Binomial Theorem PYQs Set 7 (PDF)

Set 7 of 14 · Questions 181-210 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 7 (Q181-210)
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181. $The sixth term in the expansion of (3 - \frac{17}{4} + 32)^{10} is a$ (1) $positive rational number$ (2) $negative rational number$ (3) $positive irrational number$ (4) $negative irrational number$ 182. $The smallest natural number n, such that the coefficient of x in the expansion of (x^{2} + \frac{1}{x ^{3}})^{n} is n_{C_{23}}, is :$ (1) $38$ (2) $58$ (3) $23$ (4) $35$ 183. $If the coefficient of x^{15} in the expansion of (a x^{3} + \frac{1}{b x ^{1/3}})^{15} is equal to the coefficient of x^{- 15} in the expansion of (a x^{1/3} - \frac{1}{b x ^{3}})^{15}, where a and b are positive real numbers, then for each such ordered pair (a, b) :$ (1) $a = b$ (2) $ab = 1$ (3) $a = 3b$ (4) $ab = 3$ 184. $The coefficient of x^{5} in the expansion of (x^{2} + 2 x + 3)^{5}, is$ (1) $1052$ (2) $540$ (3) $480$ (4) $1020$ 185. $For any natural number n, (15 \times 5^{2 n}) + (2 \times 2^{3 n}) is divisible by -$ (1) $7$ (2) $11$ (3) $13$ (4) $17$ 186. $The coefficient of xn in the expansion of (1 + x) (1 - x)n is-$ (1) $(n - 1)$ (2) $(-1)n (1 - n)$ (3) $(-1)n-1(n - 1)2$ (4) $(-1)n - 1 n$ 187. $For the natural numbers m, n, if (1 - y)^{m} (1 + y)^{n} = 1 + a_{1} y + a_{2} y^{2} +\dots.+ a_{m + n} y^{m + n} and a_{1} = a_{2} = 10, then the value of (m + n) is equal to:$ (1) $88$ (2) $64$ (3) $100$ (4) $80$ 188. $The total number of terms in the expansion of (x + y)^{100} + (x - y)^{100} after simplification is$ (1) $100$ (2) $50$ (3) $51$ (4) $202$ 189. $Coefficient of x11 in the expansion of (1 + x2)4 (1 + x3)7 (1 + x4)12 is$ (1) $1051$ (2) $1106$ (3) $1113$ (4) $1120$ 190. $In the expansion of (\frac{x}{cos θ} + \frac{1}{x sin θ})^{16}, if l_{1} is the least value of the term independent of x when \frac{π}{8} \leq θ \leq \frac{π}{4} and l_{2} is the least value of the term independent of x when \frac{π}{16} \leq θ \leq \frac{π}{8}, then the ratio l_{2} : l_{1} is equal to:$ (1) $1 : 8$ (2) $1 : 16$ (3) $8 : 1$ (4) $16 : 1$ 191. $The coefficient of x^{5} in (1 + 2 x + 3 x^{2} + \dots)^{- 3/2} is :$ (1) $21$ (2) $25$ (3) $26$ (4) $none of these$ 192. $The digit in the unit's place of 7^{171} + (177)! is$ (1) $3$ (2) $2$ (3) $1$ (4) $0$ 193. $The remainder obtained when 5^{124} is divided by 124 is$ (1) $5$ (2) $0$ (3) $2$ (4) $1$ 194. $If n \geq 2 is a positive integer, then the sum of the series^{n + 1} C_{2} + 2 (^{2} C_{2} +^{3} C_{2} +^{4} C_{2} + \dots +^{n} C_{2}) is:$ (1) $\frac{n ( n + 1 ) ^{2} ( n + 2 )}{12}$ (2) $\frac{n ( n - 1 ) ( 2 n + 1 )}{6}$ (3) $\frac{n ( n + 1 ) ( 2 n + 1 )}{6}$ (4) $\frac{n ( 2 n + 1 ) ( 3 n + 1 )}{6}$ 195. $To find the coefficient of x^{4} in the expansion of \frac{3 x}{( x - 2 ) ( x - 1 )}, the interval in which the expansion is valid, is$ (1) $- 2 < x < \infty$ (2) $- \frac{1}{2} < x < \frac{1}{2}$ (3) $- 1 < x < 1$ (4) $- \infty < x < \infty$ 196. $In the expansion of (x - 2 y + 3 z)^{5}, if the total number of terms is p and the coefficient of x^{2} y z^{2} is q, then \frac{q}{p} =$ (1) $60$ (2) $- \frac{180}{7}$ (3) $72$ (4) $- \frac{1080}{7}$ 197. $The remainder obtained when 1! + 2! + 3! + \dots + 11! is divided by 12 is$ (1) $8$ (2) $6$ (3) $9$ (4) $7$ 198. $If some three consecutive coefficients in the binomial expansion of (x + 1)^{n} in powers of x are in the ratio 2:15:70, then the average of these three coefficients is :$ (1) $964$ (2) $232$ (3) $227$ (4) $625$ 199. $The sum of the series 2 ∙ 20_{C_{0}} + 5 ∙ 20_{C_{1}} + 8 ∙ 20_{C_{2}} + 11 ∙ 20_{C_{3}} + \dots + 62 ∙ 20_{C_{20}} is equal to :$ (1) $2^{26}$ (2) $2^{25}$ (3) $2^{23}$ (4) $2^{24}$ 200. $If the fourth term in the expansion of (x + x^{log_{2} x})^{7} is 4480 then the value of x where x \in N is equal to$ (1) $2$ (2) $4$ (3) $3$ (4) $1$ 201. $Let m be the smallest positive integer such that the coefficient of x2 in the expansion of (1 + x)2 + (1 + x)3 +........+ (1 + x)49 + (1 + mx)50 is (3n + 1) 51C3 for some positive integer n. Then the value of n is$ 202. $Remainder when 64^{32^{32}} is divided by 9 is equal to_____$ 203. $Let α = \sum_{r = 0}^{n} (4 r^{2} + 2 r + 1)^{n} C_{r} and β = (\sum_{r = 0}^{n} \frac{^{n} C _{r}}{r + 1}) + \frac{1}{n + 1} . If 140 < \frac{2 α}{β} < 281, then the value of n is_____$ 204. $The remainder on dividing 1 + 3 + 3^{2} + 3^{3} + ... + 3^{2021} by 50 is ________ .$ 205. $Let for the 9^{t h} term in the binomial expansion of (3 + 6 x)^{n}, in the increasing powers of 6x, to be the 3 greatest for x = \frac{3}{2}, the least value of n is n0. If k is the ratio of the coefficient of x^{6} to the coefficient of x^{3}, then k + n0 is equal to:$ 206. $The natural number m, for which the coefficient of x in the binomial expansion of (x^{m} + \frac{1}{x ^{2}})^{22} is 1540,is$ 207. $The constant term in the expansion of (2 x + \frac{1}{x ^{7}} + 3 x^{2})^{5} is _________$ 208. $If { x } denotes the fractional part of ' x ', then 82 =$ 209. $The coefficients of three consecutive terms of (1 + x)n+5 are in the ratio 5 : 10 : 14. Then n =$ 210. $The remainder when 428^{2024} is divided by 21 is____$