Binomial Theorem PYQs Set 5 (PDF)

Set 5 of 14 · Questions 121-150 of 391

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Binomial Theorem · Set 5 (Q121-150)
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121. $If the 17^{th} and the 18^{th} terms in the expansion of (2 + a)^{50} are equal, then the coefficient of x^{35} in the expansion of (a + x)^{- 2} is$ (1) $-35$ (2) $35$ (3) $36$ (4) $-36$ 122. $For \| x \| < \frac{4}{3}, the approximate value of \frac{1}{( 4 - 3 x ) ^{\frac{1}{2}}} is$ (1) $\frac{1}{4} - \frac{2 x}{3} + \frac{12 x ^{2}}{39}$ (2) $1 - \frac{3 x}{16} - \frac{15}{256} x^{2}$ (3) $\frac{1}{2} + \frac{3 x}{16} + \frac{27 x ^{2}}{256}$ (4) $\frac{1}{2} - \frac{3 x}{16} + \frac{15}{256} x^{2}$ 123. $The number of terms in the expansion of (x + y + z)^{10} is$ (1) $142$ (2) $11$ (3) $110$ (4) $66$ 124. $If the coefficients of x3 and x4 in the expansion of (1 + ax + bx2) (1 - 2x)18 in powers of x are both zero, then (a, b) is equal to$ (1) $(14, \frac{272}{3})$ (2) $(16, \frac{272}{3})$ (3) $(16, \frac{251}{3})$ (4) $(14, \frac{251}{3})$ 125. $If C_{j} =^{n} C_{j}, then C_{0} C_{r} + C_{1} C_{r + 1} + C_{2} C_{r + 2} + \dots + C_{n - r} C_{n} =$ (1) $\frac{( 2 n ) !}{( n - 2 r ) ! ( n + 2 r ) !}$ (2) $\frac{( 2 n ) !}{( n - r ) ! ( n + r ) !}$ (3) $2 n_{C_{r}}$ (4) $2 n_{C_{r + 1}}$ 126. $The coefficient of x^{4} in the expansion of (1 - x + x^{2} - x^{3})^{4} is$ (1) $31$ (2) $30$ (3) $25$ (4) $-14$ 127. $\sum_{r = 0}^{10}^{40 - r} C_{5} is equal to$ (1) $^{41} C_{5} -^{30} C_{5}$ (2) $^{41} C_{6} -^{30} C_{6}$ (3) $^{41} C_{5} +^{30} C_{5}$ (4) $^{41} C_{6}$ 128. $For 0 < x < 1, the expansion of (1 + \frac{1}{x})^{\frac{1}{2}} is .$ (1) $1 + \frac{1}{2 x} - \frac{1}{2 !} (\frac{1}{2 x})^{2} + \frac{1.3}{3 !} (\frac{1}{2 x})^{3} - \frac{1.3 . 5}{4 !} (\frac{1}{2 x})^{4} + \dots \infty$ (2) $\frac{1}{x} + \frac{1}{2} x - \frac{1}{2 !} \frac{x x}{2 ^{2}} + \frac{1.3}{3 !} \frac{x ^{2} x}{2 ^{3}} - \dots \infty$ (3) $1 + \frac{1}{x} + \frac{1}{2} x x + \frac{1}{2 !} \frac{x ^{2} x}{2 ^{3}} + \frac{1.3}{3 !} \frac{x ^{3} x}{2 ^{4}} + \dots \infty$ (4) $\frac{1}{x} + \frac{1}{2 x x} - \frac{1}{2 !} (\frac{1}{2 x})^{2} \frac{1}{x} + \frac{1.3}{3 !} (\frac{1}{2 x})^{3} \frac{1}{x} - \dots \infty$ 129. $If f (n) is the coefficient of x^{n} in the expansion of (1 + x) (1 - x)^{n}, then f (2021) is equal to$ (1) $-2019$ (2) $2020$ (3) $2021$ (4) $-2022$ 130. $The last digit of number 7^{886} is$ (1) $1$ (2) $9$ (3) $7$ (4) $3$ 131. $If a, b and c are the greatest values of^{19} C_{p},^{20} C_{q} and^{21} C_{r} respectively, then$ (1) $\frac{a}{11} = \frac{b}{22} = \frac{c}{42}$ (2) $\frac{a}{22} = \frac{b}{11} = \frac{c}{42}$ (3) $\frac{a}{22} = \frac{b}{42} = \frac{c}{11}$ (4) $\frac{a}{21} = \frac{b}{11} = \frac{c}{22}$ 132. $The sum of the co-efficients of all odd degree terms in the expansion of (x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, (x > 1) is :$ (1) $1$ (2) $2$ (3) $-1$ (4) $0$ 133. $Let α > 0, β > 0 be such that α^{3} + β^{2} = 4 . If the maximum value of the term independent of x in the binomial expansion of (α x^{\frac{1}{9}} + β x^{- \frac{1}{6}})^{10} is 10 k then k is equal to$ (1) $84$ (2) $176$ (3) $336$ (4) $352$ 134. $The number of integral terms in the expansion of (3 + 85)^{256} is$ (1) $32$ (2) $33$ (3) $34$ (4) $35$ 135. $If n is a positive integer, then (3 + 1)^{2 n} - (3 - 1)^{2 n} is :$ (1) $an irrational number$ (2) $an odd positive integer$ (3) $an even positive integer$ (4) $a rational number other than positive integers$ 136. $If x = \frac{2.5}{3.6} - \frac{2.5 . 