JEE PYQ: Complex Numbers Question 42
Question 42 - 2019 (12 Jan Shift 1)
The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is:
(1) $55$
(2) $49$
(3) $48$
(4) $54$
Show Answer
Answer: (4)
Solution
General term: $\binom{60}{r}7^{(60-r)/5}(-3)^{r/10}$. For rational: $5 | (60-r)$ and $10 | r$. $r = 0, 10, 20, 30, 40, 50, 60$ and $60-r$ divisible by 5: all these work. So 7 rational terms. But $r$ must also be divisible by 10: $r = 0, 10, 20, 30, 40, 50, 60$. Check: $60-0=60$ (div by 5), $60-10=50$ (yes), etc. All 7 are rational. Irrational terms $= 61 - 7 = 54$.
BETA
