)

is equal to

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Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-3>Q 18

If

z_{1}

and

z_{2}

be two distinct complex numbers satisfying

|z_{1}^{2} - z_{2}^{2}| = |{\bar{z}}_{1}^{2} + {\bar{z}}_{2}^{2} - 2 {\bar{z}}_{1} {\bar{z}}_{2}|

and if

(\arg z_{1} - \arg z_{2}) = \frac{a π}{b}

, then find the least possible value of

\frac{| a + b |}{2}

(

a, b \in

Integer)

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Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-3>Q 19

The least value of

| z - 3 - 4 i |^{2} + | z + 2 - 7 i |^{2} + | z - 5 + 2 i |^{2}

occurs when

z = x + i y

then the real part of

z

is

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Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-4>Q 1

Find the modulus, argument and the principal argument of the complex numbers.
(a)

z = 1 + \cos \frac{10 π}{9} + i \sin (\frac{10 π}{9})

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Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-4>Q 2

Find the modulus, argument and the principal argument of the complex numbers.
(b)

(\tan 1 - i)^{2}

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Beta Question Bank for Engineering: Mathematics>Chapter 6 - Complex Numbers>EXERCISE-4>Q 3

Find the modulus, argument and the principal argument of the complex number $z = \frac{\sqrt{5 + 12 i} + \sqrt{5 - 12 i}}{\sqrt{5 + 12 i} - \sqrt{5 - 12 i}}$