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Chapter 2: Problem 138

The value of \((-i)^{(-1)}=\) (a) \(-(\pi / 2)\) (b) \((\pi / 2)\) (c) \(\mathrm{e}^{-(\pi / 2)}\) (b) \(\mathrm{e}^{(\pi / 2)}\)

Short Answer

Expert verified
\( \mathrm{e}^{-(\pi / 2)} \)

Step by step solution

01

Express -i in exponential form

The exponential form of a complex number is given by \(re^{i\theta}\), where r is the magnitude of the complex number and \(\theta\) is the argument (angle) of the complex number. Here, we have -i, which can be written as \(0 - i\). The magnitude of -i is: \[r = \sqrt{0^2 + (-1)^2} = 1\] Now we need to find the angle \(\theta\). Since -i is along the negative imaginary axis, the angle is \(\frac{3\pi}{2}\). So, the exponential form of -i is: \[-i = 1e^{i\frac{3\pi}{2}}\]
02

Raise the exponential form of -i to -1

Now we need to raise this to the power of -1: \[(-i)^{-1} = (1e^{i\frac{3\pi}{2}})^{-1}\] We know that \((a^b)^c = a^{bc}\), so we can multiply exponents: \[(1e^{i\frac{3\pi}{2}})^{-1} = 1e^{-i\frac{3\pi}{2}}\] Comparing this result to the given options, we can see that the answer is (c), which is \(\mathrm{e}^{-(\pi / 2)}\)

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Most popular questions from this chapter

If \(\mathrm{f}(\mathrm{x})=4 \mathrm{x}^{5}+5 \mathrm{x}^{4}-8 \mathrm{x}^{3}+5 \mathrm{x}^{2} 4 \mathrm{x}-34 \mathrm{i}\) and \(\mathrm{f}[(-1+\sqrt{3} \mathrm{i}) / 2]=\mathrm{a}+\mathrm{ib}\) then \(\mathrm{a}: \mathrm{b}=\) (a) \(1: 2\) (b) \(-2: 1\) (c) \(17: 1\) (d) \(-17: 1\)

For complex numbers \(z_{1}, z_{2}\) if \(\left|z_{1}\right|=12\) and \(\left|z_{2}-3-4 i\right|=5\) then the minimum value \(\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|\) is (a) 0 (b) 2 (c) 7 (d) 17

If \(z=\left[(1+7 i) /(2-i)^{2}\right]\) then the polar form of \(z\) is (a) \(\sqrt{2}[\cos (3 \pi / 4)+\mathrm{i} \sin (3 \pi / 4)]\) (b) \(\sqrt{2}[\cos (\pi / 4)+\mathrm{i} \sin (\pi / 4)]\) (c) \(\sqrt{2}[\cos (7 \pi / 4)+\mathrm{i} \sin (7 \pi / 4)]\) (d) \(\sqrt{2}[\cos (5 \pi / 4)+\mathrm{i} \sin (5 \pi / 4)]\)

The expression of complex number \([1 /(1+\cos \theta-i \sin \theta)]\) in the form \(\mathrm{a}+\mathrm{ib}\) is (a) \([(\sin \theta) /\\{2(1+\overline{\cos \theta)}\\}]+\mathrm{i}(1 / 2)\) (b) \((1 / 2)-\mathrm{i}[(\sin \theta) /\\{2(1+\cos \theta)\\}]\) (c) \((1 / 2)+\mathrm{i}(1 / 2) \tan (\theta / 2)\) (d) \((1 / 2) \tan (\theta / 2)-\mathrm{i}(1 / 2)\)

\(\mathrm{A}\left(\mathrm{z}_{1}\right), \mathrm{B}\left(\mathrm{z}_{2}\right)\) and \(\mathrm{C}\left(\mathrm{z}_{3}\right)\) are vertices of \(\triangle \mathrm{ABC}\) where \(\mathrm{m}\) \(\angle \mathrm{C}=(\pi / 2)\) and \(\mathrm{AC}=\mathrm{BC}, \mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3}\) are complex number if \(\left(\mathrm{z}_{1}-\mathrm{z}_{2}\right)^{2}=\mathrm{k}\left(\mathrm{z}_{1}-\mathrm{z}_{3}\right)\left(\mathrm{z}_{3}-\mathrm{z}_{2}\right)\) then \(\mathrm{k}=\) (a) 1 (b) 2 (c) 4 (d) non of these
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