Question

復(fù)數(shù)z是實(shí)數(shù)的充分而不必要條件是(    )

A.|z|=z           B.z=             C.z²是實(shí)數(shù)            D.z+是實(shí)數(shù)

Answer 1

思路解析:注意題目是求充分不必要條件而不是充要條件，即當(dāng)滿足條件時(shí)z為實(shí)數(shù)，但復(fù)數(shù)z為實(shí)數(shù)條件不一定成立.當(dāng)z=i時(shí),z²=-1,故C不成立.

當(dāng)z為虛數(shù)且非純虛數(shù)時(shí)，z+是實(shí)數(shù)，故D不成立.

若z=，設(shè)z=a+bi,則=a-bi，則復(fù)數(shù)相等得b=0,

∴復(fù)數(shù)z為實(shí)數(shù)；反之，若復(fù)數(shù)z為實(shí)數(shù)，則必有z=,故B是充要條件.

當(dāng)|z|=z，設(shè)z=a+bi,由復(fù)數(shù)相等得b=0，∴復(fù)數(shù)z為實(shí)數(shù)；反之，若復(fù)數(shù)z為實(shí)數(shù)且a＜0時(shí)得不出|z|=z.

答案:A