Binomial Theorem PYQs Set 11 (PDF)

Set 11 of 14 · Questions 301-330 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 11 (Q301-330)
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301. $Coefficient of t24 in (1 + t2)12 (1 + t12) (1 + t24) is:$ (1) $12C6 + 3$ (2) $12C6 + 1$ (3) $12C6$ (4) $12C6 + 2$ 302. $The value of (2^{1} P_{0} - 3^{2} P_{1} + 4^{3} P_{2} - \dots up to 51^{th} term) + (1! - 2! + 3! - \dots up to 51^{th} term ) is equal to :$ (1) $1$ (2) $1 + (52)!$ (3) $1 - 51 (51)!$ (4) $1 + (51)!$ 303. $If the coefficient of x in the expansion of (a x^{2} + bx + c) (1 - 2 x)^{26} is -56 and the coefficients of x^{2} and x^{3} are both zero, then a + b + c is equal to :$ (1) $1500$ (2) $1300$ (3) $1403$ (4) $1483$ 304. $The coefficient of x^{18} in the product (1 + x) (1 - x)^{10} (1 + x + x^{2})^{9} is :$ (1) $84$ (2) $-126$ (3) $-84$ (4) $126$ 305. $If 20_{C_{1}} + (2^{2}) 20_{C_{2}} + (3^{2}) 20_{C_{3}} + \dots + (20^{2}) 20_{C_{20}} = A (2^{β}), then the ordered pair (A, β) is equal to :$ (1) $(420, 19)$ (2) $(420, 18)$ (3) $(380, 18)$ (4) $(380, 19)$ 306. $The numerically greatest term in the binomial expansion of (2 a - 3 b)^{19} when a = \frac{1}{4} and b = \frac{2}{3} is$ (1) $^{19} C_{5} \cdot 2^{11}$ (2) $^{19} C_{3} \cdot \frac{1}{2 ^{11}}$ (3) $^{19} C_{4} \cdot \frac{1}{2 ^{13}}$ (4) $^{19} C_{3} \cdot 2^{13}$ 307. $For L.P.P., maximize z = 4 x_{1} + 2 x_{2} subjecto 3 x_{1} + 2 x_{2} \geq 9, x_{1} - x_{2} \leq 3, x_{1} \geq 0, x_{2} \geq 0 has$ (1) $Infinite number of optimal solutions$ (2) $Unbounded solution$ (3) $No solution$ (4) $One optimal solution$ 308. $2 5^{190} - 1 9^{190} - 8^{190} + 2^{190} is divisible by$ (1) $neither 14 nor 34$ (2) $14 but not by 34$ (3) $34 but not by 14$ (4) $both 14 and 34$ 309. $Let the number (22)^{2022} + (2022) leave the remainder a when divided by 3 and β when divided by 7. Then (α^{2} + β^{2}) is equal to$ (1) $20$ (2) $13$ (3) $5$ (4) $10$ 310. $The lowest integer which is greater than (1 + \frac{1}{1 0 ^{100}})^{1 0^{100}} is ___$ (1) $3$ (2) $4$ (3) $2$ (4) $1$ 311. $The maximum value of z = 6 x + 8 y subject to x - y \geq 0, x + 3 y \leq 12, x \geq 0, y \geq 0 is$ (1) $72$ (2) $42$ (3) $96$ (4) $24$ 312. $The coefficient of x^{101} in the expression (5 + x)^{500} + x (5 + x)^{499} + x^{2} (5 + x)^{498} +..... x^{500}, x > 0, is$ (1) $^{501} C_{101} (5)^{399}$ (2) $^{501} C_{101} (5)^{400}$ (3) $^{5 o 1} c_{l 01} (5)^{400}$ (4) $^{500} C_{101} (5)^{399}$ 313. $The remainder when (2021)^{2023} is divided by 7 is:$ (1) $1$ (2) $2$ (3) $5$ (4) $6$ 314. $The maximum value of z = 9 x + 11 y subject to 3 x + 2 y \leq 12, 2 x + 3 y \leq 12, x \geq 0, y \geq 0 is$ (1) $44$ (2) $54$ (3) $36$ (4) $48$ 315. $The objective function z = x_{1} + x_{2}, subject to x_{1} + x_{2} \leq 10, - 2 x_{1} + 3 x_{2} \leq 15, x_{1} \leq 6, x_{1}, x_{2} \geq 0 has maximum value of the feasible region.