Visual SolutionPYQ 2024 · Jan 30 Shift 1Standard

Question

Number of integral terms in the expansion of

\left\{7^{\frac{1}{2}} + 11^{\frac{1}{6}}\right\}^{824}

is equal to _____.

Solution Path

General term

T_{r+1}

needs

r

divisible by 6 for integral powers.

r = 0, 6, \ldots, 822

gives

138

terms.

01Question Setup

1/4

Count the number of integral terms in the expansion of

\{7^{1/2} + 11^{1/6}\}^{824}

\{7^{1/2} + 11^{1/6}\}^{824}

02General Term

2/4

Write

T_{r+1} = {}^{824}C_r \cdot 7^{(824-r)/2} \cdot 11^{r/6}

. For the term to be integral, both exponents must be integers.

T_{r+1} = {}^{824}C_r \cdot 7^{(824-r)/2} \cdot 11^{r/6}

03Divisibility ConditionKEY INSIGHT

3/4

r

must be divisible by 6 (for

11^{r/6}

), which also ensures

(824 - r)

is even (for

7^{(824-r)/2}

r = 0, 6, 12, \ldots, 822

04Final Answer

4/4

Number of valid values =

822/6 + 1 = 137 + 1 = 138

\boxed{138}

Concepts from this question1 concepts unlocked

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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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