Binomial Theorem

165 questions · Mathematics · JEE Main

Binomial Theorem

165 questions · Mathematics · JEE Main

The largest $n \in N$ such that $3^{n}$ divides 50 ! is :2025 · 2 Apr · Shift 1 · Q36 · MCQ
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 :2025 · 2 Apr · Shift 1 · Q44 · MCQ
$I f r = 0 \sum 10 (\frac{10 ^{r + 1} - 1}{10 ^{r}}) .^{11} C_{r + 1} = \frac{_{α} 11 - 11 ^{11}}{10 ^{10}}, t h e n α i s e q u a l t o :$ 2025 · 2 Apr · Shift 2 · Q42 · MCQ
The sum of all rational terms in the expansion of $(2 + 3)^{8}$ is :2025 · 3 Apr · Shift 1 · Q40 · MCQ
If $r = 1 \sum 9 (\frac{r + 3}{2 ^{r}}) \cdot^{9} C_{r} = α (\frac{3}{2})^{9} - β, α, β \in N$ , then $(α + β)^{2}$ is equal to2025 · 3 Apr · Shift 1 · Q41 · MCQ
Let $(1 + x + x^{2})^{10} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{20} x^{20}$ . If $(a_{1} + a_{3} + a_{5} + \dots + a_{19}) - 11 a_{2} = 121 k$ , then $k$ is equal to .2025 · 3 Apr · Shift 2 · Q48 · Numerical
For an integer $n \geq 2$ , if the arithmetic mean of all coefficients in the binomial expansion of $(x + y)^{2 n - 3}$ is 16 , then the distance of the point $P (2 n - 1, n^{2} - 4 n)$ from the line $x + y = 8$ is2025 · 4 Apr · Shift 1 · Q39 · MCQ
In the expansion of $(32 + \frac{1}{3 3})^{n}, n \in N$ , if the ratio of $1 5^{th}$ term from the beginning to the $1 5^{th}$ term from the end is $\frac{1}{6}$ , then the value of $^{n} C_{3}$ …2025 · 4 Apr · Shift 1 · Q44 · MCQ
If $1^{2} \cdot (^{15} C_{1}) + 2^{2} \cdot (^{15} C_{2}) + 3^{2} \cdot (^{15} C_{3}) + \dots + 1 5^{2} \cdot (^{15} C_{15}) = 2^{m} \cdot 3^{n} \cdot 5^{k}$ , where $m, n, k \in N$ , then $m + n + k$ …2025 · 4 Apr · Shift 2 · Q38 · MCQ
The remainder when $((64)^{(64)})^{(64)}$ is divided by 7 is equal to2025 · 7 Apr · Shift 1 · Q34 · MCQ
The sum of the series $2 \times 1 \times^{20} C_{4} - 3 \times 2 \times^{20} C_{5} + 4 \times 3 \times^{20} C_{6} - 5 \times 4 \times^{20} C_{7} + \dots\dots + 18 \times 17 \times^{20} C_{20}$ ,…2025 · 7 Apr · Shift 2 · Q46 · Numerical
The number of integral terms in the expansion of $(5^{\frac{1}{2}} + 7^{\frac{1}{8}})^{1016}$ is:2025 · 8 Apr · Shift 2 · Q45 · MCQ
The product of the last two digits of $(1919)^{1919}$ is2025 · 8 Apr · Shift 2 · Q46 · Numerical
If $\sum_{r = 0}^{5} \frac{^{11} C _{2 r + 1}}{2 r + 2} = \frac{m}{n}, gcd (m, n) = 1$ , then $m - n$ is equal to .2025 · 22 Jan · Shift 1 · Q47 · Numerical
Let $α, β, γ$ and $δ$ be the coefficients of $x^{7}, x^{5}, x^{3}$ and $x$ respectively in the expansion of $(x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, x > 1 . If u and v satisfy the equations α u + β v = 18, γ u + δ v = 20,$ …2025 · 22 Jan · Shift 2 · Q30 · MCQ
If $\sum_{r = 1}^{30} \frac{r ^{2} ( ^{30} C _{r} ) ^{2}}{^{30} C _{r - 1}} = α \times 2^{29}$ , then $α$ is equal to .2025 · 22 Jan · Shift 2 · Q48 · Numerical
The sum of all rational terms in the expansion of $(1 + 2^{1/3} + 3^{1/2})^{6}$ is equal to .2025 · 23 Jan · Shift 1 · Q46 · Numerical
If in the expansion of $(1 + x)^{p} (1 - x)^{q}$ , the coefficients of $x$ and $x^{2}$ are 1 and -2 , respectively, then $p^{2} + q^{2}$ is equal to :2025 · 23 Jan · Shift 2 · Q40 · MCQ
For some $n e q 10$ , let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of $(1 + x)^{n + 4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n + 4}$ is:2025 · 24 Jan · Shift 1 · Q29 · MCQ
