If the coefficients of the middle terms in the binomial expansions of (1 + alpha x)^26 and (1 - alpha x)^28, alpha != 0 are equal, then the value of alpha is:

The correct answer is Option 4. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q8 – JEE-MAIN 2026

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Home Maths Binomial TheoremQuestion #44

If the coefficients of the middle terms in the binomial expansions of

(1 + α x)^{26}

and

(1 - α x)^{28}

α \neq 0

are equal, then the value of

α

is:

Clue 1 of 3: Identify the Middle Term

For a binomial expansion $(a + b)^{n}$ , the middle term depends on whether $n$ is even or odd.

🧠 Full Solution Path

Step 1: Identify Middle Terms and their Coefficients✦ Active

For the expansion of $(1 + α x)^{26}$ , $n = 26$ (even). The middle term is the $(\frac{26}{2} + 1)$ -th term, which is $T_{14}$ . Its coefficient is $(\binom{26}{13}) (1)^{13} (α)^{13} = (\binom{26}{13}) α^{13}$ .

For the expansion of $(1 - α x)^{28}$ , $n = 28$ (even). The middle term is the $(\frac{28}{2} + 1)$ -th term, which is $T_{15}$ . Its coefficient is $(\binom{28}{14}) (1)^{14} (- α)^{14} = (\binom{28}{14}) α^{14}$ .

Step 2: Equate Coefficients and Solve for

α

○ Expand

Given that the coefficients of the middle terms are equal, we set up the equation:

(\binom{26}{13}) α^{13} = (\binom{28}{14}) α^{14}

Since $α \neq 0$ , we can divide both sides by $α^{13}$ :

(\binom{26}{13}) = (\binom{28}{14}) α

Solving for $α$ gives:

α = \frac{(\binom{26}{13})}{(\binom{28}{14})}

Step 3: Simplify the Expression for

α

○ Expand

Using the definition of binomial coefficients $(\binom{n}{k}) = \frac{n!}{k! (n - k)!}$ :

α = \frac{\frac{26!}{13! 13!}}{\frac{28!}{14! 14!}} = \frac{26!}{13! 13!} \times \frac{14! 14!}{28!}

Rearranging and simplifying the factorials:

α = \frac{26!}{28 \times 27 \times 26!} \times \frac{14 \times 13!}{13!} \times \frac{14 \times 13!}{13!} = \frac{1}{28 \times 27} \times 14 \times 14

Performing the multiplication and simplification:

α = \frac{14 \times 14}{28 \times 27} = \frac{196}{756} = \frac{7}{27}

💡 Teacher's Secret Hint

Remember that $n! = n \times (n - 1)!$ to simplify ratios of factorials efficiently.

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Maths Question 8 – JEE-MAIN 2026

Clue 1 of 3: Identify the Middle Term

🧠 Full Solution Path