Binomial Theorem and its Simple Applications jee mains questions | JEE Main Maths Binomial Theorem and its Simple Applications Previous Year Question

Q 21 :

The term independent of $x$ in the expansion of ${(\frac{2 \sqrt{x}}{5} - \frac{1}{2 x \sqrt{x}})}^{11}$ is

$5^{th} term$
$6^{th} term$
$11^{th} term$
$no term$

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(4)

$Let T_{r + 1} be general term of given expansion$

$∴$ $T_{r + 1} = {}^{11}C_{r} {(\frac{2 \sqrt{x}}{5})}^{11 - r} {(\frac{- 1}{2 x \sqrt{x}})}^{r}$

$= {}^{11}C_{r} {(\frac{2}{5})}^{11 - r} \times {(\frac{- 1}{2})}^{r} x^{\frac{11 - r}{2} - \frac{3 r}{2}}$

$∴$ $T_{r + 1} = {}^{11}C_{r} {(\frac{2}{5})}^{11 - r} {(\frac{- 1}{2})}^{r} x^{(11 - 4 r) / 2}$

$For a term independent of x, put \frac{11 - 4 r}{2} = 0 or r = \frac{11}{4}$

$But r can not be in fraction.$

$∴$ $There is no term independent of x .$

Q 22 :

The ninth term of the expansion ${(3 x - \frac{1}{2 x})}^{8}$ is

$\frac{1}{512 x^{9}}$
$\frac{- 1}{512 x^{9}}$
$\frac{- 1}{256 x^{8}}$
$\frac{1}{256 x^{8}}$

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(4)

$Given expansion is {(3 x - \frac{1}{2 x})}^{8}$

$T_{9} = {}^{8}C_{8} {(3 x)}^{0} {(\frac{- 1}{2 x})}^{8} = \frac{1}{256 x^{8}}$

Q 23 :

The term independent of $x$ in the expansion of ${(\frac{3}{2} x^{2} - \frac{1}{3 x})}^{6}$ is

$\frac{5}{12}$
$\frac{12}{5}$
$\frac{5}{6}$
none of these

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(1)

${(\frac{3}{2} x^{2} - \frac{1}{3 x})}^{6}$

$T_{r + 1} = {}^{6}C_{r} {(\frac{3}{2} x^{2})}^{6 - r} {(\frac{- 1}{3 x})}^{r} = {}^{6}C_{r} {(\frac{3}{2})}^{6 - r} {(\frac{- 1}{3})}^{r} x^{12 - 3 r}$

$Put 12 - 3 r = 0 \Rightarrow r = 4$

$∴$ $Term independent of x is T_{5} = {}^{6}C_{4} {(\frac{3}{2})}^{2} {(\frac{- 1}{3})}^{4} = \frac{5}{12}$

Q 24 :

The term independent of $x$ in the expansion of ${[\sqrt{\frac{x}{3}} + \frac{3}{2 x^{2}}]}^{10}$ is:

$\frac{5}{4}$
$\frac{7}{4}$
$\frac{9}{4}$
$45$

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(1)

$Let {(r + 1)}^{th} term in the expansion of$

${[\sqrt{\frac{x}{3}} + \frac{3}{2 x^{2}}]}^{10} is independent of x$

$So T_{r + 1} = {}^{10}C_{r} {(\sqrt{\frac{x}{3}})}^{10 - r} {(\frac{3}{2 x^{2}})}^{r}$

$= {}^{10}C_{r} {(\frac{1}{\sqrt{3}})}^{10 - r} {(\frac{3}{2})}^{r} x^{\frac{10 - 5 r}{2}}$

$Putting x^{(\frac{10 - 5 r}{2})} equal to x^{0} \Rightarrow \frac{10 - 5 r}{2} = 0 \Rightarrow r = 2$

$∴$ $T_{2 + 1} = {}^{10}C_{2} {(\frac{1}{\sqrt{3}})}^{10 - 2} {(\frac{3}{2})}^{2} = \frac{5}{4}$

Q 25 :

If $x^{r}$ occurs in the expansion of ${(x + \frac{1}{x})}^{n},$ then its coefficient is

$\frac{n!}{{(r!)}^{2}}$
$\frac{n!}{(r + 1)! (r - 1)!}$
$\frac{n!}{(\frac{n + r}{2})! (\frac{n - r}{2})!}$
$\frac{n!}{{[(\frac{r}{2})!]}^{2}}$

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(3)

$T_{m + 1} = {}^{n}C_{m} {(x)}^{n - m} {(\frac{1}{x})}^{m} = {}^{n}C_{m} x^{n - 2 m}$

$If x^{r} occur in expansion, then$

$\Rightarrow n - 2 m = r \Rightarrow m = \frac{n - r}{2}$

$Coefficient of x^{r} = {}^{n}C_{(\frac{n - r}{2})} = \frac{n!}{(\frac{n - r}{2})! (\frac{n + r}{2})!}$

Q 26 :

If $A$ and $B$ are coefficients of $x^{n}$ in the expansions of ${(1 + x)}^{2 n}$ and ${(1 + x)}^{2 n - 1}$ respectively, then $\frac{B}{A}$ is equal to

$\frac{1}{2}$
$2$
$1$
$\frac{1}{n}$

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(1)

$Now, A = {}^{2 n}C_{n}$ $and$ $B = {}^{2 n - 1}C_{n}$

$Thus, \frac{B}{A} = \frac{(2 n - 1)!}{n! (n - 1)!} \times \frac{n! n!}{(2 n)!} = \frac{n}{2 n} = \frac{1}{2}$

Q 27 :

The coefficient of the term independent of $x$ in the expansion of ${(\sqrt{x} + \frac{1}{\sqrt{x}})}^{10}$ is equal to

10
252
20
256

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(2)

$General term of expansion {(\sqrt{x} + \frac{1}{\sqrt{x}})}^{10} is$

$T_{r + 1} = {}^{10}C_{r} {(\sqrt{x})}^{10 - r} {(\frac{1}{\sqrt{x}})}^{r} = {}^{10}C_{r} x^{5 - r}$

$Term independent of x, means power of x is 0$

$\Rightarrow 5 - r = 0 \Rightarrow r = 5$

$Hence, T_{5 + 1} = {}^{10}C_{5} = 252$

Q 28 :

The $13^{th}$ term in the expansion of ${(x^{2} + \frac{2}{x})}^{n}$ is independent of $x$ , then the sum of the divisors of $n$ is

39
36
37
38

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(1)

$T_{13} = {}^{n}C_{12} {(x^{2})}^{n - 12} \cdot {(\frac{2}{x})}^{12}$

$\Rightarrow x^{2 n - 24 - 12} = x^{0} \Rightarrow n = 18$

$∴$ $The sum of the divisors of n is 39$

Q 29 :

If the coefficient of $x^{8}$ in ${(a x^{2} + \frac{1}{b x})}^{13}$ is equal to the coefficient of $x^{- 8}$ in ${(a x - \frac{1}{b x^{2}})}^{13},$ then $a$ and $b$ will satisfy the relation

$a b + 1 = 0$
$a b = 1$
$a = 1 - b$
$a + b = - 1$

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(1)

$∵$ $T_{r + 1} in {(a x^{2} + \frac{1}{b x})}^{13}$

$= {}^{13}C_{r} {(a x^{2})}^{13 - r} {(\frac{1}{b x})}^{r} = {}^{13}C_{r} \frac{a^{13 - r}}{b^{r}} x^{26 - 3 r}$

$For coefficient of x^{8}, we have 26 - 3 r = 8 \Rightarrow r = 6$

$\Rightarrow Coefficient of x^{8} = {}^{13}C_{6} \frac{a^{7}}{b^{6}}$

$∵$ $T_{r + 1} in {(a x - \frac{1}{b x^{2}})}^{13} = {}^{13}C_{r} {(a x)}^{13 - r} {(- \frac{1}{b x^{2}})}^{r}$

$= {}^{13}C_{r} \frac{a^{13 - r}}{b^{r}} x^{13 - 3 r} {(- 1)}^{r}$

$For coefficient of x^{- 8}, we have 13 - 3 r = - 8 \Rightarrow r = 7$

$∴$ $Coefficient of x^{- 8} = - {}^{13}C_{7} \frac{a^{6}}{b^{7}}$

$Now, {}^{13}C_{6} \frac{a^{7}}{b^{6}} = - {}^{13}C_{7} \frac{a^{6}}{b^{7}} \Rightarrow a b = - 1 \Rightarrow a b + 1 = 0$

Q 30 :

The value of $({}^{21}C_{1} - {}^{10}C_{1}) + ({}^{21}C_{2} - {}^{10}C_{2}) + ({}^{21}C_{3} - {}^{10}C_{3}) + ({}^{21}C_{4} - {}^{10}C_{4}) + \dots + ({}^{21}C_{10} - {}^{10}C_{10})$ is

$2^{21} - 2^{10}$
$2^{20} - 2^{9}$
$2^{20} - 2^{10}$
$2^{21} - 2^{11}$

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(3)

$({}^{21}C_{1} - {}^{10}C_{1}) + ({}^{21}C_{2} - {}^{10}C_{2}) + \dots + ({}^{21}C_{10} - {}^{10}C_{10})$

$= ({}^{21}C_{1} + {}^{21}C_{2} + \dots + {}^{21}C_{10}) - ({}^{10}C_{1} + {}^{10}C_{2} + \dots + {}^{10}C_{10})$

$= \frac{2^{21}}{2} - 2^{10} = 2^{20} - 2^{10}$

Q 31 :

Let $X = {({}^{10}C_{1})}^{2} + 2 {({}^{10}C_{2})}^{2} + 3 {({}^{10}C_{3})}^{2} + \dots + 10 {({}^{10}C_{10})}^{2},$ where ${}^{10}C_{r}$ , $r \in {1, 2, \dots, 10}$ , denote binomial coefficients. Then, the value of $\frac{1}{1430} X$ is ______ .

(646)

$X = \sum_{r = 1}^{10} r {({}^{10}C_{r})}^{2} = \sum_{r = 1}^{10} r \cdot {}^{10}C_{r} \cdot {}^{10}C_{r}$

$= 10 \sum_{r = 1}^{10} {}^{9}C_{r - 1} \cdot {}^{10}C_{10 - r} = 10 \cdot {}^{19}C_{9}$

$∴$ $\frac{X}{1430} = \frac{10 \cdot {}^{19}C_{9}}{1430} = \frac{{}^{19}C_{9}}{143} = 646$

Q 32 :

What is the value of the expression ${(1 + \sqrt{2})}^{4} - {(1 - \sqrt{2})}^{4} ?$

$34$
$10 \sqrt{2}$
$12 + 8 \sqrt{2}$
$24 \sqrt{2}$

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(4)

$Consider, {(1 + \sqrt{2})}^{4} - {(1 - \sqrt{2})}^{4}$

$= 2 [{}^{4}C_{1} \sqrt{2} + {}^{4}C_{3} {(\sqrt{2})}^{3}]$ $[∵ {(a + b)}^{4} - {(a - b)}^{4} = 2 ({}^{4}C_{1} a^{3} b + {}^{4}C_{3} a b^{3})]$

$= 2 [4 \sqrt{2} + 4 \times 2 \sqrt{2}] = 2 (12 \sqrt{2}) = 24 \sqrt{2}$

Q 33 :

The number 81 is the coefficient of $x^{k}$ in the binomial expansion of ${(x^{2} + \frac{3}{x})}^{4}, x \neq 0 .$ Then the value of $k$ equals

- 2
2
- 4
4

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(3)

${(x^{2} + \frac{3}{x})}^{4} = {}^{4}C_{0} {(\frac{3}{x})}^{4} + {}^{4}C_{1} x^{2} {(\frac{3}{x})}^{3}$ $+ {}^{4}C_{2} x^{4} {(\frac{3}{x})}^{2} + {}^{4}C_{3} x^{6} {(\frac{3}{x})}^{1} + {}^{4}C_{4} x^{8} {(\frac{3}{x})}^{0}$

$= \frac{81}{x^{4}} + 4 \times x^{2} \times \frac{27}{x^{3}} + 6 \times 9 \times x^{2} + 12 x^{5} + x^{8}$

$= \frac{81}{x^{4}} + \frac{108}{x} + 54 x^{2} + 12 x^{5} + x^{8}$

$Here, 81 is coefficient of x^{- 4}$ $∴$ $k = - 4$

Q 34 :

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of ${(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6},$ then

$α + β = - 30$
$α - β = - 132$
$α + β = 60$
$α - β = 60$

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(2)

${(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6}$

$= 2 [{}^{6}C_{0} x^{6} + {}^{6}C_{2} x^{4} (x^{2} - 1) + {}^{6}C_{4} x^{2} {(x^{2} - 1)}^{2} + {}^{6}C_{6} {(x^{2} - 1)}^{3}]$

$= 2 [x^{6} + 15 (x^{6} - x^{4}) + 15 x^{2} (x^{4} - 2 x^{2} + 1) + (- 1 + 3 x^{2} - 3 x^{4} + x^{6})]$

$= 2 (32 x^{6} - 48 x^{4} + 18 x^{2} - 1)$

$∴$ $α = - 96$ $and$ $β = 36$

$∴$ $α - β = - 132$

Q 35 :

Let $m, n \in N$ and $\gcd (2, n) = 1$ . If $30 (\binom{30}{0}) + 29 (\binom{30}{1}) + \dots + 2 (\binom{30}{28}) + 1 (\binom{30}{29}) = n \cdot 2^{m},$ then $n + m$ is equal to _________. $(Here, (\binom{n}{k}) = {}^{n}C_{k})$

(45)

$We have,$

$30 (\binom{30}{0}) + 29 (\binom{30}{1}) + \dots + 2 (\binom{30}{28}) + 1 (\binom{30}{29})$

$a = 30 (\binom{30}{30}) + 29 (\binom{30}{29}) + \dots + 2 (\binom{30}{2}) + 1 (\binom{30}{1})$

$= \sum_{r = 1}^{30} r (\binom{30}{r}) = \sum_{r = 1}^{30} r \cdot \frac{30}{r} (\binom{29}{r - 1})$

$= 30 [(\binom{29}{0}) + (\binom{29}{1}) + (\binom{29}{2}) + \dots + (\binom{29}{29})]$

$= 30 \cdot 2^{29} = 15 \times 2^{30} = n \cdot 2^{m}$ $(given)$

$∴$ $n + m = 15 + 30 = 45$

Q 36 :

$\sum_{k = 0}^{20} {({}^{20}C_{k})}^{2}$ is equal to

${}^{40}C_{21}$
${}^{41}C_{20}$
${}^{40}C_{20}$
${}^{40}C_{19}$

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(3)

We know that

${(1 + x)}^{n} = {}^{n}C_{0} + {}^{n}C_{1} x + {}^{n}C_{2} x^{2} + \dots + {}^{n}C_{n} x^{n}$ ...(i)

Also, ${(x + 1)}^{n} = {}^{n}C_{0} x^{n} + {}^{n}C_{1} x^{n - 1} + {}^{n}C_{2} x^{n - 2} + \dots + {}^{n}C_{n}$ ...(ii)

Multiply (i) & (ii), we get

${(1 + x)}^{2 n} = ({}^{n}C_{0} + {}^{n}C_{1} x + \dots + {}^{n}C_{n} x^{n}) ({}^{n}C_{0} x^{n} + {}^{n}C_{1} x^{n - 1} + \dots + {}^{n}C_{n})$

Comparing coefficient of $x^{n}$ on both sides, we get

${}^{2 n}C_{n} = {({}^{n}C_{0})}^{2} + {({}^{n}C_{1})}^{2} + {({}^{n}C_{2})}^{2} + \dots + {({}^{n}C_{n})}^{2}$

$\Rightarrow \sum_{r = 0}^{n} {({}^{n}C_{r})}^{2} = {}^{2 n}C_{n}$

$∴$ $\sum_{k = 0}^{20} {({}^{20}C_{k})}^{2} = {}^{40}C_{20}$ $[∵ n = 20]$

Q 37 :

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

$\frac{1}{15}$
$\frac{4}{15}$
$\frac{14}{15}$
$\frac{8}{15}$

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(1)

$We have,$ ${\frac{4^{2022}}{15}} = {\frac{{(4^{2})}^{1011}}{15}} = {\frac{{(1 + 15)}^{1011}}{15}}$

$= {\frac{1}{15} + \frac{15^{1010}}{15} + \dots} = {\frac{1}{15} + 0 + \dots} = {\frac{1}{15}}$

Q 38 :

The remainder, when $7^{103}$ is divided by 17, is ______.

(12)

$7^{103} = 7 \cdot 7^{102}$

$7 \cdot 7^{102} = 7 {(7^{2})}^{51} = 7 {(49)}^{51} = 7 {(51 - 2)}^{51}$

$Remainder = 7 {(- 2)}^{51} = - 7 {(2)}^{51} = - 7 (2^{48 + 3}) = - 7 \cdot 8 (2^{48})$

$= - 56 {(2^{4})}^{12} = - 56 {(17 - 1)}^{12}$

$Again Remainder = - 56 \times {(- 1)}^{12} = (- 56)$

$Remainder = 12$

Q 39 :

The coefficient of $x^{17}$ in ${(1 - x)}^{13} {(1 + x + x^{2})}^{12}$ is

${}^{12}C_{6}$
${}^{9}C_{7}$
$0$
$1$

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(3)

$We have,$ ${(1 - x)}^{13} {(1 + x + x^{2})}^{12}$

$= (1 - x) {[(1 - x) (1 + x + x^{2})]}^{12} = (1 - x) {(1 - x^{3})}^{12}$

In the expansion, all the powers of $x$ is either in the form of $3 k$ or $(3 k + 1)$ for some $k \in ℤ$ . But 17 has the form $(3 k + 2)$ .

$∴$ Coefficient of $x^{17}$ in the given expansion is 0.

Q 40 :

The remainder when $428^{2024}$ is divided by 21 is _______.

(1)

$We have,$ $428^{2024} = {(420 + 8)}^{2024}$

$= {}^{2024}C_{0} {(420)}^{2024} + {}^{2024}C_{1} {(420)}^{2023} {(8)}^{1} + \dots + {}^{2024}C_{2023} (420) 8^{2023} + {}^{2024}C_{2024} 8^{2024}$

$= 420 [Some integer] + 64^{1012}$

$= 21 \times 20 [Some integer] + {(63 + 1)}^{1012}$

$= 21 \times 20 [Some integer] + 63 \times Some integer + 1$

$= 21 \times 20 [Some integer] + 21 \times 3 \times Some integer + 1$

$Hence, remainder = 1$

Q 41 :

The middle term of expansion ${(\frac{10}{x} + \frac{x}{10})}^{10}$ is

${}^{8}C_{5}$
${}^{10}C_{5}$
${}^{7}C_{5}$
${}^{9}C_{5}$

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(2)

$Given expansion is {(\frac{10}{x} + \frac{x}{10})}^{10}$

Since, $n = 10$ is even

$∴$ $Middle term = t_{\frac{n}{2} + 1} = t_{6}$

$∴$ $t_{6} = {}^{10}C_{5} {(\frac{10}{x})}^{5} {(\frac{x}{10})}^{5} = {}^{10}C_{5}$

Q 42 :

The constant term in the expansion of ${(x^{2} - \frac{1}{x^{2}})}^{16}$ is

${}^{16}C_{8}$
${}^{16}C_{7}$
${}^{16}C_{9}$
${}^{16}C_{10}$

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(1)

$Let {(r + 1)}^{th} term be the constant term in the expansion of {(x^{2} - \frac{1}{x^{2}})}^{16}$

$∴$ $T_{r + 1} = {}^{16}C_{r} x^{2 r} {(- \frac{1}{x^{2}})}^{16 - r}$

$= {(- 1)}^{16 - r} {}^{16}C_{r} x^{2 r} \cdot x^{- 32 + 2 r} = {(- 1)}^{16 - r} {}^{16}C_{r} x^{4 r - 32}$

For constant term, $4 r - 32 = 0 \Rightarrow r = 8$

$∴$ $T_{9} = {(- 1)}^{8} {}^{16}C_{8} = {}^{16}C_{8}$

Q 43 :

If the fourth term in the binomial expansion of ${(\frac{2}{x} + x^{\log_{8} x})}^{6}$ $(x > 0)$ is $20 \times 8^{7}$ , then a value of $x$ is

$8$
$8^{- 2}$
$8^{3}$
$8^{2}$

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(4)

Q 44 :

The positive value of $λ$ for which the coefficient of $x^{2}$ in the expression $x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}$ is 720, is

$\sqrt{5}$
$4$
$2 \sqrt{2}$
$3$

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(2)

$T_{r + 1} term of {(\sqrt{x} + \frac{λ}{x^{2}})}^{10} is$

${}^{10}C_{r} {(\sqrt{x})}^{10 - r} {(\frac{λ}{x^{2}})}^{r} = {}^{10}C_{r} x^{5 - \frac{5 r}{2}} {(λ)}^{r}$

$For this to be independent of x,$ $5 - \frac{5 r}{2} = 0 \Rightarrow r = 2$

$∴$ ${}^{10}C_{2} λ^{2} = 720 \Rightarrow 45 λ^{2} = 720$

$\Rightarrow$ $λ^{2} = 16 \Rightarrow λ = \pm 4$

$∴$ $The positive value of λ for which the coefficient of x^{2} is 720 is 4 .$

Q 45 :

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 ________.

(55)

Q 46 :

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 ________.

(210)

$The term independent of x in the expansion of$

${(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}$

$= {[(x^{1 / 3} + 1) - \frac{(x^{1 / 2} + 1)}{x^{1 / 2}}]}^{10} = {(x^{1 / 3} - x^{- 1 / 2})}^{10}$

$T_{r + 1} = {}^{10}C_{r} {(x^{1 / 3})}^{r} \cdot {(- x^{- 1 / 2})}^{10 - r}$

$= {}^{10}C_{r} {(- 1)}^{10 - r} \cdot x^{r / 3 + r / 2 - 5}$

$For this term to be independent,$ $\frac{r}{3} + \frac{r}{2} = 5$

$\Rightarrow r = 6$

$∴$ $Term independent is T_{7} = {}^{10}C_{6} = 210$

Q 47 :

Let the coefficients of third, fourth and fifth terms in the expansion of ${(x + \frac{a}{x^{2}})}^{n},$ $x \neq 0$ , be in the ratio 12 : 8 : 3. Then the term independent of $x$ in the expansion, is equal to _______.

(4)

$T_{r + 1} = {}^{n}C_{r} {(x)}^{n - r} {(\frac{a}{x^{2}})}^{r} = {}^{n}C_{r} a^{r} x^{n - 3 r}$

$We have,$ $\frac{{}^{n}C_{2} a^{2}}{{}^{n}C_{3} a^{3}} = \frac{12}{8} \Rightarrow 2 = a (n - 2)$ ...(i)

$and$ $\frac{{}^{n}C_{3} a^{3}}{{}^{n}C_{4} a^{4}} = \frac{8}{3} \Rightarrow 3 = 2 a (n - 3)$ ...(ii)

$Dividing (i) by (ii), we get$

$\frac{2}{3} = \frac{n - 2}{2 (n - 3)} \Rightarrow n = 6$

$From (i),$ $2 = a (6 - 2) \Rightarrow a = \frac{1}{2}$

$For the term independent of x,$ $n - 3 r = 0$

$\Rightarrow$ $6 - 3 r = 0 \Rightarrow r = 2$

$So,$ $T_{3} = {}^{6}C_{2} a^{2} = \frac{15}{4} \approx 4$ $(Approx.)$

Q 48 :

If the constant term, in binomial expansion of ${(2 x^{r} + \frac{1}{x^{2}})}^{10}$ is 180, then $r$ is equal to ________.

(8)

Q 49 :

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}})}^{n}$ , in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6} : 1$ . If the sixth term from the beginning is $\frac{α}{\sqrt[4]{3}}$ , then $α$ is equal to _______.

(84)

$We have,$ ${(2^{1 / 4} + \frac{1}{3^{1 / 4}})}^{n}$

$Given that$ $\frac{T_{5}}{T_{n - 3}} = \frac{6^{1 / 4}}{1}$

$\Rightarrow$ $\frac{{}^{n}C_{4} {(2^{1 / 4})}^{n - 4} {(3^{- 1 / 4})}^{4}}{{}^{n}C_{n - 4} {(2^{1 / 4})}^{4} {(3^{- 1 / 4})}^{n - 4}} = \frac{6^{1 / 4}}{1}$

$\Rightarrow$ ${(2^{1 / 4})}^{n - 8} {(3^{- 1 / 4})}^{8 - n} = 6^{1 / 4}$

$\Rightarrow$ ${(6^{1 / 4})}^{n - 8} = 6^{1 / 4} \Rightarrow n - 8 = 1 \Rightarrow n = 9$

$Now,$ $T_{6} = {}^{9}C_{5} {(2^{1 / 4})}^{4} {(3^{- 1 / 4})}^{5}$

$= 126 \times 2 \times (3^{- 1}) (3^{- 1 / 4})$

$= \frac{252}{3} \frac{1}{3^{1 / 4}} = \frac{84}{\sqrt[4]{3}} = \frac{α}{\sqrt[4]{3}}$

$Thus,$ $α = 84$

Q 50 :

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of ${(a x^{2} + \frac{70}{27 b x})}^{4}$ is equal to the coefficient of $x^{- 5}$ in the expansion of ${(a x - \frac{1}{b x^{2}})}^{7}$ , then the value of $2 b$ is __________.

(3)

$Given,$ ${(a x^{2} + \frac{70}{27 b x})}^{4}$

$The general term is given by$

$T_{r + 1} = {}^{4}C_{r} {(a x^{2})}^{4 - r} {(\frac{70}{27 b x})}^{r}$ ...(1)

$For coefficient of x^{5}, put 8 - 2 r - r = 5 \Rightarrow r = 1$

$∴$ $The coefficient of x^{5} in {(a x^{2} + \frac{70}{27 b x})}^{4} = {}^{4}C_{1} \cdot \frac{a^{3} (70)}{27 \cdot b}$

$General term of {(a x - \frac{1}{b x^{2}})}^{7} is given by$

$T_{r + 1} = {}^{7}C_{r} {(a x)}^{7 - r} {(- \frac{1}{b x^{2}})}^{r}$

$For coefficient of x^{- 5}, put 7 - r - 2 r = - 5 \Rightarrow r = 4$

$∴$ $The coefficient of x^{- 5} in {(a x - \frac{1}{b x^{2}})}^{7} = {}^{7}C_{4} \frac{a^{3} {(- 1)}^{4}}{b^{4}}$

$Now,$ ${}^{4}C_{1} \frac{a^{3} (70)}{27 (b)} = {}^{7}C_{4} \frac{a^{3}}{b^{4}}$

$\Rightarrow$ $\frac{4 \times 70}{27 (b)} = \frac{35}{b^{4}} \Rightarrow b^{3} = \frac{27}{8} \Rightarrow b = \frac{3}{2}$

$∴$ $2 b = 3$

Topic Question Set

Q 21 :

The term independent of $x$ in the expansion of ${(\frac{2 \sqrt{x}}{5} - \frac{1}{2 x \sqrt{x}})}^{11}$ is

Q 22 :

The ninth term of the expansion ${(3 x - \frac{1}{2 x})}^{8}$ is

Q 23 :

The term independent of $x$ in the expansion of ${(\frac{3}{2} x^{2} - \frac{1}{3 x})}^{6}$ is

Q 24 :

The term independent of $x$ in the expansion of ${[\sqrt{\frac{x}{3}} + \frac{3}{2 x^{2}}]}^{10}$ is:

Q 25 :

If $x^{r}$ occurs in the expansion of ${(x + \frac{1}{x})}^{n},$ then its coefficient is

Q 26 :

If $A$ and $B$ are coefficients of $x^{n}$ in the expansions of ${(1 + x)}^{2 n}$ and ${(1 + x)}^{2 n - 1}$ respectively, then $\frac{B}{A}$ is equal to

Q 27 :

The coefficient of the term independent of $x$ in the expansion of ${(\sqrt{x} + \frac{1}{\sqrt{x}})}^{10}$ is equal to

Q 28 :

The $13^{th}$ term in the expansion of ${(x^{2} + \frac{2}{x})}^{n}$ is independent of $x$ , then the sum of the divisors of $n$ is

Q 29 :

If the coefficient of $x^{8}$ in ${(a x^{2} + \frac{1}{b x})}^{13}$ is equal to the coefficient of $x^{- 8}$ in ${(a x - \frac{1}{b x^{2}})}^{13},$ then $a$ and $b$ will satisfy the relation

Q 30 :

The value of $({}^{21}C_{1} - {}^{10}C_{1}) + ({}^{21}C_{2} - {}^{10}C_{2}) + ({}^{21}C_{3} - {}^{10}C_{3}) + ({}^{21}C_{4} - {}^{10}C_{4}) + \dots + ({}^{21}C_{10} - {}^{10}C_{10})$ is

Q 31 :

Let $X = {({}^{10}C_{1})}^{2} + 2 {({}^{10}C_{2})}^{2} + 3 {({}^{10}C_{3})}^{2} + \dots + 10 {({}^{10}C_{10})}^{2},$ where ${}^{10}C_{r}$ , $r \in {1, 2, \dots, 10}$ , denote binomial coefficients. Then, the value of $\frac{1}{1430} X$ is ______ .

Q 32 :

What is the value of the expression ${(1 + \sqrt{2})}^{4} - {(1 - \sqrt{2})}^{4} ?$

Q 33 :

The number 81 is the coefficient of $x^{k}$ in the binomial expansion of ${(x^{2} + \frac{3}{x})}^{4}, x \neq 0 .$ Then the value of $k$ equals

Q 34 :

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of ${(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6},$ then

Q 35 :

Let $m, n \in N$ and $\gcd (2, n) = 1$ . If $30 (\binom{30}{0}) + 29 (\binom{30}{1}) + \dots + 2 (\binom{30}{28}) + 1 (\binom{30}{29}) = n \cdot 2^{m},$ then $n + m$ is equal to _________. $(Here, (\binom{n}{k}) = {}^{n}C_{k})$

Q 36 :

$\sum_{k = 0}^{20} {({}^{20}C_{k})}^{2}$ is equal to

Q 37 :

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

Q 38 :

The remainder, when $7^{103}$ is divided by 17, is ______.

Q 39 :

The coefficient of $x^{17}$ in ${(1 - x)}^{13} {(1 + x + x^{2})}^{12}$ is

Q 40 :

The remainder when $428^{2024}$ is divided by 21 is _______.

Q 41 :

The middle term of expansion ${(\frac{10}{x} + \frac{x}{10})}^{10}$ is

Q 42 :

The constant term in the expansion of ${(x^{2} - \frac{1}{x^{2}})}^{16}$ is

Q 43 :

If the fourth term in the binomial expansion of ${(\frac{2}{x} + x^{\log_{8} x})}^{6}$ $(x > 0)$ is $20 \times 8^{7}$ , then a value of $x$ is

Q 44 :

The positive value of $λ$ for which the coefficient of $x^{2}$ in the expression $x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}$ is 720, is

Q 45 :

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 ________.

Q 46 :

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 ________.

Q 47 :

Let the coefficients of third, fourth and fifth terms in the expansion of ${(x + \frac{a}{x^{2}})}^{n},$ $x \neq 0$ , be in the ratio 12 : 8 : 3. Then the term independent of $x$ in the expansion, is equal to _______.

Q 48 :

If the constant term, in binomial expansion of ${(2 x^{r} + \frac{1}{x^{2}})}^{10}$ is 180, then $r$ is equal to ________.

Q 50 :

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of ${(a x^{2} + \frac{70}{27 b x})}^{4}$ is equal to the coefficient of $x^{- 5}$ in the expansion of ${(a x - \frac{1}{b x^{2}})}^{7}$ , then the value of $2 b$ is __________.

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Topic Question Set

Q 21 : The term independent of x in the expansion of (2x5-12xx)11 is

Q 22 : The ninth term of the expansion (3x-12x)8 is

Q 23 : The term independent of x in the expansion of (32x2-13x)6 is

Q 24 : The term independent of x in the expansion of [x3+32x2]10 is:

Q 25 : If xr occurs in the expansion of (x+1x)n, then its coefficient is

Q 26 : If A and B are coefficients of xn in the expansions of (1+x)2n and (1+x)2n-1 respectively, then BA is equal to

Q 27 : The coefficient of the term independent of x in the expansion of (x+1x)10 is equal to

Q 28 : The 13th term in the expansion of (x2+2x)n is independent of x, then the sum of the divisors of n is

Q 29 : If the coefficient of x8 in (ax2+1bx)13 is equal to the coefficient of x-8 in (ax-1bx2)13, then a and b will satisfy the relation

Q 30 : The value of (C121-C110)+(C221-C210)+(C321-C310)+(C421-C410)+⋯+(C1021-C1010) is

Q 31 : Let X=(C110)2+2(C210)2+3(C310)2+⋯+10(C1010)2, where Cr10, r∈{1,2,…,10}, denote binomial coefficients. Then, the value of 11430X is ______ .

Q 32 : What is the value of the expression (1+2)4-(1-2)4 ?

Q 33 : The number 81 is the coefficient of xk in the binomial expansion of (x2+3x)4, x≠0. Then the value of k equals

Q 34 : If α and β be the coefficients of x4 and x2 respectively in the expansion of (x+x2-1)6+(x-x2-1)6, then

Q 35 : Let m,n∈N and gcd(2,n)=1. If 30(300)+29(301)+⋯+2(3028)+1(3029)=n·2m, then n+m is equal to _________. (Here, (nk)=Ckn)

Q 36 : ∑k=020(Ck20)2 is equal to

Q 37 : Fractional part of the number 4202215 is equal to

Q 38 : The remainder, when 7103 is divided by 17, is ______.

Q 39 : The coefficient of x17 in (1-x)13(1+x+x2)12 is

Q 40 : The remainder when 4282024 is divided by 21 is _______.

Q 41 : The middle term of expansion (10x+x10)10 is

Q 42 : The constant term in the expansion of (x2-1x2)16 is

Q 43 : If the fourth term in the binomial expansion of (2x+xlog8x)6 (x>0) is 20×87, then a value of x is

Q 44 : The positive value of λ for which the coefficient of x2 in the expression x2(x+λx2)10 is 720, is

Q 45 : If (3644)k is the term, independent of x, in the binomial expansion of (x4-12x2)12, then k is equal to ________.

Q 46 : The term independent of 'x' in the expansion of (x+1x2/3-x1/3+1-x-1x-x1/2)10, where x≠0,1 is equal to ________.

Q 47 : Let the coefficients of third, fourth and fifth terms in the expansion of (x+ax2)n, x≠0, be in the ratio 12 : 8 : 3. Then the term independent of x in the expansion, is equal to _______.

Q 48 : If the constant term, in binomial expansion of (2xr+1x2)10 is 180, then r is equal to ________.

Q 49 : Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of (24+134)n, in the increasing powers of 134 be 64:1. If the sixth term from the beginning is α34, then α is equal to _______.

Q 50 : Let a and b be two nonzero real numbers. If the coefficient of x5 in the expansion of (ax2+7027bx)4 is equal to the coefficient of x-5 in the expansion of (ax-1bx2)7, then the value of 2b is __________.

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Q 21 :

The term independent of $x$ in the expansion of ${(\frac{2 \sqrt{x}}{5} - \frac{1}{2 x \sqrt{x}})}^{11}$ is

Q 22 :

The ninth term of the expansion ${(3 x - \frac{1}{2 x})}^{8}$ is

Q 23 :

The term independent of $x$ in the expansion of ${(\frac{3}{2} x^{2} - \frac{1}{3 x})}^{6}$ is

Q 24 :

The term independent of $x$ in the expansion of ${[\sqrt{\frac{x}{3}} + \frac{3}{2 x^{2}}]}^{10}$ is:

Q 25 :

If $x^{r}$ occurs in the expansion of ${(x + \frac{1}{x})}^{n},$ then its coefficient is

Q 26 :

If $A$ and $B$ are coefficients of $x^{n}$ in the expansions of ${(1 + x)}^{2 n}$ and ${(1 + x)}^{2 n - 1}$ respectively, then $\frac{B}{A}$ is equal to

Q 27 :

The coefficient of the term independent of $x$ in the expansion of ${(\sqrt{x} + \frac{1}{\sqrt{x}})}^{10}$ is equal to

Q 28 :

The $13^{th}$ term in the expansion of ${(x^{2} + \frac{2}{x})}^{n}$ is independent of $x$ , then the sum of the divisors of $n$ is

Q 29 :

If the coefficient of $x^{8}$ in ${(a x^{2} + \frac{1}{b x})}^{13}$ is equal to the coefficient of $x^{- 8}$ in ${(a x - \frac{1}{b x^{2}})}^{13},$ then $a$ and $b$ will satisfy the relation

Q 30 :

The value of $({}^{21}C_{1} - {}^{10}C_{1}) + ({}^{21}C_{2} - {}^{10}C_{2}) + ({}^{21}C_{3} - {}^{10}C_{3}) + ({}^{21}C_{4} - {}^{10}C_{4}) + \dots + ({}^{21}C_{10} - {}^{10}C_{10})$ is

Q 31 :

Let $X = {({}^{10}C_{1})}^{2} + 2 {({}^{10}C_{2})}^{2} + 3 {({}^{10}C_{3})}^{2} + \dots + 10 {({}^{10}C_{10})}^{2},$ where ${}^{10}C_{r}$ , $r \in {1, 2, \dots, 10}$ , denote binomial coefficients. Then, the value of $\frac{1}{1430} X$ is ______ .

Q 32 :

What is the value of the expression ${(1 + \sqrt{2})}^{4} - {(1 - \sqrt{2})}^{4} ?$

Q 33 :

The number 81 is the coefficient of $x^{k}$ in the binomial expansion of ${(x^{2} + \frac{3}{x})}^{4}, x \neq 0 .$ Then the value of $k$ equals

Q 34 :

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of ${(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6},$ then

Q 35 :

Let $m, n \in N$ and $\gcd (2, n) = 1$ . If $30 (\binom{30}{0}) + 29 (\binom{30}{1}) + \dots + 2 (\binom{30}{28}) + 1 (\binom{30}{29}) = n \cdot 2^{m},$ then $n + m$ is equal to _________. $(Here, (\binom{n}{k}) = {}^{n}C_{k})$

Q 36 :

$\sum_{k = 0}^{20} {({}^{20}C_{k})}^{2}$ is equal to

Q 37 :

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

Q 38 :

The remainder, when $7^{103}$ is divided by 17, is ______.

Q 39 :

The coefficient of $x^{17}$ in ${(1 - x)}^{13} {(1 + x + x^{2})}^{12}$ is

Q 40 :

The remainder when $428^{2024}$ is divided by 21 is _______.

Q 41 :

The middle term of expansion ${(\frac{10}{x} + \frac{x}{10})}^{10}$ is

Q 42 :

The constant term in the expansion of ${(x^{2} - \frac{1}{x^{2}})}^{16}$ is

Q 43 :

If the fourth term in the binomial expansion of ${(\frac{2}{x} + x^{\log_{8} x})}^{6}$ $(x > 0)$ is $20 \times 8^{7}$ , then a value of $x$ is

Q 44 :

The positive value of $λ$ for which the coefficient of $x^{2}$ in the expression $x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}$ is 720, is

Q 45 :

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 ________.

Q 46 :

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 ________.

Q 47 :

Let the coefficients of third, fourth and fifth terms in the expansion of ${(x + \frac{a}{x^{2}})}^{n},$ $x \neq 0$ , be in the ratio 12 : 8 : 3. Then the term independent of $x$ in the expansion, is equal to _______.

Q 48 :

If the constant term, in binomial expansion of ${(2 x^{r} + \frac{1}{x^{2}})}^{10}$ is 180, then $r$ is equal to ________.

Q 50 :

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^{5}$ in the expansion of ${(a x^{2} + \frac{70}{27 b x})}^{4}$ is equal to the coefficient of $x^{- 5}$ in the expansion of ${(a x - \frac{1}{b x^{2}})}^{7}$ , then the value of $2 b$ is __________.