Binomial Theorem PYQs Set 2 (PDF)

Set 2 of 14 · Questions 31-60 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 2 (Q31-60)
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31. $If n is an odd positive integer and (1 + x + x^{2} + x^{3})^{n} = \sum_{r = 0}^{3 n} a_{r} x^{r}, then a_{0} - a_{1} + a_{2} - a_{3} + \dots .. - a_{3 n} is$ (1) $4^{n}$ (2) $1$ (3) $-1$ (4) $0$ 32. $If the term independent of x in the expansion of (a x^{2} + \frac{1}{2 x ^{3}})^{10} is 105 , then a^{2} is equal to :$ (1) $2$ (2) $4$ (3) $6$ (4) $9$ 33. $For n, p \in N - 1, the coefficient of x^{3} in \frac{( 1 - x ) ^{- 1/ p}}{( 1 - x ) ^{n}} is equal to$ (1) $\frac{( np + 1 ) ( np + p + 1 ) ( np + 2 p + 1 )}{p ^{3} \times 3 !}$ (2) $\frac{( np + 1 ) ( np + p ) ( np + 2 p )}{3 ! p ^{3}}$ (3) $\frac{( np + p ) ( np + 2 p ) ( np + 3 p )}{3 ! p ^{3}}$ (4) $\frac{( np + 1 ) ( np + 2 ) ( np + 3 )}{3 ! p ^{3}}$ 34. $If 21^{st} and 22^{nd} terms in the expansion of (1 + x)^{44} are equal, then x is equal to$ (1) $\frac{8}{7}$ (2) $\frac{21}{22}$ (3) $\frac{7}{8}$ (4) $\frac{23}{24}$ 35. $The largest n \in N such that 3^{n} divides 50! is:$ (1) $22$ (2) $20$ (3) $21$ (4) $23$ 36. $Let x \in R be so small that the powers of x beyond two are insignificant and negligibly small. For such x, if (1 - x)^{3} (2 + x)^{6} is approximated by a + bx + c x^{2}, then a + b + c =$ (1) $-80$ (2) $144$ (3) $80$ (4) $127$ 37. $Let S1 = \sum j (j - 1) 10Cj , S2 = \sum j 10Cj and S3 = \sum j^{2} 10Cj. Statement -1 : S3 = 55 \times 2^{9} . Statement -2 : S1 = 90 \times 2^{8} and S2 = 10 \times 2^{8} .$ (1) $Statement -1 is true, Statement-2 is true ; Statement -2 is not a correct explanation for Statement -1.$ (2) $Statement-1 is true, Statement-2 is false.$ (3) $Statement -1 is false, Statement -2 is true.$ (4) $Statement -1 is true, Statement -2 is true; Statement-2 is a correct explanation for Statement-1.$ 38. $Let (1 + x + 2 x^{2})^{20} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{40} x^{40} then a_{1} + a_{3} + a_{5} + \dots + a_{37} is equal to$ (1) $2^{20} (2^{20} - 21)$ (2) $2^{19} (2^{20} - 21)$ (3) $2^{19} (2^{20} + 21)$ (4) $2^{20} (2^{20} + 21)$ 39. $For r = 0, 1, ...., 10, let Ar, Br and Cr denote, respectively, the coefficient of xr in the expansions of (1 + x)10 , (1 + x)20 and (1 + x)30 . Then \sum A_{r} (B_{10} B_{r} - C_{10} A_{r}) is equal to$ (1) $B10 - C10$ (2) $A10 (B210 - C10 A10)$ (3) $0$ (4) $C10 - B10$ 40. $If in the expansion of (1 + x)^{p} (1 - x)^{q}, the coefficients of x and x^{2} are 1 and -2 , respectively, then p^{2} + q^{2} is equal to:$ (1) $18$ (2) $13$ (3) $8$ (4) $20$ 41. $Suppose l, m, n respectively represent the coefficient of x^{10}, the constant term and the coefficient of x^{- 10} in the expansion of (a x^{2} + \frac{b}{x ^{3}})^{15} . If \frac{l}{m} + \frac{m}{n} = \frac{26}{11}, then a^{2} : b^{2} =$ (1) $16 : 9$ (2) $9 : 4$ (3) $4 : 1 .