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Binomial Theorem Previous Year Questions typically cover topics like the general term, middle term, finding specific coefficients, and properties of binomial expansions. Examples include finding the term in an expansion, calculating the coefficient of a particular term in , and using the binomial expansion to simplify expressions or solve identities. Solutions involve applying formulas such as the general term , understanding symmetry in coefficients, and identifying patterns. Practicing these questions helps in mastering expansion techniques and enhances algebraic manipulation skills critical for JEE-level problems. JEE Main Previous Year Solved Questions on Binomial Theorem JEE Adv Previous Year Solved Questions on Binomial Theorem Binomial Theorem Previous Year Questions for JEE with Solutions JEE questions in Binomial Theorem often test concepts related to binomial expansion, general term, middle term, and coefficients of specific terms. Some common types of problems include: General and Middle Term: Questions on finding the r^th term or the middle term in the expansion of expressions like Specific Coefficients: Problems that involve finding the coefficient of a particular term, such as xr in the expansion of or Greatest/Least Term and Term Independent of x: Questions involving identification of terms with maximum numerical value or the term independent of the variable. Properties and Identities: Applications of binomial identities, symmetry of binomial coefficients, and problems involving expansions with negative or fractional exponents (for JEE Advanced). These questions are aimed at testing algebraic manipulation skills and a strong understanding of the binomial expansion formula. Note: In the JEE Main Mathematics exam, you can generally expect 1 to 2 questions from the Binomial Theorem chapter. Key Concepts to Remember – Binomial Theorem (for JEE) 1. Binomial Expansion Formula: Where is the binomial coefficient. 2. General Term: The term in the expansion of is given by: Used to find a specific term or coefficient in the expansion. 3. Middle Term(s): For even n: There is 1 middle term, at position . For odd n: There are 2 middle terms, at positions . 4. Term Independent of x: To find the term independent of xx in an expression like , set the power of x in the general term to zero and solve for r. 5. Greatest Term: Used when finding the numerically greatest term in an expansion for given values of x. Use approximation or comparison between consecutive terms. 6. Properties of Binomial Coefficients: Symmetry: Pascal’s Identity: Sum of all coefficients in 7. Special Expansions: where n is a positive integer. where n is negative or fractional: Use binomial series expansion (for JEE Advanced only). 8. Applications in Inequalities and Approximations: Used in approximation of expressions like for small x, or bounding polynomial values. 9. Negative and Fractional Index: For |x| < 1, Used when n is not a positive integer (JEE Advanced level). Mastering these key concepts is essential to solving both direct and application-based problems from the Binomial Theorem in JEE. JEE Mains Past Year Questions with Solutions on Binomial Theorem 1. If the coefficients of x4, x5 and x6 in the expansion of (1 + x)n are in the arithmetic progression, then the maximum value of n is : (1) 14 (2) 21 (3) 28 (4) 7 Ans. (1) Sol. Coeff. of x4 = nC4 Coeff. of x5 = nC5 Coeff. of x6 = nC6 nC4, nC5, nC6 …. AP 2.nC5 = nC4 + nC6 12(n – 4) = 30 + n2 – 9n + 20 n2 – 21n + 98 = 0 (n – 14) (n – 7) = 0 nmax = 14 nmin = 7 2. The sum of all rational terms in the expansion of \left( \frac{1}{2^{5}} + \frac{1}{5^3} \right)^{15}is equal to: (1) 3133 (2) 633 (3) 931 (4) 6131 Ans. (1) Sol. R = 3, 15µ r = 0, 15 2 rational terms = 8 + 3125 = 3133 3. Let Then is equal to ______ Ans. (8) Sol. 4. If the constant term in the expansion of , x ≠ 0, is α×28× , then 25α is equal to : (1) 639 (2) 724 (3) 693 (4) 742 Ans. (3) Sol. 5. If the constant term in the expansion of is p, then 108p is equal to Ans. (54) Sol. 6. If the second, third and fourth terms in the expansion of (x + y)n are 135, 30 and , respectively, then is equal to _____. Ans. (806) Sol. 7. Let and . If 140 < < 281, then the value of n is ______. Ans. (5) Sol. 8. The sum of the coefficient of x2/3 and x–2/5 in the binomial expansion of is : (1) 21/4 (2) 69/16 (3) 63/16 (4) 19/4 Ans. (1) Sol. 9. The remainder when 4282024 is divided by 21 is ________. Ans. (1) Sol. (428)2024 = (420 + 8)2024 = (21 × 20 + 8)2024 = 21m + 82024 Now 82024 = (82)1012 = (64)1012 = (63 + 1)1012 = (21 × 3 + 1)1012 = 21n + 1 ⇒ The remainder is 1. 10. If the Coefficient of x30 in the expansion of (1 + x2)7 (1 – x3)8 ; x ≠ 0 is α, then |α| equals ___________. Ans. (678) Sol. 11. The coefficient of x2012 in the expansion of is equal to Ans. (0) Sol. 12. Remainder when is divided by 9 is equal to ____. Ans. 1 Sol. 13. Number of integral terms in the expansion of is equal to ________. Ans. (138) Sol. General term in expansion of is tr + 1 = 824Cr For integral terms, r must be multiple of 6. Hence r = 0, 6, 12, ……….822 14. In the expansion of , the sum of the coefficient of x3 and x–13 is equal to ___