Binomial Theorem PYQs Set 1 (PDF)

Set 1 of 14 · Questions 1-30 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 1 (Q1-30)
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1. $The coefficient of x^{3} in the expansion of (1 - x)^{3/2}, (\| x \| < 1) is$ (1) $- \frac{3}{16}$ (2) $\frac{1}{16}$ (3) $\frac{1}{8}$ (4) $\frac{3}{16}$ 2. $If A denotes the sum of all the coefficients in the expansion of (1 - 3 x + 10 x^{2})^{n} and B denotes the sum of all the coefficients in the expansion of (1 + x^{2})^{n}, then :$ (1) $A = B^{3}$ (2) $3 A = B$ (3) $B = A^{3}$ (4) $A = 3 B$ 3. $If the ratio of the 7th term from the beginning to the 7 th term from the end in the expansion of (32 + \frac{1}{3 3})^{n} is \frac{1}{6}, then n =$ (1) $6$ (2) $8$ (3) $9$ (4) $12$ 4. $The remainder when ((64)^{(64)})^{(64)} is divided by 7 is equal to$ (1) $4$ (2) $6$ (3) $3$ (4) $1$ 5. $Let m = (9 n^{2} + 54 n + 80) (9 n^{2} + 45 n + 54) (9 n^{2} + 36 n + 35) . The greatest positive integer which divides m, for all positive integers n, is$ (1) $720$ (2) $724$ (3) $696$ (4) $842$ 6. $Let S = {x : x ϵ R and (3 + 2)^{x^{2} - 4} + (3 - 2)^{x^{2} - 4} - 10} Then n (S) eqaual to$ (1) $2$ (2) $4$ (3) $6$ (4) $0$ 7. $The sum of all rational terms in the expansion of (2 + 3)^{8} is:$ (1) $16923$ (2) $18817$ (3) $3763$ (4) $33845$ 8. $The sum of the real values of x for which the middle term in the binomial expansion of (\frac{x ^{3}}{3} + \frac{3}{x})^{8} equals 5670 is :$ (1) $0$ (2) $6$ (3) $4$ (4) $8$ 9. $The sum of the coefficients in the expansion of (1 + \frac{x}{2})^{12} is$ (1) $0$ (2) $2^{11}$ (3) $(\frac{3}{2})^{12}$ (4) $2^{12}$ 10. $For x = \frac{5}{7}, if t_{k} is the first negative term in the expansion of (1 + x)^{7/5}, then t_{1} + t_{2} + \dots t_{k} =$ (1) $\frac{13}{7}$ (2) $\frac{107}{14}$ (3) $\frac{104}{49}$ (4) $\frac{921}{28}$ 11. $In the third term in binomial expansion of (1 + x^{log_{2} x})^{5} equals 2560, then a possible value of x is :$ (1) $\frac{1}{4}$ (2) $42$ (3) $\frac{1}{8}$ (4) $22$ 12. $If (1 - x + x^{2})^{10} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{20} x^{20}, then 2 a_{2} + 3 a_{3} + 4 a_{4} + \dots + 20 a_{20} =$ (1) $0$ (2) $10$ (3) $20$ (4) $-20$ 13. $If the 4^{th} term in the expansion of (\frac{x}{2} - \frac{2 y}{3})^{6} is -20 , then xy =$ (1) $2$ (2) $3$ (3) $8$ (4) $27$ 14. $If x is so small that x3 and higher powers of x may be neglected, then \frac{( 1 + x ) ^{3/2} - ( 1 + \frac{1}{2} x ) ^{3}}{( 1 - x ) ^{1/2}} may be approximated as :$ (1) $\frac{x}{2} - \frac{3}{8} x^{2}$ (2) $- \frac{3}{8} x^{2}$ (3) $3 x + \frac{3}{8} x^{2}$ (4) $1 - \frac{3}{8} x^{2}$ 15. $The sum of coefficients of integral powers of x in the binomial expansion of (1 - 2 x)^{50} is$ (1) $\frac{1}{2} (3^{50} + 1)$ (2) $\frac{1}{2} (3^{50})$ (3) $\frac{1}{2} (3^{50} - 1)$ (4) $\frac{1}{2} (2^{50} + 1)$ 16. $The value of the numerically greatest term in the expansion of (2 x + 3 y)^{11}, when x = \frac{1}{2} and y = \frac{1}{3} is$ (1) $462$ (2) $^{11} C_{5} (\frac{2}{3})^{6}$ (3) $^{11} C_{6} (\frac{3}{2})^{5}$ (4) $576$ 17. $If \sum_{r = 0}^{10} (\frac{10 ^{r + 1} - 1}{10 ^{r}}) \cdot^{11} C_{r + 1} = \frac{α 11 - 11 ^{11}}{10 ^{10}}, then α,is equal to:$ (1) $11$ (2) $20$ (3) $24$ (4) $15$ 18. $The coefficients of x^{50} in (1 + x)^{101} (1 - x + x^{2})^{100} is \dots \dots \dots$ (1) $1$ (2) $-1$ (3) $0$ (4) $2$ 19. $If P (n) : 2^{n} < n! then the smallest positive integer for which P (n) is true, is$ (1) $2$ (2) $4$ (3) $3$ (4) $5$ 20. $Statement-1 : \sum (r + 1) nCr = (n + 2) 2^{n - 1} Statement-2 : \sum (r + 1) nCr xr = (1 + x)^{n} + n x (1 + x)^{n - 1}$ (1) $Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.$ (2) $Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1$ (3) $Statement-1 is True, Statement-2 is False$ (4) $Statement-1 is False, Statement-2 is True$ 21. $If α and β are the least positive integers such that for all n \in N, n^{3} + αn is divisible by 3 and n^{3} - βn is divisible by 6 , then α + β =$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 22. $The coefficient of the middle term in the binomial expansion in powers of x of (1 + αx)4 and of (1 - αx)6 is the same, if α equals :$ (1) $- \frac{5}{3}$ (2) $\frac{10}{3}$ (3) $- \frac{3}{10}$ (4) $\frac{3}{5}$ 23. $The term independent of x in the expansion of (\frac{1}{60} - \frac{x ^{8}}{81}) . (2 x^{2} - \frac{3}{x ^{2}})^{6} is equal to:$ (1) $-72$ (2) $36$ (3) $-36$ (4) $-108$ 24. $The term independent of x in the expansion of (\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}})^{10}, x \neq = 1, is equal to$ 25. $In the expansion of (3^{\frac{- x}{4}} + 3^{\frac{5 x}{4}})^{n}, the sum of the binomial coefficients is 256 and four times the term with greatest binomial coefficient exceeds the square of the third term by 21n, then find 4x.$ 26. $If \sum \frac{( - 1 ) ^{r} . C _{r}}{( r + 1 ) ( r + 2 ) ( r + 3 )} = \frac{1}{a ( n + b )}, then a + b is$ 27. $The product of the last two digits of (1919)^{1919} is$ 28. $If \sum K^{2} (1 0_{C_{K}})^{2} = 22000 L, then L is equal to___.$ 29. $If the second, third and fourth terms in the expansion of (x + y)^{n} are 135,30 and \frac{10}{3}, respectively, then 6 (n^{3} + x^{2} + y) is equal to______$ 30. $The remainder when (2023)^{2023} is divided by 35$