If 26((2^3/3)(^12C_2) + (2^5/5)(^12C_4) + (2^7/7)(^12C_6) + ... + (2^13/13)(^12C_12)) = 3^13 - alpha, then alpha is equal to :

The correct answer is Option 3. 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 #43

26 (\frac{2^{3}}{3} (^{12} C_{2}) + \frac{2^{5}}{5} (^{12} C_{4}) + \frac{2^{7}}{7} (^{12} C_{6}) + \dots + \frac{2^{13}}{13} (^{12} C_{12})) = 3^{13} - α

, then

α

is equal to :

Clue 1 of 3: Conceptual Principle

Recognize that the given sum resembles the result of integrating a binomial expansion term by term.

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Step 1: Express the sum using binomial expansion and integration✦ Active

Let the given sum be $P$ . We know that $(1 + x)^{12} + (1 - x)^{12} = 2 (^{12} C_{0} +^{12} C_{2} x^{2} +^{12} C_{4} x^{4} + \dots +^{12} C_{12} x^{12})$ . Let $f (x) = \frac{1}{2} ((1 + x)^{12} + (1 - x)^{12})$ . Integrating $f (x)$ from $0$ to $2$ gives:

\int_{0}^{2} f (x) d x = {[^{12} C_{0} x +^{12} C_{2} \frac{x^{3}}{3} +^{12} C_{4} \frac{x^{5}}{5} + \dots +^{12} C_{12} \frac{x^{13}}{13}]}_{0}^{2}

Evaluating the definite integral at the limits:

\int_{0}^{2} f (x) d x = (^{12} C_{0} (2) +^{12} C_{2} \frac{2^{3}}{3} +^{12} C_{4} \frac{2^{5}}{5} + \dots +^{12} C_{12} \frac{2^{13}}{13}) - (0)

This expression is $2 \cdot^{12} C_{0} + P$ . Since $^{12} C_{0} = 1$ , we have $\int_{0}^{2} f (x) d x = 2 + P$ . Therefore, $P = \int_{0}^{2} f (x) d x - 2$ .

Step 2: Evaluate the definite integral○ Expand

Now, we evaluate the integral $I = \int_{0}^{2} f (x) d x$ :

I = \int_{0}^{2} \frac{1}{2} ((1 + x)^{12} + (1 - x)^{12}) d x = \frac{1}{2} {[\frac{(1 + x)^{13}}{13} - \frac{(1 - x)^{13}}{13}]}_{0}^{2}

Substitute the limits of integration:

I = \frac{1}{26} [((1 + 2)^{13} - (1 - 2)^{13}) - ((1 + 0)^{13} - (1 - 0)^{13})]

I = \frac{1}{26} [(3^{13} - (- 1)^{13}) - (1^{13} - 1^{13})]

I = \frac{1}{26} [(3^{13} - (- 1)) - (1 - 1)]

I = \frac{1}{26} [(3^{13} + 1) - 0] = \frac{3^{13} + 1}{26}

💡 Teacher's Secret Hint

Be careful with the signs when evaluating $(1 - x)^{13}$ at the limits, especially for $x = 0$ and $x = 2$ .

Step 3: Calculate

α

○ Expand

Substitute the value of $I$ back into the expression for $P$ :

P = I - 2 = \frac{3^{13} + 1}{26} - 2

The given equation is $26 P = 3^{13} - α$ . Multiply $P$ by $26$ :

26 P = 26 (\frac{3^{13} + 1}{26} - 2)

26 P = (3^{13} + 1) - 26 \times 2

26 P = 3^{13} + 1 - 52

26 P = 3^{13} - 51

Comparing this with $3^{13} - α$ , we find that $α = 51$ .

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Maths Question 8 – JEE-MAIN 2026

Clue 1 of 3: Conceptual Principle

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