Suppose $A$ and $B$ are the coefficients of $3 0^{th}$ and $1 2^{th}$ terms respectively in the binomial expansion of $(1 + x)^{2 n - 1}$ . If $2 A = 5 B$ , then n is equal to:2025 · 24 Jan · Shift 2 · Q34 · MCQ
If $α = 1 + r = 1 \sum 6 (- 3)^{r - 1}^{12} C_{2 r - 1}$ , then the distance of the point $(12, 3)$ from the line $α x - 3 y + 1 = 0$ is .2025 · 28 Jan · Shift 1 · Q48 · Numerical
Let the coefficients of three consecutive terms $T_{r}$ , $T_{r + 1}$ and $T_{r + 2}$ in the binomial expansion of $(a + b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$ . Let $q$ be the sum of all rational terms in…2025 · 28 Jan · Shift 2 · Q32 · MCQ
The least value of n for which the number of integral terms in the Binomial expansion of $(37 + 1211)^{n}$ is 183, is :2025 · 29 Jan · Shift 1 · Q34 · MCQ
The remainder, when $7^{103}$ is divided by 23, is equal to:2025 · 29 Jan · Shift 2 · Q35 · MCQ
If the Coefficient of $x^{30}$ in the expansion of $(1 + \frac{1}{x})^{6} (1 + x^{2})^{7} (1 - x^{3})^{8}; x e q 0$ is $α$ , then $∣ α ∣$ equals .2024 · 1 Feb · Shift 1 · Q53 · Numerical
Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $(\frac{1}{3} x^{\frac{1}{3}} + \frac{1}{2 x ^{\frac{2}{3}}})^{18}$ . Then $(\frac{n}{m})^{\frac{1}{3}}$ …2024 · 1 Feb · Shift 2 · Q49 · MCQ
The sum of all rational terms in the expansion of $(2^{\frac{1}{5}} + 5^{\frac{1}{3}})^{15}$ is equal to :2024 · 4 Apr · Shift 1 · Q45 · MCQ
Let $a = 1 + \frac{^{2} C _{2}}{3 !} + \frac{^{3} C _{2}}{4 !} + \frac{^{4} C _{2}}{5 !} + ...., b = 1 + \frac{^{1} C _{0} + ^{1} C _{1}}{1 !} + \frac{^{2} C _{0} + ^{2} C _{1} + ^{2} C _{2}}{2 !} + \frac{^{3} C _{0} + ^{3} C _{1} + ^{3} C _{2} + ^{3} C _{3}}{3 !} + ....$ …2024 · 4 Apr · Shift 1 · Q60 · Numerical
If the coefficients of $x^{4}, x^{5}$ and $x^{6}$ in the expansion of $(1 + x)^{n}$ are in the arithmetic progression, then the maximum value of $n$ is:2024 · 4 Apr · Shift 2 · Q32 · MCQ
If the constant term in the expansion of $(1 + 2 x - 3 x^{3}) (\frac{3}{2} x^{2} - \frac{1}{3 x})^{9}$ is $p$ , then $108 p$ is equal to .2024 · 5 Apr · Shift 1 · Q59 · Numerical
If the constant term in the expansion of $(\frac{5 3}{x} + \frac{2 x}{3 5})^{12}, x e q 0$ , is $α \times 2^{8} \times 53$ , then $25 α$ is equal to :2024 · 5 Apr · Shift 2 · Q36 · MCQ
If the second, third and fourth terms in the expansion of $(x + y)^{n}$ are 135, 30 and $\frac{10}{3}$ , respectively, then $6 (n^{3} + x^{2} + y)$ is equal to .2024 · 6 Apr · Shift 1 · Q53 · Numerical
If the term independent of $x$ in the expansion of $(a x^{2} + \frac{1}{2 x ^{3}})^{10}$ is 105 , then $a^{2}$ is equal to :2024 · 8 Apr · Shift 2 · Q50 · MCQ
The coefficient of $x^{70}$ in $x^{2} (1 + x)^{98} + x^{3} (1 + x)^{97} + x^{4} (1 + x)^{96} + \dots + x^{54} (1 + x)^{46}$ is $^{99} C_{p} -^{46} C_{q}$ . Then a possible value of $p + q$ is :2024 · 9 Apr · Shift 1 · Q38 · MCQ
The remainder when $42 8^{2024}$ is divided by 21 is .2024 · 9 Apr · Shift 1 · Q54 · Numerical
The sum of the coefficient of $x^{2/3}$ and $x^{- 2/5}$ in the binomial expansion of $(x^{2/3} + \frac{1}{2} x^{- 2/5})^{9}$ is2024 · 9 Apr · Shift 2 · Q49 · MCQ
$^{n - 1} C_{r} = (k^{2} - 8)^{n} C_{r + 1}$ if and only if :2024 · 27 Jan · Shift 1 · Q45 · MCQ
If A denotes the sum of all the coefficients in the expansion of $(1 - 3 x + 10 x^{2})^{n}$ and B denotes the sum of all the coefficients in the expansion of $(1 + x^{2})^{n}$ , then :2024 · 27 Jan · Shift 1 · Q49 · MCQ
