If the term without in the expansion of is , then is equal to:

1 Explanation: Solution 1. General Term Formula In the binomial expansion of , the general term is given by: For the given expression : Substituting these into the formula: 2. Simplify the Power of Combine the exponents of : 3. Condition for the Term Independent of For the term to be "without " (independent of ), the exponent of must be zero: 4. Calculate The value of this term ( ) is given as : Calculate : So the equation becomes: Final Answer: The value of is .

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the term without $x$ in the expansion of $\left(x^{\frac{2}{3}} + \frac{a}{x^3}\right)^{22}$ is $7315$ , then $|a|$ is equal to:

Choose the correct answer:

Explanation

Solution

1. General Term Formula

In the binomial expansion of $(A + B)^n$ , the general term $T_{r+1}$ is given by:

$T_{r+1} = \binom{n}{r} A^{n-r} B^r$

For the given expression $\left(x^{2/3} + \frac{a}{x^3}\right)^{22}$ :

$n = 22$

$A = x^{2/3}$

$B = \frac{a}{x^3} = a \cdot x^{-3}$

Substituting these into the formula:

$T_{r+1} = \binom{22}{r} (x^{2/3})^{22-r} (ax^{-3})^r$

$T_{r+1} = \binom{22}{r} x^{\frac{44-2r}{3}} \cdot a^r \cdot x^{-3r}$

2. Simplify the Power of $x$

Combine the exponents of $x$ :

$T_{r+1} = \binom{22}{r} a^r \cdot x^{\left(\frac{44-2r}{3} - 3r\right)}$

$T_{r+1} = \binom{22}{r} a^r \cdot x^{\left(\frac{44-2r-9r}{3}\right)}$

$T_{r+1} = \binom{22}{r} a^r \cdot x^{\left(\frac{44-11r}{3}\right)}$

3. Condition for the Term Independent of $x$

For the term to be "without $x$ " (independent of $x$ ), the exponent of $x$ must be zero:

$\frac{44 - 11r}{3} = 0$

$44 - 11r = 0$

$11r = 44 \implies r = 4$

4. Calculate $|a|$

The value of this term ( $T_{4+1}$ ) is given as $7315$ :

$\binom{22}{4} a^4 = 7315$

Calculate $\binom{22}{4}$ :

$\binom{22}{4} = \frac{22 \times 21 \times 20 \times 19}{4 \times 3 \times 2 \times 1} = 7315$

So the equation becomes:

$7315 \cdot a^4 = 7315$

$a^4 = 1$

$a^2 = 1 \text{ or } a^2 = -1 \text{ (not possible for real } a \text{)}$

$|a| = \sqrt{1} = 1$

Final Answer:

The value of $|a|$ is $1$ .

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