Binomial Theorem PYQs Set 13 (PDF)

Set 13 of 14 · Questions 361-390 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 13 (Q361-390)
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361. $\frac{k ^{5}}{5} + \frac{k ^{3}}{3} + \frac{7 k}{15} is, if k \in N$ (1) $a natural number$ (2) $an integer$ (3) $a positive fraction$ (4) $equal to zero$ 362. $If (^{30} C_{1})^{2} + 2 (^{30} C_{2})^{2} + 3 (^{30} C_{3})^{2} + \dots + 30 (^{30} C_{30})^{2} = \frac{α 60 !}{( 30 ! ) ^{2}}, then α is equal to$ (1) $30$ (2) $60$ (3) $15$ (4) $10$ 363. $If the fourth term in the binomial expansion of (x^{\frac{1}{1 + l o g _{10} x}} + x^{\frac{1}{12}})^{6} is equal to 200, and x > 1, then the value of x is :$ (1) $100$ (2) $10$ (3) $10^{3}$ (4) $10^{4}$ 364. $If \sum_{i = 1}^{20} (\frac{^{20} C _{i - 1}}{^{20} C _{i} + ^{20} C _{i - 1}})^{3} = \frac{k}{21}, then k equals :$ (1) $400$ (2) $50$ (3) $200$ (4) $100$ 365. $The total number of irrational terms in the binomial expansion of (7^{1/5} - 3^{1/10})^{60} is :$ (1) $55$ (2) $49$ (3) $48$ (4) $54$ 366. $If p and q are respectively the coefficient of x^{- 3} and x^{- 5} in the expansion of (x^{1/3} + \frac{1}{2 x ^{1/3}})^{21}, x > 0, then \frac{5 p}{4 q} equal to$ (1) $102$ (2) $408$ (3) $182$ (4) $468$ 367. $\sum_{k = 0}^{6}^{51 - k} C_{3} is equal to$ (1) $^{51} C_{4} -^{45} C_{4}$ (2) $^{51} C_{3} -^{45} C_{3}$ (3) $^{52} C_{4} -^{45} C_{4}$ (4) $^{52} C_{3} -^{45} C_{3}$ 368. $If the integer represented by 100! has K consecutive zeroes at the end, then K =$ (1) $24$ (2) $36$ (3) $64$ (4) $128$ 369. $Consider the following statements I : The number of non-trivial even divisors of the number 2^{α_{1}} 3^{α_{2}} 4^{α_{3}} 5^{α_{4}} 6^{α_{5}}, is (α_{1} + 1) (α_{2} + 1) (α_{3} + 1) (α_{4} + 1) (α_{5} + 1) - 2 II : The number of non-trivial odd divisors of the number 2^{α_{1}} \cdot 3^{α_{2}} \cdot 4^{α_{3}} \cdot 5^{α_{4}} \cdot 6^{α_{5}}, is α_{2} + α_{4} + α_{5} + α_{2} α_{4} + α_{4} α_{5} . Then$ (1) $I is false and II is false$ (2) $I is true and II is true$ (3) $I is false and II is true$ (4) $I is true and II is false$ 370. $x^{n} + y^{n} is divisible by$ (1) $x - y for all n\inN$ (2) $x + y for all n \in N$ (3) $x+y for all n = 2 m - 1, m \in N$ (4) $x+y for all n = 2 m, m \in N$ 371. $A furniture manufacturer produces tables and bookshelves made up of wood and steel. The weekly requirement of wood and steel is given as below. Product Wood Steel Table (x) 82 Bookshelf (y) 113 The weekly availability of wood and steel is 450 and 100 units respectively. Profit on a table is ₹ 1000 and that on a bookshelf is ₹ 1200 . To determine the number of tables and bookshelves to be produced every week in order to maximize the total profit, formulate the problem as L.P.P.$ (1) $Maximize Z=1000x+1200y Subject to 8x+11y\geq450,2x+3y\leq100,x\geq0,y\geq0$ (2) $Maximize Z=1000x+1200y Subject to 8x+11y\leq450,2x+3y\leq100,x\geq0,y\geq0$ (3) $Maximize Z=1000x+1200y Subject to 8x+11y\leq450,2x+3y\geq100,x\geq0,y\geq0$ (4) $Maximize Z=1000x+1200y Subject to 8x+11y\geq450,2x+3y\geq100,x\geq0,y\geq0$ 372. $The digit in the unit's place of 5^{834} is$ (1) $3$ (2) $5$ (3) $0$ (4) $1$ 373. $If \frac{1}{n + 1}^{n} C_{n} + \frac{1}{n}^{n} C_{n - 1} + \frac{1}{2}^{n} C_{1} +^{n} C_{0} = \frac{1023}{10} then