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Consider the following statements in respect of an arbitrary complex number Z: 1. The difference of Z and its conjugate is an imaginary number. 2. The sum of Z and its conjugate is a real number. Which of the above statements is/are correct?

Consider the following statements in respect of an arbitrary complex number Z: 1. The difference of Z and its conjugate is an imaginary number. 2. The sum of Z and its conjugate is a real number. Which of the above statements is/are correct? Correct Answer Both 1 and 2

Concept:

Let z = x + iy be a complex number,

Where x is called the real part of the complex number or Re (z) and y is called the Imaginary part of the complex number or Im (z)

Conjugate of z = z̅ = x - iy  

Calculation:

1. The difference of Z and its conjugate is an imaginary number.

Consider z = a + ib           ....(i)

conjugate of z = z̅ = a - ib      ....(ii)

eq(i) - eq (ii)

z - z̅ = a + ib - a + ib

⇒ 2ib

Thus it is clear that the difference of z and its conjugate is an imaginary number.

2. The sum of Z and its conjugate is a real number.

eq (i) + eq(ii)

z + z̅ = a + ib + a - ib

⇒ 2a

Thus it is clear that the sum of Z and its conjugate is a real number.

So, Both 1 and 2 are correct.

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