Binomial Theorem PYQs Set 6 (PDF)

Set 6 of 14 · Questions 151-180 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 6 (Q151-180)
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151. $If the k^{th} term in the expansion of (\frac{3}{2} x^{2} - \frac{1}{3 x})^{6} is independent of x, then the numerically greatest term in the expansion of (\frac{3}{2} x^{2} - \frac{1}{3 x})^{k} when x = \frac{2}{3}, is$ (1) $\frac{40}{81}$ (2) $(\frac{7}{6})^{5}$ (3) $\frac{20}{27}$ (4) $(\frac{7}{6})^{4}$ 152. $The remainder, when 7^{103} is divided by 23, is equal to:$ (1) $6$ (2) $17$ (3) $9$ (4) $14$ 153. $If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1 + x)^{n + 5} are in the ratio 5 : 10 : 14, then the largest coefficientin this expansion is:$ (1) $252$ (2) $462$ (3) $792$ (4) $330$ 154. $If the coefficients of x^{9} and x^{10} in the binomial expansion of (3 + \frac{x}{2})^{n} are equal, then n =$ (1) $69$ (2) $96$ (3) $66$ (4) $99$ 155. $For z \in C, if (1 + z)^{n} = 1 +^{n} C_{1} z +^{n} C_{2} z^{2} + \dots .^{n} C_{n} z^{n} and \sum_{r = 0}^{100} 100 c_{r} (sin rx) = (2 cos \frac{x}{2})^{100} sin kx, then k =$ (1) $25$ (2) $100$ (3) $50$ (4) $75$ 156. $If x is so small, that x^{5} and higher power of x may be neglected, then the coefficient of x^{4} in the expansion of x^{2} + 4 - x^{2} + 9, is$ (1) $\frac{19}{1728}$ (2) $\frac{- 19}{1728}$ (3) $\frac{43}{1728}$ (4) $\frac{- 43}{1728}$ 157. $The number of terms in the expansion of (x^{2} + y^{2})^{25} - (x^{2} - y^{2})^{25} after simplification is$ (1) $0$ (2) $26$ (3) $13$ (4) $50$ 158. $If r^{th} and (r + 1)^{th} terms in the expansion of (p + q)^{n} are equal, then \frac{( n + 1 ) q}{r ( p + q )} is$ (1) $0$ (2) $1$ (3) $¼$ (4) $1/2$ 159. $The value of \frac{1}{1 ! 50 !} + \frac{1}{3 ! 48 !} + \frac{1}{5 ! 46 !} + \dots \dots + \frac{1}{49 ! 2 !} + \frac{1}{51 ! 1 !} is$ (1) $\frac{2 ^{50}}{50 !}$ (2) $\frac{2 ^{50}}{51 !}$ (3) $\frac{2 ^{51}}{51 !}$ (4) $\frac{2 ^{51}}{50 !}$ 160. $If \| x \| < 1, then the coefficient of x^{5} in the expansion of \frac{3 x}{( x - 2 ) ( x + 1 )} is$ (1) $\frac{33}{32}$ (2) $\frac{- 33}{32}$ (3) $\frac{31}{32}$ (4) $\frac{- 31}{32}$ 161. $If k is the coefficient of x^{5} in the expansion of (2 x^{2} - \frac{1}{3 x ^{3}})^{5} then \frac{3 k}{2} =$ (1) $-20$ (2) $-40$ (3) $20$ (4) $40$ 162. $The remainder when 3^{100} \times 2^{50} is divided by 5 is$ (1) $3$ (2) $4$ (3) $1$ (4) $2$ 163. $The remainder when 3^{2022} is divided by 5 is$ (1) $1$ (2) $2$ (3) $3$ (4) $4$ 164. $The number of integral terms in the expansion of (5^{\frac{1}{2}} + 7^{\frac{1}{8}})^{1016} is:$ (1) $127$ (2) $128$ (3) $130$ (4) $129$ 165. $The value of^{10} C_{1} +^{10} C_{2} +^{10} C_{3} + \dots . +^{10} C_{9} is$ (1) $2^{10} - 1$ (2) $2^{10}$ (3) $2^{11}$ (4) $2^{10} - 2$ 166. $The middle term of expansion (\frac{10}{x} + \frac{x}{10})^{10} is$ (1) $^{8} C_{5}$ (2) $^{10} C_{5}$ (3) $^{7} C_{5}$ (4) $^{9} C_{5}$ 167. $The least positive integer x satisfying 2^{2010} \equiv 3 x (mod 5) is$ (1) $4$ (2) $3$ (3) $2$ (4) $1$ 168. $The value of (0 30) (10 30) - (1 30) (11 30) + (2 30) (12 30) - \dots + (20 30) (30 30) is :$ (1) $^{60} C_{20}$ (2) $^{30} C_{10}$ (3) $^{30} C_{15}$ (4) $None of these$ 169. $Assertion (A) : If \|x\|<1, then \sum (- 1)^{n} x^{n + 1} = \frac{x}{x + 1} Reason (R): If \|x\|<1, then (1 + x)^{- 1} = 1 - x + x^{2} - x^{3} + \dots Which one of the following is true?$ (1) $(A) and (R) are true, (R) is a correct explanation of (A)$ (2) $(A) and (R) are true but (R) is not a correct explanation of (A)$ (3) $(A) is true, but (R) is false$ (4) $(A) is false but (R) is true$ 170. $If the number of integral terms in the expansion of (3^{\frac{1}{2}} + 5^{\frac{1}{8}})^{n} is exactly 33, then the least value of n is:$ (1) $264$ (2) $128$ (3) $256$ (4) $248$ 171. $The term independent of x in the expansion of (\frac{( x + 1 )}{( x ^{2/3} + 1 - x ^{1/3} )} - \frac{( x - 1 )}{( x - x ^{1/2} )})^{10}, x > 1, is:$ (1) $240$ (2) $120$ (3) $150$ (4) $210$ 172. $Suppose 2 - p, p, 2 - α, α are the coefficient of four consecutive terms in the expansion of (1 + x)^{n} . Then the value of p^{2} - α^{2} + 6 α + 2 p_{equals}$ (1) $4$ (2) $10$ (3) $8$ (4) $none of these$ 173. $If n is the number of irrational terms in theexpansion of (3^{1/4} + 5^{1/8})^{60}, then (n - 1) is divisible by:$ (1) $26$ (2) $30$ (3) $8$ (4) $7$ 174. $Suppose A and B are the coefficients of 30^{th} and 12^{th} terms respectively in the binomial expansion of (1 + x)^{2 n - 1} . If 2 A = 5 B, then n is equal to:$ (1) $22$ (2) $20$ (3) $21$ (4) $19$ 175. $In the expansion of (1 + x) (1 - x^{2}) (1 + \frac{3}{x} + \frac{3}{x ^{2}} + \frac{1}{x ^{3}})^{5}, x \neq = 0, the sum of the coefficient of x^{3} and x^{- 13} is equal to_______$ 176. $The coefficient of x^{- 6}, in the expansion of (\frac{4 x}{5} + \frac{5}{2 x ^{2}})^{9}, is ___________$ 177. $The term independent of x in the expansion of (\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}})^{10}, where x \neq = 0, 1 is equal to$ 178. $If \sum_{r = 0}^{5} \frac{^{11} C _{22 r}}{2 r + 2} = \frac{m}{n}, gcd (m, n) = 1, then m - n is equal to .$ 179. $The remainder, when 7^{103} is divided by 17, is$ 180. $If the coefficient of x^{10} in the binomial expansion of (\frac{x}{5 ^{\frac{1}{4}}} + \frac{5}{x ^{\frac{1}{3}}})^{60} is 5^{k} l, where l, k \in n and l is prime to 5, then k is equal to_____.$