Binomial Theorem PYQs Set 9 (PDF)

Set 9 of 14 · Questions 241-270 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 9 (Q241-270)
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241. $In solving the LPP : "minimize f = 6 x + 10 y subject to constraints x \geq 6, y \geq 2, 2 x + y \geq 10, x \geq 0, y \geq 0 " redundant constraints are$ (1) $x\geq6,y\geq2$ (2) $2x+y\geq10,x\geq0,y\geq0$ (3) $x\geq6$ (4) $none of these$ 242. $The digit in the unit place of the number 2009! + 3^{7886} is$ (1) $1$ (2) $9$ (3) $7$ (4) $3$ 243. $The value of \frac{^{100} C _{50}}{51} + \frac{^{100} C _{51}}{52} + \dots + \frac{^{100} C _{100}}{101} is :$ (1) $\frac{2 ^{101}}{100}$ (2) $\frac{2 ^{100}}{100}$ (3) $\frac{2 ^{101}}{101}$ (4) $\frac{2 ^{100}}{101}$ 244. $If (n - 1)Cr = (k2 - 3) nCr+1, then an interval in which k lies is$ (1) $(2, \infty)$ (2) $(- \infty, - 2)$ (3) $(- 3, 3)$ (4) $(3, 2)$ 245. $The greatest value of x satisfying 21 \equiv 385 (mod x) and 587 \equiv 167 (mod x) is$ (1) $32$ (2) $156$ (3) $56$ (4) $28$ 246. $Let S_{n} = 1 + q + q^{2} + \dots + q^{n} and T_{n} = 1 + (\frac{q + 1}{2}) + (\frac{q + 1}{2})^{2} + \dots + (\frac{q + 1}{2})^{n} where q is a real number and q \neq = 1 . If 101_{C_{1}} + 101_{C_{2}} . S_{1} + \dots + 101_{C_{101}} . S_{100} = α T_{100}, then α is equal to :$ (1) $2^{99}$ (2) $202$ (3) $200$ (4) $2^{100}$ 247. $The sum of the series 20C0 - 20C1 + 20C2 - 20C3 + ..... + 20C10 is$ (1) $-20C10$ (2) $\frac{1}{2} 20C10$ (3) $0$ (4) $20C10$ 248. $If x = \frac{2.5}{2 ! 3} + \frac{2.5 . 7}{3 ! 3 ^{2}} + \frac{2 \cdot 5 \cdot 7 \cdot 9}{4 ! 3 ^{3}} + \dots, then x^{2} + 8 x + 8 =$ (1) $108$ (2) $54$ (3) $100$ (4) $144$ 249. $If denotes the fractional part of the number , then is equal to$ (1) $\frac{5}{8}$ (2) $\frac{1}{8}$ (3) $\frac{7}{8}$ (4) $\frac{3}{8}$ 250. $If the coefficients of x^{4}, x^{5} and x^{6} in the expansion of (1 + x)^{n} are in the arithmetic progression, then th maximum value of n is:$ (1) $7$ (2) $21$ (3) $28$ (4) $14$ 251. $The sum of the co-efficient all even degree terms in x in the expansion of (x + x^{3} - 1)^{6} + (x - x^{3} - 1)^{6}, (x > 1) is equal to :$ (1) $29$ (2) $32$ (3) $26$ (4) $24$ 252. $In the expansion of (53 + 32)^{15}$ (1) $Number of rational terms is 3$ (2) $Sum of all rational terms is 58$ (3) $Sum of all rational terms is greater than the sum of all irrational terms$ (4) $Sum of all irrational terms is greater than the sum of all rational terms$ 253. $If (2 - 5 x)^{\frac{- 1}{5}} = a_{0} + a_{1} x + a_{2} x^{2} + \dots, then \frac{a _{1}}{a _{2}} =$ (1) $\frac{1}{3}$ (2) $- \frac{2}{3}$ (3) $- \frac{1}{3}$ (4) $\frac{2}{3}$ 254. $Let x be a real number and - 2 < x < 2 . When \frac{x + 1}{( x + 3 ) ( x - 2 )} is expanded in powers of x, then the coefficient of x^{3} is$ (1) $- \frac{55}{1296}$ (2) $- \frac{97}{216}$ (3) $- \frac{13}{216}$ (4) $- \frac{119}{1800}$ 255. $The sum of the