Binomial Theorem PYQs Set 8 (PDF)

Set 8 of 14 · Questions 211-240 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 8 (Q211-240)
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211. $If ' n' is a positive integer, then n^{3} + 2 n is divisible by$ (1) $15$ (2) $3$ (3) $2$ (4) $6$ 212. $If L and M are respectively the coefficient of x^{- 7} in (ax + \frac{b}{x ^{2}})^{11} and the coefficient of x^{7} in (b x^{2} + \frac{a}{x})^{11} then L + M =$ (1) $\frac{1}{b} [coefficient of x^{- 6} in ((ax + \frac{b}{x ^{2}})^{12})$ (2) $\frac{1}{a} [coefficient of x^{6} in ((a x^{2} + \frac{b}{x})^{12})$ (3) $a[coefficient of x^{- 10} in ((ax + \frac{b}{x ^{2}})^{11})$ (4) $b[coefficient of x^{4} in ((a x^{2} + \frac{b}{x})^{11})$ 213. $In the expansion of (32 + \frac{1}{3 3})^{n}, n \in N, if the ratio of 15^{th} term from the beginning to the 15^{th} term from the end is \frac{1}{6}, then the value of^{n} C_{3} is$ (1) $4960$ (2) $2300$ (3) $1040$ (4) $4060$ 214. $If the set of all values of x for which the expansion of (7 - 5 x)^{- \frac{2}{3}} valid is equal to (- a, a), then 5 a + 7 is equal to$ (1) $14$ (2) $21$ (3) $0$ (4) $12$ 215. $The least value of n for which the number of integral terms in the Binomial expansion of (37 + 1211)^{n} is 183, is:$ (1) $2184$ (2) $2196$ (3) $2148$ (4) $2172$ 216. $The sum, of the coefficients of the first 50 terms in the binomial expansion of (1 - x)^{100}, is equal to$ (1) $^{101} C_{50}$ (2) $^{99} C_{49}$ (3) $-^{101} C_{50}$ (4) $-^{99} C_{49}$ 217. $The value of^{16} C_{9} +^{16} C_{10} -^{16} C_{6} -^{16} C_{7} is$ (1) $1$ (2) $^{17} C_{10}$ (3) $^{17} C_{3}$ (4) $0$ 218. $Let S = \frac{1}{25 !} + \frac{1}{3 ! 23 !} + \frac{1}{5 ! 21 !} + \dots up to 13 terms. If 13 S = \frac{2 ^{k}}{n !}, k \in N, then n + k is equal to$ (1) $50$ (2) $49$ (3) $52$ (4) $51$ 219. $The positive value of λ for which the co-efficient of x^{2} in the expression x^{2} (x + \frac{λ}{x ^{2}})^{10} is 720, is :$ (1) $4$ (2) $22$ (3) $5$ (4) $3$ 220. $Let α, β, γ and δ be the coefficients of x^{7}, x^{5}, x^{3} and x respectively in the expansion of (x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, x > 1 . If u and v satisfy the equations αu + βv = 18 γu + δv = 20 then u + v equals$ (1) $5$ (2) $3$ (3) $4$ (4) $8$ 221. $The 13^{th} term in the expansion of (x^{2} + \frac{2}{x})^{n} is independent of x, then the sum of the divisors of n is$ (1) $39$ (2) $36$ (3) $37$ (4) $38$ 222. $The term independent of x in expansion of (\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}})^{10} is :$ (1) $4$ (2) $120$ (3) $210$ (4) $310$ 223. $For natural numbers m, n if (1 - y)m (1 + y)n= 1 + a1y + a2y2 + ..... and a1 = a2 = 10, then (m, n) is :$ (1) $(35, 20)$ (2) $(45, 35)$ (3) $(35, 45)$ (4) $(20, 45)$ 224. $The coefficient of x^{2} in the expansion of (1 - 3 x)^{- 1/4} is$ (1) $\frac{45}{64}$ (2) $\frac{45}{8}$ (3) $\frac{45}{16}$ (4) $\frac{45}{32}$ 225. $The coefficient of x^{4} in the expansion of (1 - x - x^{2} + x^{3})^{6} is$ (1) $120$ (2) $15$ (3) $-75$ (4) $-60$ 226. $If n is a positive integer and the coefficient of x^{10} in the expansion of (1 + x)^{15} is equal to the coefficient of x^{5} in the expansion of (1 - x)^{- n}, then n =$ (1) $15$ (2) $12$ (3) $11$ (4) $10$ 227. $(x^{4} + 1) = \frac{1}{a} (x + 1)^{4} is a reciprocal equation$ (1) $only for a = 1$ (2) $for all a \in R -{1}$ (3) $for all a \in R$ (4) $when a is an irrational number$ 228. $Let a_{0}, a_{1}, a_{2}, \dots . a_{n} \in R be in an arithmetic progression and let C_{0}, C_{1}, C_{2}, \dots, C_{n} be the binomial coefficients Then \sum_{k = 0}^{n} a_{k} \cdot C_{k} =$ (1) $\frac{1}{2} (a_{0} + a_{n})$ (2) $(a_{0} + an) \cdot 2^{n - 1}$ (3) $(a_{0} + an)$ (4) $0$ 229. $The coefficient of x^{6} in the power series expansion of \frac{x ^{4} - 12 x ^{2} + 7}{( x ^{2} + 1 ) ^{3}} is$ (1) $149$ (2) $-253$ (3) $-145$ (4) $253$ 230. $If a is the coefficient of x^{10 - r} in the Binomial expansion of (1 + x)^{10}, then \sum r^{3} (\frac{a _{r}}{a _{r - 1}})^{2} is equal to$ (1) $4895$ (2) $1210$ (3) $5445$ (4) $3025$ 231. $The coefficient of x^{70} in x^{2} (1 + x)^{98} + x^{3} (1 + x)^{97} + x^{4} (1 + x)^{96} + \dots + x^{54} (1 + x)^{46} is^{99} C_{p} -^{46} C_{q} . Then a possible value of p + q is :$ (1) $55$ (2) $83$ (3) $61$ (4) $68$ 232. $If the coefficients of x^{9}, x^{10} and x^{11} in the expansion of (1 + x)^{n} are in arithmetic progression, then n^{2} - 41 n is equal to$ (1) $399$ (2) $298$ (3) $-398$ (4) $198$ 233. $If (^{40} C_{0}) + (^{4 l} C_{1}) + (^{42} C_{2}) + ... + (^{60} C_{20}) = \frac{m}{n} 60C20, m and n are coprime, then m + n is equal to____.$ 234. $For a positive integer n, (1 + \frac{1}{x})^{n} is expanded in increasing powers of x . If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12 then n is equal to$ 235. $Number of integral terms in the expansion of (7^{(\frac{1}{2})} + 11^{(\frac{1}{6})})^{824} is equal to$ 236. $Let C_{r} denote the binomial coefficient of x^{r} in the expansion of (1 + x)^{10} . If α, β \in R . C_{1} +3. 2 C_{2} +5. 3 C_{3} +\dotsupto 10 terms = \frac{α \times 2 ^{11}}{2 ^{β} - 1} (C_{0} + \frac{C _{1}}{2} + \frac{C _{2}}{3} + ... upto 10 terms) then the value of α + β is equal to$ 237. $If \sum_{r = 1}^{30} \frac{r ^{2} ( ^{30} C _{r} ) ^{2}}{^{30} C _{r - 1}} = α \times 2^{29}, then α is equal to .$ 238. $The number of values of ' x ' for which the fourth term in the expansion, (5^{\frac{2}{5} log_{5} 4^{x} + 44} + \frac{1}{5 ^{log_{5} 3 2^{x - 1} + 7}})^{8} is 336, is :$ 239. $If (\frac{3 ^{6}}{4 ^{4}}) k is the term, independent of x, in the binomial expansion of (\frac{x}{4} - \frac{12}{x ^{2}})^{12}, then k is equal to .$ 240. $The coefficient of x^{4} in the expansion of (1 + x + x^{2} + x^{3})^{6} in powers of x, is$