Binomial Theorem PYQs Set 10 (PDF)

Set 10 of 14 · Questions 271-300 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 10 (Q271-300)
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271. $If the fractional part of the number is , then k is equal to :$ (1) $14$ (2) $8$ (3) $6$ (4) $4$ 272. $In the binomial expansion of (1 + x)^{15}, the coefficients of x^{r} and x^{r + 3} are equal. Then r is$ (1) $7$ (2) $8$ (3) $6$ (4) $4$ 273. $Let S_{n} = 1 + q + q^{2} + \dots. + q^{n} and T_{n} = 1 + (\frac{q + 1}{2}) + (\frac{q + 1}{2})^{2} + \dots. + (\frac{q + 1}{2})^{n} . where q is a real number and q \neq = 1 . If 101C1 + 101C2 . S_{1} + \dots..+ 101C101 . S_{100} = α T_{100} then α is equal to$ (1) $200$ (2) $2^{99}$ (3) $2^{100}$ (4) $202$ 274. $If [x] denotes the greatest integer function on x, then the number of positive integral divisors of [(2 + 3)^{5}] is$ (1) $6$ (2) $4$ (3) $2$ (4) $8$ 275. $Let x = (83 + 13)^{13} and y = (72 + 9)^{9} . If [t] denotes the greatest integer \leq t, then$ (1) $[x] + [y] is even$ (2) $[x] is odd but [y] is even$ (3) $[x] is even but [y] is odd$ (4) $[x] and [y] are both odd$ 276. $If \sum_{r = 0}^{20}^{20 + r} C_{r} = \frac{p}{q}^{40} C_{20} and GCD of (p, q) = 1, then p^{2} - q^{2} =$ (1) $1302$ (2) $1220$ (3) $1240$ (4) $1364$ 277. $For an integer n \geq 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)^{2 n - 3} is 16, then the distance of the point P (2 n - 1, n^{2} - 4 n) from the line x + y = 8 is$ (1) $2$ (2) $22$ (3) $52$ (4) $32$ 278. $The remainder when, 10^{10} \cdot (10^{10} + 1) (10^{10} + 2) is divided by 6 is$ (1) $2$ (2) $4$ (3) $0$ (4) $6$ 279. $If P (n) : " 2^{2 n} - 1 is divisible by k for all n \in N " is true, then the value of ' k' is$ (1) $6$ (2) $3$ (3) $7$ (4) $2$ 280. $If the value of C_{0} + 2 \cdot C_{1} + 3 \cdot C_{2} + \dots \dots \dots + (n + 1) \cdot C_{n} = 576, then n is$ (1) $7$ (2) $5$ (3) $6$ (4) $9$ 281. $The greatest integer less than or equal to (3 + 2)^{5} is$ (1) $721$ (2) $722$ (3) $723$ (4) $724$ 282. $The remainder when (2021)^{2022} + (2022)^{2021} is divided by 7 is$ (1) $0$ (2) $1$ (3) $2$ (4) $6$ 283. $By neglecting x^{4} and higher powers of x, find approximate value of 3 x^{2} + 64 - 3 x^{2} + 27$ (1) $1 - \frac{7}{234} x^{2}$ (2) $1 - \frac{7}{432} x^{2}$ (3) $1 - \frac{7}{32} x^{2}$ (4) $1 - \frac{7}{42} x^{2}$ 284. $If \sum (50_{C_{r}} . 50 - r_{C_{25 - r}}) = K (50_{C_{25}}), then K is equal to :$ (1) $(25)^{2}$ (2) $2^{25} - 1$ (3) $2^{24}$ (4) $2^{25}$ 285. $Let the coefficients of three consecutive terms T_{r}, T_{r + 1} and T_{r + 2} in the binomial expansion of (a + b)^{12} be in a G.P. and let p be the number of all possible values of r . Let q be the sum of all rational terms in the binomial expansion of (43 + 34)^{12} . Then p + q is equal to:$ (1) $283$ (2) $287$ (3) $295$ (4) $299$ 286. $If α and β be the coefficient of x^{4} and x^{2} respectively in the expansion of (x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6} then:$ (1) $α + β = 110$ (2) $α + β = - 150$ (3) $α - β = - 132$ (4) $α - β = - 60$ 287. $The remainder when 7^{n} - 6 n - 50 (n \in N) is divided by 36 , is$ (1) $22$ (2) $23$ (3) $1$ (4) $21$ 288. $Suppose \sum r^{2}^{2023} C_{r} = 2023 \times α \times 2^{2022} . Then the value of α is _________$ 289. $Let α > 0, be the smallest number such that the expansion of (x^{\frac{2}{3}} + \frac{2}{x ^{3}})^{30} has a term β x^{- α}, β ϵ N . Then α is equal to __________$ 290. $If \frac{1}{1 ! 10 !} + \frac{1}{2 ! 9 !} + \frac{1}{3 ! 8 !} + ...... + \frac{1}{10 ! 1 !} = \frac{2}{k !} (2^{k - 1} - 1) then find the value of k.$ 291. $If \sum \frac{2 r + 3}{r + 1} .^{n} C_{r} = \frac{( n + k ) . 2 ^{n + 1} - 1}{n + 1} then 'k' is$ 292. $If (2021)^{3762} is divided by 17, then the remainder is$ 293. $If (2nC1)²+ 2. (2nC2)² + 3. (2nC3)² +... + 2n. (2nC2n)² = 18 . 4n-1C2n-1, then n is :$ 294. $The coefficient of x^{18} in the expansion of (x^{4} - \frac{1}{x ^{3}})^{15} is _____________$ 295. $If the constant term in the expansion of (1 + 2 x - 3 x^{3}) (\frac{3}{2} x^{2} - \frac{1}{3 x})^{9} is p , then 108 p is equal to$ 296. $If the 6th term in the expansion of (\frac{1}{x ^{8/3}} + x^{2} log_{10} x)^{8} is 5600, then x =$ 297. $If the sum of the coefficients of all the positive even powers of x in the binomial expansion of (2 x^{3} + \frac{3}{x})^{10} is 5^{10} - β . 3^{9}, then β is equal to_____$ 298. $If (\frac{1}{^{15} C _{0}} + \frac{1}{^{15} C _{1}}) (\frac{1}{^{15} C _{1}} + \frac{1}{^{15} C _{2}}) \dots (\frac{1}{^{15} C _{12}} + \frac{1}{^{15} C _{13}}) = \frac{α ^{13}}{^{14} C _{0} ^{14} C _{1} \dots ^{14} C _{12}}, then 30 α is equal to____.$ 299. $If the maximum value of the term independent of t in the expansion of (t^{2} x^{\frac{1}{2}} + \frac{( 1 - x ) ^{\frac{1}{10}}}{t})^{15}, x \geq 0, is K, then 8K is equal to_______.$ 300. $Let the coefficients of x^{- 1} and x^{- 3} in the expansion of (2 x^{\frac{1}{5}} - \frac{1}{x ^{\frac{1}{5}}})^{15}, x > 0, be m and n respectively. If r is a positive integer such m n^{2} = 15Cr. 2^{r}, then the value of r is equal to__.$