Q94 / 180 · COMEDK 2024 (Evening Shift) · Mathematics
$\text { If }(1-4 i)^3=a+i b \text { then the value of } \mathrm{a} \text { and } \mathrm{b} \text { is }$
- A$49,-74$
- Correct answerB$-47,52$
- C$-74,49$
- D$-48,-52$
Solution
Given the expression $(1-4i)^3 = a + ib$.
Using the identity $(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$, we substitute $x = 1$ and $y = 4i$.
$(1-4i)^3 = (1)^3 - 3(1)^2(4i) + 3(1)(4i)^2 - (4i)^3$
$= 1 - 12i + 3(16i^2) - 64i^3$
Since $i^2 = -1$ and $i^3 = -i$, we have:
$= 1 - 12i + 3(-16) - 64(-i)$
$= 1 - 12i - 48 + 64i$
$= (1 - 48) + (-12 + 64)i$
$= -47 + 52i$
Comparing this with $a + ib$, we get $a = -47$ and $b = 52$.
Answer: $-47,52$
Using the identity $(x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$, we substitute $x = 1$ and $y = 4i$.
$(1-4i)^3 = (1)^3 - 3(1)^2(4i) + 3(1)(4i)^2 - (4i)^3$
$= 1 - 12i + 3(16i^2) - 64i^3$
Since $i^2 = -1$ and $i^3 = -i$, we have:
$= 1 - 12i + 3(-16) - 64(-i)$
$= 1 - 12i - 48 + 64i$
$= (1 - 48) + (-12 + 64)i$
$= -47 + 52i$
Comparing this with $a + ib$, we get $a = -47$ and $b = 52$.
Answer: $-47,52$
Hence, the correct answer is option: 'B'.