Question 18 Find the number of non-zero integral solutions of the equation | 1-i | ^ x = 2^x.
Class X1 - Maths -Complex Numbers and Quadratic Equations Page 113
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| 1 -i|^x = 2^x
we know,
|a + ib| = √(a² + b²) use this concept here,
so, |1 - i| = √{1² + (-1)²}
= √(1 + 1) = √2
now,
| 1 - i|^x = 2^x
(√2)^x = 2^x
(2½)^x = 2^x
2^(x/2) = 2^x
x/2 = x
x/2 - x = 0
x/2 = 0
x = 0 but we need non-zero solutions
here, we see , there is no non-zero solution.
so, number of solutions is zero.
we know,
|a + ib| = √(a² + b²) use this concept here,
so, |1 - i| = √{1² + (-1)²}
= √(1 + 1) = √2
now,
| 1 - i|^x = 2^x
(√2)^x = 2^x
(2½)^x = 2^x
2^(x/2) = 2^x
x/2 = x
x/2 - x = 0
x/2 = 0
x = 0 but we need non-zero solutions
here, we see , there is no non-zero solution.
so, number of solutions is zero.
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