Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Binomial Theorem question

2019 · 12 Jan · Shift 2 · Q31
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Binomial Theorem
  5. /2019 · 12 Jan · Shift 2 · Q31

Binomial Theorem question

2019 · 12 Jan · Shift 2 · Q31

JEE MainMathematicsBinomial TheoremMCQ+4 / −1
The total number of irrational terms in the binomial expansion of (71/5 – 31/10)60 is :
  1. A
    54
  2. B
    55
  3. C
    49
  4. D
    48
View written solutionFree

Correct answer: A

  1. Interpret the expression correctly

The given binomial is (71/5−31/10)60.\left(7^{1/5}-3^{1/10}\right)^{60}.(71/5−31/10)60.

We need the number of irrational terms in its binomial expansion.

  1. Write the general term

For (a−b)60,(a-b)^{60},(a−b)60, the general term is Tr+1=(60r)(71/5)60−r(−31/10)r,r=0,1,2,…,60.T_{r+1}=\binom{60}{r}(7^{1/5})^{60-r}(-3^{1/10})^r, \quad r=0,1,2,\dots,60.Tr+1​=(r60​)(71/5)60−r(−31/10)r,r=0,1,2,…,60.

So, Tr+1=(60r)(−1)r760−r53r10.T_{r+1}=\binom{60}{r}(-1)^r 7^{\frac{60-r}{5}}3^{\frac{r}{10}}.Tr+1​=(r60​)(−1)r7560−r​310r​.

  1. Condition for a term to be rational

A term will be rational only if both exponents are integers:

  • 60−r5\frac{60-r}{5}560−r​ must be an integer
  • r10\frac{r}{10}10r​ must be an integer

Now,

  • 60−r5\frac{60-r}{5}560−r​ is an integer when rrr is a multiple of 555.
  • r10\frac{r}{10}10r​ is an integer when rrr is a multiple of 101010.

For both to happen together, rrr must be a multiple of 101010.

Thus rational terms occur for r=0,10,20,30,40,50,60.r=0,10,20,30,40,50,60.r=0,10,20,30,40,50,60.

That gives 777 rational terms.

  1. Total number of terms

In the expansion of (a−b)60(a-b)^{60}(a−b)60, total number of terms is 60+1=61.60+1=61.60+1=61.

  1. Number of irrational terms

Therefore, 61−7=54.61-7=54.61−7=54.

  1. Check options
  • A: 545454 ✅
  • B: 555555 ❌
  • C: 494949 ❌
  • D: 484848 ❌

Hence the correct answer is A.

PreviousNext

More from Binomial Theorem

  • If (27)999 is divided by 7, then the remainder is :2017 · MCQ
  • The coefficient of x−5 in the binomial expansion of (x32​−x31​+1x+1​−x−x21​x−1​)10, where x e 0, 1, is :2017 · MCQ
  • For x ∈ R, x e -1, if (1 + x)2016 + x(1 + x)2015 + x2(1 + x)2014 + . . . . + x2016 = i=0∑2016​ai​xi, then a17 is equal to :2016 · MCQ
  • If the coefficients of x−2 and x−4 in the expansion of (x31​+2x31​1​)18,(x>0), are m and n respectively, then nm​ is equal to :2016 · MCQ
  • The term independent of x in expansion of (x2/3−x1/3+1x+1​−x−x1/2x−1​)10 is2013 · MCQ
  • If n is a positive integer, then (3​+1)2n−(3​−1)2n is :2012 · MCQ
  • The remainder left out when 82n−(62)2n+1 is divided by 9 is :2009 · MCQ
  • In the binomial expansion of (a−b)n,n≥5, the sum of 5th and 6th terms is zero, then a/b equals2007 · MCQ