Visual SolutionPYQ 2024 · Apr 4 Shift 1Standard

Question

The sum of all rational terms in the expansion of

\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}

is equal to:

(A)

3133

(B)

931

(C)

6131

(D)

633

Solution Path

Rational when

5 \mid (15-r)

and

3 \mid r

: only

r=0,15

T_1 = 2^3 = 8

T_{16} = 5^5 = 3125

. Sum

= 3133

01Question Setup

1/4

Find the sum of all rational terms in

(2^{1/5} + 5^{1/3})^{15}

Sum of rational terms

= \;?

02General Term

2/4

T_{r+1} = {}^{15}C_r \cdot 2^{(15-r)/5} \cdot 5^{r/3}

. Rational when

5 \mid (15-r)

and

3 \mid r

. Common:

r = 0, 15

r = 0

and

r = 15

03Compute TermsKEY INSIGHT

3/4

r = 0

T_1 = 2^3 = 8

r = 15

T_{16} = 5^5 = 3125

. Sum

= 8 + 3125 = 3133

8 + 3125 = 3133

04Final Answer

4/4

Sum of all rational terms is

3133

\boxed{3133}

Concepts from this question1 concepts unlocked

★

EASY

T(r+1) = C(n,r) * a^(n-r) * b^r gives any specific term without expanding fully

T_{r+1} = {}^nC_r\, a^{n-r}\, b^r

Core formula for finding specific terms, term independent of x, rational terms, etc. Used in 4-5 questions every JEE paper.

Specific termIndependent termRational/integral termsCoefficient finding

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