Binomial Theorem PYQs Set 12 (PDF)

Set 12 of 14 · Questions 331-360 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 12 (Q331-360)
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331. $The coefficient of x^{256} in the expansion of (1 - x)^{101} (x^{2} + x + 1)^{100} is:$ (1) $^{100} C_{16}$ (2) $^{100} C_{15}$ (3) $-^{100} C_{16}$ (4) $-^{100} C_{15}$ 332. $For the following shaded region, the linear constraints are$ (1) $x+2y\geq6,5x+3y\geq15,x\leq7,y\leq6,x,y\geq0$ (2) $x+2y\leq6,5x+3y\leq15,x\leq7,y\leq6,x,y\geq0$ (3) $x+2y\leq6,5x+3y\geq15,x\leq7,y\leq6,x,y\geq0$ (4) $x+2y\geq6,5x+3y\leq15,x\leq7,y\leq6,x,y\geq0$ 333. $If b is very small as compared to the value of a, so that the cube and other higher powers of \frac{b}{a} can be neglected in the identity \frac{1}{a - b} + \frac{1}{a - 2 b} + \frac{1}{a - 3 b} + .... + \frac{1}{a - nb} = α n + β n^{2} + γ n^{3},then the value of γ is :$ (1) $\frac{a ^{2} + b}{3 a ^{3}}$ (2) $\frac{a + b}{3 a ^{2}}$ (3) $\frac{b ^{2}}{3 a ^{3}}$ (4) $\frac{a + b ^{2}}{3 a ^{3}}$ 334. $Let (a + b x + c x^{2})^{10} = \sum_{i = 10}^{20} p_{i} x_{i}, a, b, c \in N . If p_{1} = 20 and p_{2} = 210, then 2(a + b + c) is equal to$ (1) $6$ (2) $15$ (3) $12$ (4) $8$ 335. $If x + y \leq 2, x \geq 0, y \geq 0 the point at which maximum value of 3 x + 2 y attained will be$ (1) $(0, 2)$ (2) $(0, 0)$ (3) $(2, 0)$ (4) $(\frac{1}{2}, \frac{1}{2})$ 336. $If the coefficients of three consecutive terms in the expansion of (1 + x)^{n} are the ratio 1 : 5 : 20 then the coefficient of the fourth term is$ (1) $2436$ (2) $5481$ (3) $1827$ (4) $3654$ 337. $The minimum value of z = 10 x + 25 y subject to 0 \leq x \leq 3, 0 \leq y \leq 3, x + y \geq 5 is$ (1) $80$ (2) $95$ (3) $105$ (4) $30$ 338. $The feasible region of an LPP is shown in the figure. If z = 3 x + 9 y, then the minimum value of z occurs at$ (1) $(5, 5)$ (2) $(0, 10)$ (3) $(0, 20)$ (4) $(15, 15)$ 339. $Maximize Z = 7 x + 11 y, subject to 3 x + 5 y \leq 26, 5 x + 3 y \leq 30, x \geq 0, y \geq 0 .$ (1) $59 at (\frac{9}{2}, \frac{5}{2})$ (2) $42 at (6, 0)$ (3) $49 at (7, 0)$ (4) $57.2 at (0, 5.2)$ 340. $The absolute difference of the coefficients of x^{10} and x^{7} in the expansion of (2 x^{2} + \frac{1}{2 x})^{11} is equal to$ (1) $1 3^{3} -13$ (2) $1 1^{3} -11$ (3) $1 0^{3} - 10$ (4) $1 2^{3} - 12$ 341. $The objective function z = 4 x_{1} + 5 x_{2}, subject to 2 x_{1} + x_{2} \geq 7, 2 x_{1} + 3 x_{2} \leq 15, x_{2} \leq 3, x_{1}, x_{2} \geq 0 has minimum value at the point$ (1) $On x -axis$ (2) $On y -axis$ (3) $At the origin$ (4) $On the line parallel to x -axis$ 342. $Let [x] denote greatest integer less than or equal to x . If for n \in N, (1 - x + x^{3})^{n} = \sum_{j = 0}^{3 n} a_{j} x^{j} then \sum_{j = 0}^{[3 n /2]} a_{2 j} + 4 \sum_{j = 0}^{[(3 n - 1) /2]} a_{2 j + 1} is equal to:$ (1) $2$ (2) $2^{n - 1}$ (3) $1$ (4) $n$ 343. $The value of \sum_{r = 0}^{6} (^{6} C_{r} \cdot^{6} C_{6 - r}) is equal to$ (1) $1124$ (2) $1324$ (3) $1024$ (4) $924$ 344. $Let C_{r} denote the coefficient of x^{r} in the binomial expansion of (1 + x)^{n}, n \in N, 0 \leq r \leq n . If P_{n} = C_{0} - C_{1} + \frac{2 ^{2}}{3} C_{2} - \frac{2 ^{3}}{4} C_{3} + \dots + \frac{( - 2 ) ^{n}}{n + 1} C_{n}, then the value of \sum_{n = 1}^{25} \frac{1}{P _{2 n}} equals.$ (1) $650$ (2) $675$ (3) $525$ (4) $580$ 345. $The value of \sum_{r = 0}^{20}^{50 - r} C_{6} is equal to:$ (1) $^{51} C_{7} -^{30} C_{7}$ (2) $^{50} C_{7} -^{30} C_{7}$ (3) $^{51} C_{7} +^{30} C_{7}$ (4) $^{50} C_{6} -^{30} C_{6}$ 346. $If \sum_{r = 1}^{9} (\frac{r + 3}{2 ^{r}}) \cdot^{9} C_{r} = α (\frac{3}{2})^{9} - β, α, β \in N, then (α + β)^{2} is equal to$ (1) $27$ (2) $81$ (3) $18$ (4) $9$ 347. $The total number of terms in the expansion of (x + a)^{47} - (x - a)^{47} after simplification is$ (1) $24$ (2) $96$ (3) $47$ (4) $48$ 348. $The number (49^{2} - 4) (49^{3} - 49) is divisible by$ (1) $9!$ (2) $7!$ (3) $5 !$ (4) $6 !$ 349. $If the coefficient of x^{3} in the binomial expansion of x^{3} (23 x^{2} + \frac{1}{k x})^{12} is 880, then k is equal to$ (1) $22$ (2) $43$ (3) $23$ (4) $3$ 350. $The coefficient of x^{7} in (1 - x + 2 x^{3})^{10} is__________.$ 351. $The number of integral terms in the expansion of (3^{\frac{1}{2}} + 5^{\frac{1}{4}})^{680} is equal to$ 352. $The largest natural number n such that 3n divides 66! is________$ 353. $Let the co-efficients of x^{n} in (1 + x)^{2 n} and (1 + x)^{2 n - 1} be P and Q respectively, then (\frac{P + Q}{Q})^{5} =$ 354. $The value of p, for which coefficient of x^{50} in the expression (1 + x)^{1000} + 2 x (1 + x)^{999} + 3 x^{2} (1 + x)^{998} + \dots + 1001 x^{1000} is equal to^{1002} C_{p}, is$ 355. $The mean of the coefficients of x, x^{2}, \dots, x^{7} in the binomial expression of (2 + x)^{9} is_______$ 356. $The value of λ if \sum_{m = 97}^{100}^{100} C_{m} \cdot^{m} C_{97} = 2^{λ} \cdot^{100} C_{97}, is :$ 357. $If the term without x in the expansion of (x^{\frac{2}{3}} + \frac{α}{x ^{3}})^{22} is 7315, then ∣ α ∣ is equal to ________$ 358. $If \sum_{k = 1}^{19} \frac{( - 2 ) ^{k}}{k ! ( 19 - k )!} = \frac{- λ}{19 !} then find λ .$ 359. $If the co-efficient of x^{9} in (α x^{3} + \frac{1}{β x})^{11} and the co-efficient of x^{- 9} in (α x - \frac{1}{β x ^{3}})^{11} are equal, then (α β)^{2} is equal to ________.$ 360. $The sum of all rational terms in the expansion of (1 + 2^{1/3} + 3^{1/2})^{6} is equal to$