Given that z is a real number and z= +4 i 1+ i where R , then the possible value of is

Step-by-step solution

Given z = + 4i 1 + i , where z R . To ensure z is real, the imaginary part of z must be zero. Multiply the numerator and denominator by the conjugate of the denominator, 1 - i : z = ( + 4i)(1 - i) (1 + i)(1 - i) z = - ^2 i + 4i - 4 i^2 1^2 + ^2 Since i^2 = -1 , we have: z = + 4 i + (4 - ^2)i 1 + ^2 = 1 + ^2 + i 4 - ^2 1 + ^2 For z to be real, the imaginary part must be zero: 4 - ^2 1 + ^2 = 0 4 - ^2 = 0 ^2 = 4 = 2 . Checking the options provided, -2 is a valid value for . Answer: -2