n is equal to$ (1) $9$ (2) $8$ (3) $7$ (4) $6$ 374. $The remainder when 7^{2022} + 3^{2022} is divided by 5 is:$ (1) $0$ (2) $2$ (3) $3$ (4) $4$ 375. $The value of^{50} C_{4} + \sum_{r = 1}^{6}^{56 - r} C_{3} is :$ (1) $56 C4$ (2) $56 C3$ (3) $55C3$ (4) $55C4$ 376. $The smallest positive divisor greater than 1 of a composite number ' a' is$ (1) $< a$ (2) $= a$ (3) $> a$ (4) $\leq a$ 377. $If. the middle term in the expansion of (1 + x)^{2 n} is the greatest term, then x lies in the interval$ (1) $(\frac{n}{n + 1}, \frac{n + 1}{n})$ (2) $(\frac{n + 1}{n}, \frac{n}{n + 1})$ (3) $(n - 2, n)$ (4) $(n - 1, n)$ 378. $The objective function of LPP defined over the convex set attains its optimum value at$ (1) $At least two of the corner points$ (2) $All the corner points$ (3) $At least one of the corner points$ (4) $None of the corner points$ 379. $The remainder when (11)^{1011} + (1011)^{11} is divided by 9 is$ (1) $1$ (2) $4$ (3) $6$ (4) $8$ 380. $The ten's digit in 1! + 4! + 7! + 10! + 12! + 13! + 15! + 16! + 17! is divisible by$ (1) $5$ (2) $7$ (3) $4$ (4) $3!$ 381. $The sum of all the coefficient in the binomial expansion of (1 + 2 x)^{n} is 6561 . Let R = (1 + 2 x)^{n} = I + F where I \in N and 0 < F < 1 . If x = \frac{1}{2}, then 1 - \frac{F}{1 + ( 2 - 1 ) ^{4}} =$ (1) $(32 - 4)$ (2) $4 (32 + 4)$ (3) $(2 - 1)^{4}$ (4) $1$ 382. $If C, denotes the binomial coefficient^{n} C_{r}, then (- 1) C_{0}^{2} + 2 C_{1}^{2} + 5 C_{2}^{2} + \dots + (3 n - 1) C_{n}^{2} is equal to$ (1) $(3 n - 2)^{2 n} C_{n}$ (2) $(\frac{3 n - 2}{2})^{2 n} C_{n}$ (3) $(5 + 3 n)^{2 n} C_{n}$ (4) $(\frac{3 n - 5}{2})^{2 n} C_{n + 1}$ 383. $Let^{n} C_{r} denote the binomial coefficient of x^{r} inthe expansion of (1 + x)^{n} . If \sum_{k = 0}^{10} (2^{2} + 3 k)^{n} C_{k} = α . 3^{10} + β . 2^{10}, α, β \in then α + β is equal to$ 384. $For integers n and r, let (r n) = {^{n} C_{r}, 0, if n \geq r \geq 0 otherwise . The maximum value of k for which the sum \sum_{i = 0}^{k} (i 10) (k - i 15) + \sum_{i = 0}^{k + 1} (i 12) (k + 1 - i 13) exists, is equal to$ 385. $Let (k n) denote^{n} C_{k} and [n k] = {(k n), 0, if 0 \leq k \leq n otherwise . If A_{k} = \sum_{i = 0}^{9} (i 9) [12 12 - k + i] + \sum_{i = 0}^{8} (i 8) [13 13 - k + i] and A_{4} - A_{3} = 190 p, then p is equal to :$ 386. $Let n be a positive integer. Let A = \sum_{k = 0}^{n} (- 1)^{k}^{n} C_{k} ((\frac{1}{2})^{k} + (\frac{3}{4})^{k} + (\frac{7}{8})^{k} + (\frac{15}{16})^{k} + (\frac{31}{32})^{k}) . If 63 A = 1 - \frac{1}{2 ^{30}} then n is equal to$ 387. $Let f : R \to R be a function defined as f (x) = {3 (1 - \frac{∣ x ∣}{2}), 0, if ∣ x ∣ \leq 2 if ∣ x ∣ > 2 Let g : R \to R be given by g (x) = f (x + 2) - f (x - 2) If n and m denote the number of points in R where g is not continuous and not differentiable, respectively, then n + m is equal to ____.$ 388. $\sum_{k = 1}^{3 n}^{6 n} C_{2 k - 1} (- 3)^{k} is equal to :$ 389. $If 1 + (2 +^{49} C_{1} +^{49} C_{2} + ... +^{49} C_{49}) (50 C_{2} +^{50} C_{4} + .... +^{50} C_{50}) is equal to 2^{n} .m, where m is odd, then n+ m is equal to_____$ 390. $Let α be the constant term in the binomial expansion of (x - \frac{6}{x ^{\frac{3}{2}}})^{n}, n \leq 15 . If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x^{- n} is λ α, then λ is equal to_________.$