The coefficient of $x^{2012}$ in the expansion of $(1 - x)^{2008} (1 + x + x^{2})^{2007}$ is equal to .2024 · 27 Jan · Shift 2 · Q52 · Numerical
$If \frac{^{11} C _{1}}{2} + \frac{^{11} C _{2}}{3} + \dots + \frac{^{11} C _{9}}{10} = \frac{n}{m} with gcd (n, m) = 1, then n + m is equal to$ .2024 · 29 Jan · Shift 1 · Q55 · Numerical
Remainder when $6 4^{3 2^{32}}$ is divided by 9 is equal to .2024 · 29 Jan · Shift 2 · Q58 · Numerical
$Number of integral terms in the expansion of {7^{(\frac{1}{2})} + 1 1^{(\frac{1}{6})}}^{824} is equal to .$ 2024 · 30 Jan · Shift 1 · Q55 · Numerical
Suppose $2 - p, p, 2 - α, α$ are the coefficients of four consecutive terms in the expansion of $(1 + x)^{n}$ . Then the value of $p^{2} - α^{2} + 6 α + 2 p$ equals2024 · 30 Jan · Shift 2 · Q45 · MCQ
Let $α = \sum_{k = 0}^{n} (\frac{( ^{n} C _{k} ) ^{2}}{k + 1})$ and $β = \sum_{k = 0}^{n - 1} (\frac{^{n} C _{k} ^{n} C _{k + 1}}{k + 2})$ If $5 α = 6 β$ , then $n$ equals .2024 · 30 Jan · Shift 2 · Q59 · Numerical
Let $a$ be the sum of all coefficients in the expansion of $(1 - 2 x + 2 x^{2})^{2023} (3 - 4 x^{2} + 2 x^{3})^{2024}$ and $b = lim_{x \to 0} (\frac{\int _{0}^{x} \frac{l o g ( 1 + t )}{t ^{2024} + 1} d t}{x ^{2}})$ . If the…2024 · 31 Jan · Shift 1 · Q47 · MCQ
In the expansion of $(1 + x) (1 - x^{2}) (1 + \frac{3}{x} + \frac{3}{x ^{2}} + \frac{1}{x ^{3}})^{5}, x e q 0$ , the sum of the coefficients of $x^{3}$ and $x^{- 13}$ is equal to .2024 · 31 Jan · Shift 1 · Q52 · Numerical
Let the coefficient of $x^{r}$ in the expansion of $(x + 3)^{n - 1} + (x + 3)^{n - 2} (x + 2) + (x + 3)^{n - 3} (x + 2)^{2} + \dots\dots\dots . + (x + 2)^{n - 1}$ be $α_{r}$ . If $\sum_{r = 0}^{n} α_{r} = β^{n} - γ^{n}, β, γ \in N$ , then the…2024 · 31 Jan · Shift 2 · Q58 · Numerical
The remainder, when $1 9^{200} + 2 3^{200}$ is divided by 49 , is .2023 · 1 Feb · Shift 1 · Q38 · Numerical
Let the sixth term in the binomial expansion of $(2^{l o g_{2} (10 - 3^{x})} + 5 2^{(x - 2) l o g_{2} 3})^{m}$ in the increasing powers of $2^{(x - 2) l o g_{2} 3}$ , be 21 . If the…2023 · 1 Feb · Shift 2 · Q42 · Numerical
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 .2023 · 1 Feb · Shift 2 · Q45 · Numerical
If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $(42 + \frac{1}{4 3})^{n}$ is $6 : 1$ , then the third term from the beginning is :2023 · 6 Apr · Shift 1 · Q34 · MCQ
If the coefficient of $x^{7}$ in $(a x^{2} + \frac{1}{2 b x})^{11}$ and $x^{- 7}$ in $(a x - \frac{1}{3 b x ^{2}})^{11}$ are equal, then :2023 · 6 Apr · Shift 2 · Q24 · MCQ
Among the statements : (S1) : $202 3^{2022} - 199 9^{2022}$ is divisible by 8 (S2) : $13 (13)^{n} - 12 n - 13$ is divisible by 144 for infinitely many $n \in N$ 2023 · 6 Apr · Shift 2 · Q29 · MCQ
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 .2023 · 8 Apr · Shift 1 · Q45 · Numerical
The largest natural number $n$ such that $3^{n}$ divides $66!$ is .2023 · 8 Apr · Shift 1 · Q47 · Numerical
$2 5^{190} - 1 9^{190} - 8^{190} + 2^{190}$ is divisible by :2023 · 8 Apr · Shift 2 · Q23 · MCQ
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 :2023 · 8 Apr · Shift 2 · Q28 · MCQ
If the coefficient of $x^{7}$ in $(a x - \frac{1}{b x ^{2}})^{13}$ and the coefficient of $x^{- 5}$ in $(a x + \frac{1}{b x ^{2}})^{13}$ are equal, then $a^{4} b^{4}$ is equal to :2023 · 10 Apr · Shift 1 · Q29 · MCQ
The coefficient of $x^{7}$ in $(1 - x + 2 x^{3})^{10}$ is .2023 · 10 Apr · Shift 1 · Q44 · Numerical
Let the number $(22)^{2022} + (2022)^{22}$ leave the remainder $α$ when divided by 3 and $β$ when divided by 7. Then $(α^{2} + β^{2})$ is equal to :2023 · 10 Apr · Shift 2 · Q23 · MCQ
If the coefficients of $x$ and $x^{2}$ in $(1 + x)^{p} (1 - x)^{q}$ are 4 and $-$ 5 respectively, then $2 p + 3 q$ is equal to :2023 · 10 Apr · Shift 2 · Q25 · MCQ
The mean of the coefficients of $x, x^{2}, \dots, x^{7}$ in the binomial expansion of $(2 + x)^{9}$ is .2023 · 11 Apr · Shift 1 · Q46 · Numerical
The number of integral terms in the expansion of $(3^{\frac{1}{2}} + 5^{\frac{1}{4}})^{680}$ is equal to .2023 · 11 Apr · Shift 1 · Q48 · Numerical
The sum of the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n + 2}$ , which are in the ratio $1 : 3 : 5$ , is equal to :2023 · 11 Apr · Shift 2 · Q24 · MCQ
If the $101 1^{th}$ term from the end in the binominal expansion of $(\frac{4 x}{5} - \frac{5}{2 x})^{2022}$ is 1024 times $101 1^{th}$ R term from the beginning, then $∣ x ∣$ is equal to2023 · 11 Apr · Shift 2 · Q29 · MCQ
Fractional part of the number $\frac{4 ^{2022}}{15}$ is equal to2023 · 13 Apr · Shift 1 · Q33 · MCQ
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…2023 · 13 Apr · Shift 1 · Q44 · Numerical
The coefficient of $x^{5}$ in the expansion of $(2 x^{3} - \frac{1}{3 x ^{2}})^{5}$ is :2023 · 13 Apr · Shift 2 · Q27 · MCQ
The remainder, when $7^{103}$ is divided by 17, is 2023 · 13 Apr · Shift 2 · Q38 · Numerical
Let $(a + b x + c x^{2})^{10} = i = 0 \sum 20 p_{i} x^{i}, a, b, c \in N$ . If $p_{1} = 20$ and $p_{2} = 210$ , then $2 (a + b + c)$ is equal to :2023 · 15 Apr · Shift 1 · Q31 · MCQ
Let the sum of the coefficients of the first three terms in the expansion of $(x - \frac{3}{x ^{2}})^{n}, x e 0. n \in N$ , be 376. Then the coefficient of $x^{4}$ is .2023 · 24 Jan · Shift 2 · Q40 · Numerical
The constant term in the expansion of $(2 x + \frac{1}{x ^{7}} + 3 x^{2})^{5}$ is .2023 · 25 Jan · Shift 1 · Q39 · Numerical
The remainder when (2023) $^{2023}$ is divided by 35 is .2023 · 25 Jan · Shift 2 · Q41 · Numerical
Let the coefficients of three consecutive terms in the binomial expansion of $(1 + 2 x)^{n}$ be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of those three terms, is .2023 · 29 Jan · Shift 1 · Q43 · Numerical
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…2023 · 29 Jan · Shift 1 · Q44 · Numerical
Let K be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1 + x)^{99}$ . Let $a$ be the middle term in the expansion of $(2 + \frac{1}{2})^{200}$ . If $\frac{^{200} C _{99} K}{a} = \frac{2 ^{l} m}{n}$ …2023 · 29 Jan · Shift 2 · Q35 · MCQ
If the coefficient of $x^{15}$ in the expansion of $(a x^{3} + \frac{1}{b x ^{1/3}})^{15}$ is equal to the coefficient of $x^{- 15}$ in the expansion of $(a x^{1/3} - \frac{1}{b x ^{3}})^{15}$ ,…2023 · 30 Jan · Shift 1 · Q27 · MCQ
Let $x = (83 + 13)^{13}$ and $y = (72 + 9)^{9}$ . If $[t]$ denotes the greatest integer $\leq t$ , then :2023 · 30 Jan · Shift 2 · Q35 · MCQ
$5 0^{th}$ root of a number $x$ is 12 and $5 0^{th}$ root of another number $y$ is 18 . Then the remainder obtained on dividing $(x + y)$ by 25 is .2023 · 30 Jan · Shift 2 · Q44 · Numerical
The remainder on dividing $5^{99}$ by 11 is .2023 · 31 Jan · Shift 1 · Q40 · Numerical
Let $α > 0$ , be the smallest number such that the expansion of $(x^{\frac{2}{3}} + \frac{2}{x ^{3}})^{30}$ has a term $β x^{- α}, β \in N$ . Then $α$ is equal to .2023 · 31 Jan · Shift 1 · Q42 · Numerical
The coefficient of $x^{- 6}$ , in the expansion of $(\frac{4 x}{5} + \frac{5}{2 x ^{2}})^{9}$ , is2023 · 31 Jan · Shift 2 · Q39 · Numerical
If the constant term in the binomial expansion of $(\frac{x ^{\frac{5}{2}}}{2} - \frac{4}{x ^{l}})^{9}$ is $- 84$ and the coefficient of $x^{- 3 l}$ is $2^{α} β$ , where $β < 0$ is an odd number, then $∣ α l - β ∣$ …2023 · 31 Jan · Shift 2 · Q44 · Numerical
The remainder when 32022 is divided by 5 is :2022 · 24 Jun · Shift 1 · Q23 · MCQ
The remainder on dividing 1 + 3 + 32 + 33 + ..... + 32021 by 50 is .2022 · 24 Jun · Shift 2 · Q41 · Numerical
If the maximum value of the term independent of $t$ in the expansion of $(t^{2} x^{\frac{1}{5}} + \frac{( 1 - x ) ^{\frac{1}{10}}}{t})^{15}, x ⩾ 0$ , is $K$ , then $8 K$ is equal to …2022 · 25 Jul · Shift 1 · Q38 · Numerical
The remainder when $(11)^{1011} + (1011)^{11}$ is divided by 9 is2022 · 25 Jul · Shift 2 · Q26 · MCQ
If $\frac{1}{2 . 3 ^{10}} + \frac{1}{2 ^{2} . 3 ^{9}} + ..... + \frac{1}{2 ^{10} . 3} = \frac{K}{2 ^{10} . 3 ^{10}}$ , then the remainder when K is divided by 6 is :2022 · 25 Jun · Shift 1 · Q28 · MCQ
The coefficient of x101 in the expression $(5 + x)^{500} + x (5 + x)^{499} + x^{2} (5 + x)^{498} + ..... + x^{500}$ , x > 0, is2022 · 25 Jun · Shift 2 · Q29 · MCQ
If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of $(2 x^{3} + \frac{3}{x})^{10}$ is $5^{10} - β . 3^{9}$ , then $β$ is equal to .2022 · 25 Jun · Shift 2 · Q40 · Numerical
The remainder when (2021)2023 is divided by 7 is :2022 · 26 Jun · Shift 1 · Q24 · MCQ
The remainder when $(2021)^{2022} + (2022)^{2021}$ is divided by 7 is2022 · 27 Jul · Shift 1 · Q28 · MCQ
Let for the $9^{th}$ term in the binomial expansion of $(3 + 6 x)^{n}$ , in the increasing powers of $6 x$ , to be the greatest for $x = \frac{3}{2}$ , the least value of $n$ is $n_{0}$ . If $k$ is the…2022 · 27 Jul · Shift 2 · Q35 · Numerical
If the coefficient of x10 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 co-prime to 5, then k is equal…2022 · 27 Jun · Shift 1 · Q39 · Numerical
The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is :2022 · 28 Jul · Shift 1 · Q33 · MCQ
Let the coefficients of the middle terms in the expansion of $(\frac{1}{6} + β x)^{4}, (1 - 3 β x)^{2}$ and $(1 - \frac{β}{2} x)^{6}, β > 0$ , respectively form the first three terms of an A.P. If…2022 · 28 Jul · Shift 2 · Q35 · Numerical
The number of positive integers k such that the constant term in the binomial expansion of $(2 x^{3} + \frac{3}{x ^{k}})^{12}$ , x $e$ 0 is 28 . l, where l is an odd integer, is .2022 · 28 Jun · Shift 1 · Q41 · Numerical
The term independent of x in the expansion of $(1 - x^{2} + 3 x^{3}) (\frac{5}{2} x^{3} - \frac{1}{5 x ^{2}})^{11}, x e 0$ is :2022 · 28 Jun · Shift 2 · Q25 · MCQ
Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $(42 + \frac{1}{4 3})^{n}$ , in the increasing powers of $\frac{1}{4 3}$ be $46 : 1$ …2022 · 29 Jul · Shift 1 · Q43 · Numerical
If the constant term in the expansion of $(3 x^{3} - 2 x^{2} + \frac{5}{x ^{5}})^{10}$ is 2k.l, where l is an odd integer, then the value of k is equal to:2022 · 29 Jun · Shift 1 · Q32 · MCQ
Let the coefficients of x $-$ 1 and x $-$ 3 in the expansion of $(2 x^{\frac{1}{5}} - \frac{1}{x ^{\frac{1}{5}}})^{15}, x > 0$ , be m and n respectively. If r is a positive integer such that $m n^{2} =^{15} C_{r} . 2^{r}$ …2022 · 29 Jun · Shift 2 · Q40 · Numerical
For two positive real numbers a and b such that $\frac{1}{a ^{2}} + \frac{1}{b ^{3}} = 4$ , then minimum value of the constant term in the expansion of $(a x^{\frac{1}{8}} + b x^{- \frac{1}{12}})^{10}$ is :2022 · 30 Jun · Shift 1 · Q27 · MCQ
If n is the number of irrational terms in the expansion of $(3^{1/4} + 5^{1/8})^{60}$ , then (n $-$ 1) is divisible by :2021 · 16 Mar · Shift 1 · Q29 · MCQ
If the fourth term in the expansion of $(x + x^{l o g_{2} x})^{7}$ is 4480, then the value of x where x $\in$ N is equal to :2021 · 17 Mar · Shift 1 · Q31 · MCQ
If (2021)3762 is divided by 17, then the remainder is .2021 · 17 Mar · Shift 1 · Q37 · Numerical
Let the coefficients of third, fourth and fifth terms in the expansion of $(x + \frac{a}{x ^{2}})^{n}, x e 0$ , be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to …2021 · 17 Mar · Shift 2 · Q35 · Numerical
The term independent of x in the expansion of $[\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}}]^{10}$ , x $e$ 1, is equal to .2021 · 18 Mar · Shift 2 · Q39 · Numerical
The number of rational terms in the binomial expansion of $(4^{\frac{1}{4}} + 5^{\frac{1}{6}})^{120}$ is .2021 · 20 Jul · Shift 1 · Q39 · Numerical
If the constant term, in binomial expansion of $(2 x^{r} + \frac{1}{x ^{2}})^{10}$ is 180, then r is equal to .2021 · 22 Jul · Shift 2 · Q45 · Numerical
If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)2022 is divided by 8 is .2021 · 25 Feb · Shift 2 · Q41 · Numerical
The total number of two digit numbers 'n', such that 3n + 7n is a multiple of 10, is .2021 · 25 Feb · Shift 2 · Q42 · Numerical
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 $\frac{1}{a - b} + \frac{1}{a - 2 b} + \frac{1}{a - 3 b} + ..... + \frac{1}{a - nb} = α n + β n^{2} + γ n^{3}$ …2021 · 25 Jul · Shift 1 · Q29 · MCQ
The ratio of the coefficient of the middle term in the expansion of (1 + x)20 and the sum of the coefficients of two middle terms in expansion of (1 + x)19 is .2021 · 25 Jul · Shift 1 · Q41 · Numerical
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 $e$ 0, 1 is equal to .2021 · 25 Jul · Shift 1 · Q44 · Numerical
The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 is :2021 · 25 Jul · Shift 2 · Q24 · MCQ
If the greatest value of the term independent of 'x' in the expansion of $(x sin α + a \frac{c o s α}{x})^{10}$ is $\frac{10 !}{( 5 !) ^{2}}$ , then the value of 'a' is equal to :2021 · 25 Jul · Shift 2 · Q27 · MCQ
The lowest integer which is greater than $(1 + \frac{1}{10 ^{100}})^{10^{100}}$ is .2021 · 25 Jul · Shift 2 · Q29 · MCQ
If the co-efficient of x7 and x8 in the expansion of $(2 + \frac{x}{3})^{n}$ are equal, then the value of n is equal to .2021 · 25 Jul · Shift 2 · Q46 · Numerical
The maximum value of the term independent of 't' in the expansion of $(t x^{\frac{1}{5}} + \frac{( 1 - x ) ^{\frac{1}{10}}}{t})^{10}$ where x $\in$ (0, 1) is :2021 · 26 Feb · Shift 1 · Q23 · MCQ
3 $\times$ 722 + 2 $\times$ 1022 $-$ 44 when divided by 18 leaves the remainder .2021 · 27 Aug · Shift 2 · Q39 · Numerical
If the coefficients of x7 in $(x^{2} + \frac{1}{b x})^{11}$ and x $-$ 7 in $(x - \frac{1}{b x ^{2}})^{11}$ , b $e$ 0, are equal, then the value of b is equal to :2021 · 27 Jul · Shift 1 · Q29 · MCQ
A possible value of 'x', for which the ninth term in the expansion of ${3^{l o g_{3} 25^{x - 1} + 7} + 3^{(- \frac{1}{8}) l o g_{3} (5^{x - 1} + 1)}}^{10}$ in the increasing powers…2021 · 27 Jul · Shift 2 · Q24 · MCQ
If $(\frac{3 ^{6}}{4 ^{4}}) k$ is the term, independent of x, in the binomial expansion of $(\frac{x}{4} - \frac{12}{x ^{2}})^{12}$ , then k is equal to .2021 · 31 Aug · Shift 1 · Q44 · Numerical
If the coefficient of a7b8 in the expansion of (a + 2b + 4ab)10 is K.216, then K is equal to .2021 · 31 Aug · Shift 2 · Q37 · Numerical
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…2020 · 2 Sep · Shift 1 · Q24 · MCQ
If the number of integral terms in the expansion of (31/2 + 51/8)n is exactly 33, then the least value of n is :2020 · 3 Sep · Shift 1 · Q24 · MCQ
If the term independent of x in the expansion of $(\frac{3}{2} x^{2} - \frac{1}{3 x})^{9}$ is k, then 18 k is equal to :2020 · 3 Sep · Shift 2 · Q39 · MCQ
Let $(2 x^{2} + 3 x + 4)^{10} = r = 0 \sum 20 a_{r} x^{r}$ Then $\frac{a _{7}}{a _{13}}$ is equal to .2020 · 4 Sep · Shift 1 · Q24 · Numerical
The natural number m, for which the coefficient of x in the binomial expansion of $(x^{m} + \frac{1}{x ^{2}})^{22}$ is 1540, is .............2020 · 5 Sep · Shift 1 · Q22 · Numerical
If {p} denotes the fractional part of the number p, then ${\frac{3 ^{200}}{8}}$ , is equal to :2020 · 6 Sep · Shift 1 · Q32 · MCQ
If the constant term in the binomial expansion of $(x - \frac{k}{x ^{2}})^{10}$ is 405, then |k| equals :2020 · 6 Sep · Shift 2 · Q34 · MCQ
The greatest positive integer k, for which 49k + 1 is a factor of the sum 49125 + 49124 + ..... + 492 + 49 + 1, is:2020 · 7 Jan · Shift 1 · Q39 · MCQ
The coefficient of x7 in the expression (1 + x)10 + x(1 + x)9 + x2(1 + x)8 + ......+ x10 is:2020 · 7 Jan · Shift 2 · Q22 · MCQ
If $α$ and $β$ be the coefficients of x4 and x2 respectively in the expansion of $(x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6}$ , then2020 · 8 Jan · Shift 2 · Q31 · MCQ
The coefficient of x4 is the expansion of (1 + x + x2)10 is .2020 · 9 Jan · Shift 1 · Q32 · Numerical
In the expansion of $(\frac{x}{c o s θ} + \frac{1}{x s i n θ})^{16}$ , if $ℓ_{1}$ is the least value of the term independent of x when $\frac{π}{8} \leq θ \leq \frac{π}{4}$ and $ℓ_{2}$ is the…2020 · 9 Jan · Shift 2 · Q23 · MCQ
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 :2019 · 8 Apr · Shift 2 · Q34 · MCQ
If the fourth term in the binomial expansion of $(\frac{2}{x} + x^{l o g_{8} x})^{6}$ (x > 0) is 20 × 87, then a value of x is :2019 · 9 Apr · Shift 1 · Q42 · MCQ
If the fractional part of the number ${\frac{2 ^{403}}{15}} i s \frac{k}{15}$ , then k is equal to :2019 · 9 Jan · Shift 1 · Q37 · MCQ
The smallest natural number n, such that the coefficient of x in the expansion of $(x^{2} + \frac{1}{x ^{3}})^{n}$ is nC23, is :2019 · 10 Apr · Shift 2 · Q42 · MCQ
If the third term in the binomial expansion of $(1 + x^{l o g_{2} x})^{5}$ equals 2560, then a possible value of x is -2019 · 10 Jan · Shift 1 · Q33 · MCQ
The positive value of $λ$ for which the co-efficient of x2 in the expression x2 $(x + \frac{λ}{x ^{2}})^{10}$ is 720, is -2019 · 10 Jan · Shift 2 · Q29 · MCQ