$ (4) $1 : 25$ 42. $If \frac{( 1 - px ) ^{- 1}}{( 1 - qx )} = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots \dots \dots .. then a_{n} =$ (1) $\frac{p ^{n + 1} - q ^{n + 1}}{q - p}$ (2) $\frac{p ^{n + 1} - q ^{n + 1}}{p - q}$ (3) $\frac{p ^{n} - q ^{n}}{q - p}$ (4) $\frac{p ^{n} - q ^{n}}{p - q}$ 43. $If the greatest divisor of 30 \cdot 5^{2 n} + 4 \cdot 2^{3 n} is p, \forall n \in N and that of 2^{2 n + 1} - 6 n - 2 is q, \forall n \in N, then p + q =$ (1) $26$ (2) $52$ (3) $104$ (4) $13$ 44. $The maximum value of the term independent of ’ t ’ in the expansion of (t x^{\frac{1}{5}} + \frac{( 1 - x ) ^{\frac{1}{10}}}{t})^{10} where x \in (0, 1) is:$ (1) $\frac{10 !}{3 ( 5 ! ) ^{2}}$ (2) $\frac{2.10 !}{3 ( 5 ! ) ^{2}}$ (3) $\frac{10 !}{3 ( 5 ! ) ^{2}}$ (4) $\frac{2.10 !}{3 3 ( 5 ! ) ^{2}}$ 45. $When 3^{2023} is divided by 16 , the remainder obtained is$ (1) $15$ (2) $11$ (3) $9$ (4) $7$ 46. $The coefficient of x^{301} in (1 + x)^{500} + x (1 + x)^{499} + x^{2} (1 + x)^{498} + \dots + x^{500} is$ (1) $^{501} C_{302}$ (2) $^{500} C_{301}$ (3) $^{500} C_{300}$ (4) $^{501} C_{200}$ 47. $For all positive integers k, if the greatest divisor of 25^{k} + 12 k - 1 is d, then 4 d =$ (1) $36$ (2) $8$ (3) $20$ (4) $24$ 48. $The constant term in the expansion of (x^{2} - \frac{1}{x ^{2}})^{16} is$ (1) $^{16} C_{8}$ (2) $^{16} C_{7}$ (3) $^{16} C_{9}$ (4) $^{16} C_{10}$ 49. $The coefficient of x7 in the expansion of (1 - x - x2 + x3)6 is :$ (1) $144$ (2) $- 132$ (3) $- 144$ (4) $132$ 50. $A ratio the 5th term from the beginning to the 5th term from the end in the binomial expansion of (2^{\frac{1}{3}} + \frac{1}{2 ( 3 ) ^{\frac{1}{3}}})^{10} is:$ (1) $1 : 2 (6)^{\frac{1}{3}}$ (2) $1 : 4 (16)^{\frac{1}{3}}$ (3) $4 (36)^{\frac{1}{3}} : 1$ (4) $2 (36)^{\frac{1}{3}} : 1$ 51. $If for some m, n;^{6} C_{m} + 2 (^{6} C_{m + 1}) +^{6} C_{m + 2} >^{8} C_{3} and^{n - 1} P_{3} :^{n} P_{4} = 1 : 8, then^{n} P_{m + 1} +^{n + 1} C_{m} is equal to$ (1) $380$ (2) $376$ (3) $384$ (4) $372$ 52. $If the coefficients of (2 r + 1)^{th} term and (r + 1)^{th} term in the expansion of (1 + x)^{42} are equal then r can be$ (1) $12$ (2) $14$ (3) $16$ (4) $20$ 53. $If x is positive, the first negative term in the expansion of (1 + x)^{\frac{27}{5}} is :$ (1) $7th term$ (2) $5th term$ (3) $8th term$ (4) $6th term.$ 54. $The value of r for which^{20} C_{r}^{20} C_{0} +^{20} C_{r - 1}^{20} C_{1} +^{20} C_{r - 2}^{20} C_{2} + \dots +^{20} C_{0}^{20} C_{r} is maximum is :$ (1) $15$ (2) $20$ (3) $11$ (4) $10$ 55. $The number of positive integers k such that the constant term in the binomial expansion of (2 x^{3} + \frac{3}{x ^{k}})^{12}, x \neq = 0 is 2^{8} . l, where l is an odd integer, is________.$ 56. $The number of rational terms in the binomial expansion of (4^{\frac{1}{4}} + 5^{\frac{1}{6}})^{120} is ___$ 57. $Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of (42 + \frac{1}{4 3})^{n}, in the increasing powers of \frac{1}{4 3} be 46 : 1. If the sixth term from the beginning is \frac{α}{4 3}, then a is equal to_______.$ 58. $The coefficient of x9 in the expansion of (1 + x) (1 + x2) (1 + x3) ......(1 + x100) is$ 59. $If (1 + x + x² +... + xp)6 = a0 + a1x + a2x²+...+a6p x6p, then the value of : \frac{1}{p ( p + 1 ) ^{6}} [a1 + 2a2 + 3a3 +.... + 6p a6p] is :$ 60. $Find the value of 6C0. 12C6.- 6C1 11C6+ 6C2 10C6 - 6C3 . 9C6 + 6C4 . 8C6 - 6C5 . 7C6 + 6C6 . 6C6$