Theory of Equations 1 Question 4

4. If α and β are the roots of the equation x2−2x+2=0, then the least value of n for which αβn=1 is

(a) 2

(b) 5

(c) 4

(d) 3

(2019 Main, 8 April I)

Show Answer

Answer:

Correct Answer: 4. (c)

Solution:

  1. Given, α and β are the roots of the quadratic equation,

⇒x2−2x+2=0(x−1)2+1=0

⇒(x−1)2=−1⇒x−1=±i⇒x=(1+i) or (1−i)

Clearly, if α=1+i, then β=1−i

According to the question αβn=1

⇒1+in1−i=1

⇒(1+i)(1+i)(1−i)(1+i)n=1 [by rationalization]

⇒1+i2+2i1−i2n=1⇒2i2n=1⇒in=1

So, minimum value of n is 4 .

[∵i4=1]

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