Question

The cubic equation \( 2 z^{3}-5 z^{2}+c z-5=0, c \in R \). If one of the root is \( z=1-2 i \). Find the value of \( c \). 

a. \( -4 \) 

b. 10 

c. 12 

d. 2

Answer 1

2z³ - 5z² + cz - 5 = 0, c \(\in \) R----(1)

\(\because\) 1 - 2i is a root  of equation (1)

\(\therefore\) 1 + 2i (complex conjugate of 1 - 2i) is also a root of given cubic equation

Let z be its 3rd root

\(\therefore\) sum of roots  = \(\frac{-(-5)}2\)

⇒ z + (1 - 2i) + (1 + 2i) = 5/2

⇒ z = 5/2 - 2 = 1/2

sum of product of two roots = C/2

⇒ \(\frac12((1 + 2i) + (1 - 2i)) + (1 + 2i) (1 - 2i)=\frac c2\)

⇒ 1 + 1 - 4c² = c/2

⇒ c/2 = 6 

⇒ c = 12

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