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NDA 1 2025 Maths, question 20

Topic: Algebra

Q20

What is the number of rational terms in the expansion of (31/2+51/4)12\left( 3^{1/2} + 5^{1/4} \right)^{12}?

  1. (a) 2
  2. (b) 3
  3. (c) 4Correct answer
  4. (d) 6

Solution.

Answer: (c) 4

  1. The general term is Tr+1=(12r)3(12−r)/25r/4T_{r + 1} = \binom{12}{r}3^{(12 - r)/2}5^{r/4}, with r=0,1,…,12r = 0,1,\ldots,12.
  2. Since 3 and 5 are distinct primes, both exponents must be integers for the term to be rational.
  3. Thus rr must be a multiple of 4.
  4. The possibilities are r=0,4,8,12r = 0,4,8,12, giving four rational terms.