$ (1) $at only one point$ (2) $at only two points$ (3) $at every point of the segment joining two points$ (4) $at every point of the line joining two points$ 316. $If^{2 n} C_{3} :^{n} C_{3} = 10 : 1, then the ratio (n^{2} + 3n): (n^{2} - 3n + 4) is$ (1) $35 : 16$ (2) $27 : 11$ (3) $65 : 37$ (4) $2 : 1$ 317. $If the coefficients of x and x^{2} in (1 + x)^{p} (1 - x)^{q} are 4 and -5 respectively, then 2p + 3q is equal to$ (1) $60$ (2) $69$ (3) $66$ (4) $63$ 318. $If the 101 1^{t h} term from the end in the binomial expansion of (\frac{4 x}{5} - \frac{5}{2 x})^{2022} is 1024 times 101 1^{t h} term from the beginning, then 32 \|x\| is equal to$ (1) $15$ (2) $10$ (3) $12$ (4) $8$ 319. $Let n \geq 5 be an integer. If 9^{n} - 8n - 1 = 64 α and 6^{n} - 5n - 1 = 25 β, then α - β is equal to:$ (1) $1 +^{n} C_{2} (8-5) +^{n} C_{3} (8^{2} - 5^{2}) +...+^{n} C_{n} (8^{n - 1} - 5^{n - 1})$ (2) $1 +^{n} C_{3} (8-5) +^{n} C_{4} (8^{2} - 5^{2}) +...+^{n} C_{n} (8^{n - 2} - 5^{n - 2})$ (3) $^{n} C_{3} (8-5) +^{n} C_{4} (8^{2} - 5^{2}) +...+^{n} C_{n} (8^{n - 2} - 5^{n - 2})$ (4) $^{n} C_{4} (8-5) +^{n} C_{5} (8^{2} - 5^{2}) +...+^{n} C_{n} (8^{n - 3} - 5^{n - 3})$ 320. $The coefficient of x^{5} in the expansion of (2 x^{3} - \frac{1}{3 x ^{2}})^{5} is$ (1) $\frac{80}{9}$ (2) $9$ (3) $8$ (4) $\frac{26}{3}$ 321. $The maximum value of 2 x + y subject to 3 x + 5 y \leq 26 and 5 x + 3 y \leq 30, x \geq 0, y \geq 0 is$ (1) $12$ (2) $11.5$ (3) $10$ (4) $17.33$ 322. $For the LPP, maximise z = x + 4 y subject to the constraints x + 2 y \leq 2, x + 2 y \geq 8, x, y \geq 0 .$ (1) $z_{max} = 4$ (2) $z_{max} = 8$ (3) $z_{max} = 16$ (4) $Has no feasible solution$ 323. $The remainder obtained when is divided by 10^{2} is$ (1) $27$ (2) $28$ (3) $17$ (4) $14$ 324. $The term independent of x in the expression of (1 - x^{2} + 3 x^{3}) (\frac{5}{2} x^{3} - \frac{1}{5 x ^{2}})^{11}, x \neq = 0 is$ (1) $\frac{7}{40}$ (2) $\frac{33}{200}$ (3) $\frac{39}{200}$ (4) $\frac{11}{50}$ 325. $If the constant term in the expansion of (3 x^{3} - 2 x^{2} + \frac{5}{x ^{5}})^{10} is 2^{k} . l, where l is an odd integer, then the value of k is equal to :$ (1) $6$ (2) $7$ (3) $8$ (4) $9$ 326. $Let a = 1 + \frac{^{2} C _{2}}{3 !} + \frac{^{3} C _{2}}{4 !} + \frac{^{4} C _{2}}{5 !} + \dots, b = 1 + \frac{^{1} C _{0} + ^{1} C _{1}}{1 !} + \frac{^{2} C _{0} + ^{2} C _{1} + ^{2} C _{2}}{2 !} + \frac{^{3} C _{0} + ^{3} C _{1} + ^{3} C _{2} + ^{3} C _{3}}{3 !} + \dots Then \frac{2 b}{a ^{2}} is equal to$ 327. $The index ' n ' of the binomial (\frac{x}{5} + \frac{2}{5})^{n} if the only 9th term of the expansion has numerically the greatest coefficient (n \in N), is :$ 328. $If α = 1 + \sum_{r = 1}^{6} (- 3)^{r - 1}^{12} C_{2 r - 1}, then the distance of the point (12, 3) from the line αx - 3 y + 1 = 0 is .$ 329. $If n is a positive integer & Ck = nCk, find the value of (\sum \frac{k ^{3}}{n ( n + 1 ) ^{2} . ( n + 2 )} (\frac{C _{k}}{C _{k - 1}})^{2})^{- 1} is$ 330. $The value of the expression (\sum (- 1)^{K} \frac{10 C _{K}}{2 ^{K}}) is :$