8}{3.6 . 9} (\frac{2}{5}) + \frac{2.5 . 8.11}{3.6 . 9.12} (\frac{2}{5})^{2} - \dots \infty, then 7^{2} (12 x + 55)^{3} =$ (1) $3^{8} 5^{3}$ (2) $3^{8} 5^{5}$ (3) $3^{3} 5^{5}$ (4) $3^{3} 5^{8}$ 137. $If the coefficients of x^{2} and x^{3} are both zero, in the expansion of the expression (1 + ax + b x^{2}) (1 - 3 x)^{15} in powers of x, then the ordered pair (a, b) is equal to:$ (1) $(28, 861)$ (2) $(-54, 31)$ (3) $(28, 315)$ (4) $(-21, 714)$ 138. $The sum of the coefficient of x^{2/3} and x^{- 2/5} in the binomial expansion of (x^{2/3} + \frac{1}{2} x^{- 2/5})^{9} is$ (1) $21/4$ (2) $63/16$ (3) $19/4$ (4) $69/16$ 139. $The coefficient of x^{5} in the expansion of (1 + x)^{21} + (1 + x)^{22} + \dots + (1 + x)^{30} is$ (1) $^{31} C_{6} -^{21} C_{6}$ (2) $^{51} C_{5}$ (3) $^{9} C_{5}$ (4) $^{30} C_{5} +^{20} C_{5}$ 140. $Assuming \| x \| to be so small, that x^{2} and higher powers of x can be neglected, then \frac{1 + x + ( 1 - x ) ^{3/2}}{( 1 + x ) + 1 + x} =$ (1) $1 + \frac{5 x}{4}$ (2) $1 - \frac{5 x}{4}$ (3) $1 + \frac{4 x}{5}$ (4) $1 - \frac{4 x}{5}$ 141. $The value of (^{21} C_{1} -^{10} C_{1}) + (^{21} C_{2} -^{10} C_{2}) + (^{21} C_{3} -^{10} C_{3}) + (^{21} C_{4} -^{10} C_{4}) +........+ (^{21} C_{10} -^{10} C_{10}) is$ (1) $2^{21} - 2^{11}$ (2) $2^{21} - 2^{10}$ (3) $2^{20} - 2^{9}$ (4) $2^{20} - 2^{10}$ 142. $^{34} C_{5} + \sum_{r = 0}^{4} (38 - r) C_{4} is equal to$ (1) $22 \times^{39} C_{4}$ (2) $^{39} C_{4}$ (3) $^{3 \times 39} C_{5}$ (4) $^{39} C_{5}$ 143. $If p is an integral multiple of 4 lying in between the coefficients of x^{4} and x in the expansion of (x^{2} + \frac{1}{x})^{8}, then the number of such values of p is$ (1) $3$ (2) $4$ (3) $5$ (4) $6$ 144. $If the fourth term in the Binomial expansion of (\frac{2}{x} + x^{log_{g} x})^{6} (x > 0) is 20 \times 8^{7}, then value of x is :$ (1) $8^{3}$ (2) $8^{2}$ (3) $8$ (4) $8^{- 2}$ 145. $If \sum (\frac{20 C _{i - 1}}{20 C _{i} + ^{20} C _{i - 1}})^{3} = \frac{k}{21}, then k equals :$ (1) $50$ (2) $400$ (3) $200$ (4) $100$ 146. $Let (x + 10)^{50} + (x - 10)^{50} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{50} x^{50}, for all x ϵ R; then \frac{a _{2}}{a _{0}} is equal to :$ (1) $12.50$ (2) $12.00$ (3) $12.25$ (4) $12.75$ 147. $If the sum of the coefficients in the expansion of (x + y)^{n} is 4096, then the greatest coefficient in the expansion is .$ 148. $The number of values of ' r ' satisfying the equation, 39 C_{3 r - 1} -^{39} C_{r^{2}} = 39 C_{r^{2} - 1} -^{39} C_{3 r} is :$ 149. $Let [t] denote the greatest integer \leq t. if the constant term in the expansion of (3 x^{2} - \frac{1}{2 x ^{5}})^{7} is α then [α] is equal to_____$ 150. $Let n \in N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms nC0, 3 \cdotnC1, 5 \cdotnC2, 7 \cdotnC3,\dots. . is equal to 2100.101 then 2 (\frac{n - 1}{2}) is equal to ___$