If imaginary part of multiplicative inverse of 2-3i3-2i is - ab- where a and b are coprime numbers- then

Let the multiplicative inverse be Z Z×2+3i3+2i=1 ⇒ Z=12+3i3+2i ⇒ Z=3+2i2+3i×2 3i2 3i ⇒ Z= 6 9i+4i 6i^22^2 (3i)^2 ⇒ Z=12 5i13 ⇒ Im(Z)= 513 ∴ a=5, b=13 a+b = 18, ...

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Standard VII

Mathematics

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If imaginary ...

Question

If imaginary part of multiplicative inverse of $\frac{2 + 3 i}{3 + 2 i}$ is $- \frac{a}{b}$ , where $a$ and $b$ are coprime numbers, then

b - a = 8

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a + b = - 8

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a + b = 13

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b - a = 18

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Solution

The correct option is A $b - a = 8$
Let the multiplicative inverse be $Z$
$Z \times \frac{2 + 3 i}{3 + 2 i} = 1 \Rightarrow Z = \frac{1}{\frac{2 + 3 i}{3 + 2 i}} \Rightarrow Z = \frac{3 + 2 i}{2 + 3 i} \times \frac{2 - 3 i}{2 - 3 i} \Rightarrow Z = \frac{6 - 9 i + 4 i - 6 i^{2}}{2^{2} - (3 i)^{2}} \Rightarrow Z = \frac{12 - 5 i}{13} \Rightarrow I m (Z) = - \frac{5}{13} ∴ a = 5, b = 13 a + b = 18, b - a = 8$

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Standard VII Mathematics

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If imaginary part of multiplicative inverse of 2+3i3+2i is −ab, where a and b are coprime numbers, then

The correct option is A b−a=8Let the multiplicative inverse be Z Z×2+3i3+2i=1⇒Z=12+3i3+2i⇒Z=3+2i2+3i×2−3i2−3i⇒Z=6−9i+4i−6i222−(3i)2⇒Z=12−5i13⇒Im(Z)=−513∴a=5,b=13a+b=18, b−a=8

If imaginary part of multiplicative inverse of $\frac{2 + 3 i}{3 + 2 i}$ is $- \frac{a}{b}$ , where $a$ and $b$ are coprime numbers, then