Question:
If $x,y \in \mathbb{R}$ and $x+iy=-(6+i)^{3}, i^{2}=-1$, then $x-y$ is equal to ________.
If $x,y \in \mathbb{R}$ and $x+iy=-(6+i)^{3}, i^{2}=-1$, then $x-y$ is equal to ________.
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$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
Updated On: Jun 26, 2026
- 93
- -93
- 91
- -91
- -107
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Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Concept
Expand the cubic expression and equate real and imaginary parts.
Step 2: Meaning
$(6+i)^3 = 6^3 + 3(6^2)i + 3(6)i^2 + i^3 = 216 + 108i - 18 - i = 198 + 107i$.
Step 3: Analysis
$x+iy = -(198 + 107i) = -198 - 107i$. So $x = -198$ and $y = -107$.
Step 4: Conclusion
$x-y = -198 - (-107) = -198 + 107 = -91$. Final Answer: (D)
Expand the cubic expression and equate real and imaginary parts.
Step 2: Meaning
$(6+i)^3 = 6^3 + 3(6^2)i + 3(6)i^2 + i^3 = 216 + 108i - 18 - i = 198 + 107i$.
Step 3: Analysis
$x+iy = -(198 + 107i) = -198 - 107i$. So $x = -198$ and $y = -107$.
Step 4: Conclusion
$x-y = -198 - (-107) = -198 + 107 = -91$. Final Answer: (D)
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