Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 202523 JanuaryMorning Shift
Exam
JEE Main
Year
2025
Date
23 January
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2025 · 23 January · Morning Shift · Mathematics · Binomial Theorem

The sum of all rational terms in the expansion of (1+2^ 1 / 3 +3^ 1...

The sum of all rational terms in the expansion of (1+21/3+31/2)6\left(1+2^{1 / 3}+3^{1 / 2}\right)^6 is equal to

Correct answer

0

Step-by-step solution

The general term of multinomial expansion is 6!α!β!γ!(1)α\frac{6!}{\alpha!\beta!\gamma!}(1)^\alpha\left(2^{\frac{1}{3}}\right)^\beta(312)γ\left(3^{\frac{1}{2}}\right)^\gamma For terms to be rational 3∣β3 \mid \beta and 2∣γ<tableclass="qx−match−listqx−match−table"border="1"cellpadding="6"cellspacing="0"><tr><td>β</td><td>γ</td><td>α</td><td>Term</td></tr><tr><td>0</td><td>0</td><td>6</td><td>1</td></tr><tr><td>0</td><td>2</td><td>4</td><td>15.3=45</td></tr><tr><td>0</td><td>4</td><td>2</td><td>15.32=135</td></tr><tr><td>0</td><td>6</td><td>0</td><td>1.33=27</td></tr><tr><td>3</td><td>0</td><td>3</td><td>20.2=40</td></tr><tr><td>3</td><td>2</td><td>1</td><td>60.2.3=360</td></tr><tr><td>6</td><td>0</td><td>0</td><td>1.4=4</td></tr></table>2 \mid \gamma <table class="qx-match-list qx-match-table" border="1" cellpadding="6" cellspacing="0"><tr><td>\beta</td><td>\gamma</td><td>\alpha</td><td>Term</td></tr><tr><td>0</td><td>0</td><td>6</td><td>1</td></tr><tr><td>0</td><td>2</td><td>4</td><td>15.3=45</td></tr><tr><td>0</td><td>4</td><td>2</td><td>15.3^2=135</td></tr><tr><td>0</td><td>6</td><td>0</td><td>1.3^3=27</td></tr><tr><td>3</td><td>0</td><td>3</td><td>20.2=40</td></tr><tr><td>3</td><td>2</td><td>1</td><td>60.2 .3=360</td></tr><tr><td>6</td><td>0</td><td>0</td><td>1.4=4</td></tr></table>\RightarrowSumofrationaltermsSum of rational terms=1+45+135+27+40+360+4=612
Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If 26((2^3)/(3) 12 2 + (2^5)/(5) 12 4 + (2^7)/(7) 12 6 + … + 2^ 13 13 12 12 ) = 3^ 13 - α… JEE Main 2026If (1 - x^3)^ 10 = ∑ _ r=0 ^ 10 a_r x^r (1-x)^ 30-2r , then 9a_9 a_ 10 is equal to… JEE Main 2026If the coefficients of the middle terms in the binomial expansions of (1 + α x)^ 26 and… JEE Main 2026The coefficient of x^2 in the expansion of (2x^2 + (1)/(x))^ 10 , x ≠ 0 , is : JEE Main 2026If the sum of the coefficients of x^7 and x^ 14 in the expansion of ((1)/(x^3) - x^4)^n… JEE Main 2026In the expansion of (9x- 1 3 x )^ 18 , x>0 , if the term independent of x is (221)k… JEE Main 2026Let the smallest value of k ∈ ℕ , for which the coefficient of x^3 in (1+x)^3 + (1+x)^4 +… JEE Main 2026If for 3 ≤ r ≤ 30 , 30 30-r + 3 30 31-r + 3 30 32-r + 30 33-r = m r , then m equals: JEE Main 2026The number of elements in the set S = \ (r, k) : k ∈ ℤ and ^ 36 C_ r+1 = 6( ^ 35 C_r)… JEE Main 2026Given below are two statements : Statement I: 25^ 13 +20^ 13 +8^ 13 +3^ 13 is divisible… JEE Main 2026The sum of the coefficients of x^ 499 and x^ 500 in (1+x)^ 1000 +x(1+x)^ 999 +x^ 2 (1+x)^… JEE Main 2026Let S=(1)/(25!)+(1)/(3!23!)+(1)/(5!21!)+… up to 13 terms. If 13 ~S= 2^ k n! , k ∈ ~N… JEE Main 2026The value of ^ 100 C_ 50 51 + ^ 100 C_ 51 52 +….+ ^ 100 C_ 100 101 is : JEE Main 2026The sum of all possible values of n ∈ N , so that the coefficients of x, x^ 2 and x^ 3 in… JEE Main 2026Let C_ r denote the coefficient of x^ r in the binomial expansion of (1+x)^ n , n ∈ N, 0… JEE Main 2026The coefficient of x^ 48 in (1+x)+2(1+x)^ 2 +3(1+x)^ 3 +…+100(1+x)^ 100 is equal to JEE Main 2026 Full Binomial Theorem list Binomial Theorem JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs