Q. 515.0( 2 Votes )

Show that x = 2 i

Answer :

Let


We need to find the roots of Δ = 0.


Recall that the value of a determinant remains same if we apply the operation Ri Ri + kRj or Ci Ci + kCj.


Applying R2 R2 – R1, we get





Taking the term (x – 2) common from R2, we get



Applying R3 R3 – R1, we get





Taking the term (x + 3) common from R3, we get



Applying C1 C1 + C3, we get




Expanding the determinant along C1, we have


Δ = (x – 2)(x + 3)(x – 1)[(–3)(1) – (2)(1)]


Δ = (x – 2)(x + 3)(x – 1)(–5)


Δ = –5(x – 2)(x + 3)(x – 1)


The given equation is Δ = 0.


–5(x – 2)(x + 3)(x – 1) = 0


(x – 2)(x + 3)(x – 1) = 0


Case – I:


x – 2 = 0 x = 2


Case – II:


x + 2 = 0 x = –3


Case – III:


x – 1 = 0 x = 1


Thus, 2 is a root of the equation and its other roots are –3 and 1.


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