coefficients of x^{499} and x^{500} in (1 + x)^{1000} + x (1 + x)^{999} + x^{2} (1 + x)^{998} + \dots + x^{1000} is :$ (1) $^{1002} C_{500}$ (2) $^{1002} C_{501}$ (3) $^{1001} C_{501}$ (4) $^{1000} C_{501}$ 256. $Let m and n be the coefficients of seventh and thirteenth terms respectively in the expansion of (\frac{1}{3} x^{\frac{1}{3}} + \frac{1}{2 x ^{\frac{2}{3}}})^{18} . Then (\frac{n}{m})^{\frac{1}{3}} is :$ (1) $\frac{4}{9}$ (2) $\frac{1}{9}$ (3) $\frac{1}{4}$ (4) $\frac{9}{4}$ 257. $The coefficient of the highest power of x in the expansion of (x + x^{2} - 1)^{8} + (x - x^{2} - 1)^{8} is$ (1) $64$ (2) $128$ (3) $256$ (4) $512$ 258. $If the term independent of x in the expansion of (\frac{3}{2} x^{2} - \frac{1}{3 x})^{9} is k then 18 k is equal to:$ (1) $9$ (2) $11$ (3) $5$ (4) $7$ 259. $^{n - 1} C_{r} = (k^{2} - 8)^{n} C_{r + 1} if and only if :$ (1) $22 < k \leq 3$ (2) $23 < k \leq 32$ (3) $23 < k < 33$ (4) $22 < k < 23$ 260. $The sum of all possible values of n \in N, so that the coefficients of x, x^{2} and x^{3} in the expansion of (1 + x^{2})^{2} (1 + x)^{n}, are in arithmetic progression is :$ (1) $12$ (2) $7$ (3) $3$ (4) $9$ 261. $n^{5} - 5 n^{3} + 4 n is divisible by 120 is true for$ (1) $all positive integers n$ (2) $all positive integers for n\geq3 only$ (3) $all positive integers for n\geq1 only$ (4) $all positive integers for n\geq5 only$ 262. $Let (1 + x + x^{2})^{10} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{20} x^{20} . If (a_{1} + a_{3} + a_{5} + \dots + a_{19}) - 11 a_{2} = 121 k, then k is equal to _____.$ 263. $Let (2 x^{2} + 3 x + 4)^{10} = \sum_{r = 0}^{20} a_{r} x^{r} Then \frac{a _{7}}{a _{13}} is equal to$ 264. $The ratio of the coefficient of the middle term in the expansion of (1 + x)^{20} and the sum of the coefficients of two middle terms in expansion of (1 + x)^{19} is$ 265. $If C_{r} =^{25} C_{r} and C_{0} + 5 \cdot C_{1} + 9 \cdot C_{2} + \dots + (101) \cdot C_{25} = 2^{25} \cdot k then k is equal to$ 266. $If the Coefficient of x^{30} in the expansion of (1 + \frac{1}{x})^{6} (1 + x^{2})^{7} (1 - x^{3})^{8}; x \neq = 0 is α, then \| α \| equals$ 267. $The sum of the series 2 \times 1 \times^{20} C_{4} - 3 \times 2 \times^{20} C_{5} + 4 \times 3 \times^{20} C_{6} - 5 \times 4 \times^{20} C_{7} + \dots + 18 \times 17 \times^{20} C_{20}, is equal to _______.$ 268. $Let the coefficient of x^{r} in the expansion of (x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots \dots + (x + 2)^{n - 1} be α_{r} . If \sum_{r = 0}^{n} α_{r} = β^{n} - γ^{n}, β, γ \in N, then the value of β^{2} + γ^{2} equals$ 269. $If the co-efficient of x^{7} and x^{8} in the expansion of (2 + \frac{x}{3})^{n} are equal, then the value of n is equal to ___$ 270. $Let the sum of the coefficients of the first three terms in the expansion of (x - \frac{3}{x ^{2}})^{n}, x \neq = 0, n ϵ N, be 376. Then the coefficient of x^{4} is ________$