The sum of the real values of x for which the middle term in the binomial expansion of $(\frac{x ^{3}}{3} + \frac{3}{x})^{8}$ equals 5670 is :2019 · 11 Jan · Shift 1 · Q31 · MCQ
The coefficient of x18 in the product (1 + x) (1 – x)10 (1 + x + x2)9 is :2019 · 12 Apr · Shift 1 · Q42 · MCQ
The term independent of x in the expansion of $(\frac{1}{60} - \frac{x ^{8}}{81}) . (2 x^{2} - \frac{3}{x ^{2}})^{6}$ is equal to :2019 · 12 Apr · Shift 2 · Q29 · MCQ
A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $(2^{1/3} + \frac{1}{2 ( 3 ) ^{1/3}})^{10}$ is :2019 · 12 Jan · Shift 1 · Q24 · MCQ
The total number of irrational terms in the binomial expansion of (71/5 – 31/10)60 is :2019 · 12 Jan · Shift 2 · Q31 · MCQ
If (27)999 is divided by 7, then the remainder is :2017 · 8 Apr · Shift 1 · Q28 · MCQ
The coefficient of x−5 in the binomial expansion of $(\frac{x + 1}{x ^{\frac{2}{3}} - x ^{\frac{1}{3}} + 1} - \frac{x - 1}{x - x ^{\frac{1}{2}}})^{10},$ where x $e$ 0, 1, is :2017 · 9 Apr · Shift 1 · Q25 · MCQ
For x $\in$ R, x $e$ -1, if (1 + x)2016 + x(1 + x)2015 + x2(1 + x)2014 + . . . . + x2016 = $i = 0 \sum 2016 a_{i} x^{i},$ then a17 is equal to :2016 · 9 Apr · Shift 1 · Q24 · MCQ
If the coefficients of x−2 and x−4 in the expansion of $(x^{\frac{1}{3}} + \frac{1}{2 x ^{\frac{1}{3}}})^{18}, (x > 0),$ are m and n respectively, then $\frac{m}{n}$ is equal to :2016 · 10 Apr · Shift 1 · Q23 · MCQ
The term independent of $x$ in expansion of $(\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}})^{10}$ is2013 · Shift 0 · Q39 · MCQ
If $n$ is a positive integer, then $(3 + 1)^{2 n} - (3 - 1)^{2 n}$ is :2012 · Shift 0 · Q40 · MCQ
The remainder left out when $8^{2 n} - (62)^{2 n + 1}$ is divided by 9 is :2009 · Shift 0 · Q42 · MCQ
In the binomial expansion of $(a - b)^{n}, n \geq 5,$ the sum of $5^{t h}$ and $6^{t h}$ terms is zero, then $a / b$ equals2007 · Shift 0 · Q59 · MCQ
If the expansion in powers of $x$ of the function $\frac{1}{( 1 - a x ) ( 1 - b x )}$ is $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} .....$ then $a_{n}$ is2006 · Shift 0 · Q73 · MCQ
If $x$ is so small that $x^{3}$ and higher powers of $x$ may be neglected, then $\frac{( 1 + x ) ^{\frac{3}{2}} - ( 1 + \frac{1}{2} x ) ^{3}}{( 1 - x ) ^{\frac{1}{2}}}$ may be approximated…2005 · Shift 0 · Q100 · MCQ
If the coefficients of rth, (r+1)th, and (r + 2)th terms in the binomial expansion of $(1 + y)^{m}$ are in A.P., then m and r satisfy the equation2005 · Shift 0 · Q101 · MCQ
If the coefficient of $x^{7}$ in $[a x^{2} + (\frac{1}{b x})]^{11}$ equals the coefficient of $x^{- 7}$ in $[a x - (\frac{1}{b x ^{2}})]^{11}$ , then $a$ and $b$ satisfy…2005 · Shift 0 · Q99 · MCQ
The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + α x)^{4}$ and $(1 - α x)^{6}$ is the same if $α$ equals2004 · Shift 0 · Q104 · MCQ
The coefficient of $x^{n}$ in expansion of $(1 + x) (1 - x)^{n}$ is2004 · Shift 0 · Q106 · MCQ
If $x$ is positive, the first negative term in the expansion of $(1 + x)^{27/5}$ is2003 · Shift 0 · Q104 · MCQ
The number of integral terms in the expansion of $(3 + 85)^{256}$ is2003 · Shift 0 · Q106 · MCQ
The coefficients of $x^{p}$ and $x^{q}$ in the expansion of $(1 + x)^{p + q}$ are2002 · Shift 0 · Q105 · MCQ
The positive integer just greater than $(1 + 0.0001)^{10000}$ is2002 · Shift 0 · Q106 · MCQ