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

Q 31 :

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}})}^{n}$ is $\sqrt{6} : 1$ , then the third term from the beginning is

$30 \sqrt{2}$
$60 \sqrt{3}$
$60 \sqrt{2}$
$30 \sqrt{3}$

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4

(2)

Q 32 :

If the coefficients of three consecutive terms in the expansion of ${(1 + x)}^{n}$ are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

1827
5481
2436
3654

1

2

3

4

(4)

$Given coefficients of three consecutive terms are in the ratio 1 : 5 : 20$

$i.e.,$ ${}^{n}C_{r - 1} : {}^{n}C_{r} : {}^{n}C_{r + 1} = 1 : 5 : 20$

$Now,$ $\frac{{}^{n}C_{r - 1}}{{}^{n}C_{r}} = \frac{1}{5} \Rightarrow n = 6 r - 1$ ...(i)

$Also,$ $\frac{{}^{n}C_{r}}{{}^{n}C_{r + 1}} = \frac{5}{20} \Rightarrow n = 5 r + 4$ ...(ii)

$From (i) and (ii), we get$

$6 r - 1 = 5 r + 4 \Rightarrow r = 5 \Rightarrow n = 25 + 4 = 29$

$∴$ $The coefficient of 4^{th} term = {}^{29}C_{3} = 3654$

Q 33 :

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 ________.

(1)

$T_{r + 1} = {}^{22}C_{r} {(x^{\frac{2}{3}})}^{22 - r} \cdot {(\frac{α}{x^{3}})}^{r}$

$= {}^{22}C_{r} \cdot (x^{\frac{44}{3} - \frac{2 r}{3} - 3 r}) \cdot (α^{r})$

$For the term without' x',$ $\frac{44}{3} - \frac{2 r}{3} - 3 r = 0 \Rightarrow \frac{44}{3} = \frac{11 r}{3}$

$∴$ $r = 4$

$So, the term without x is {}^{22}C_{4} α^{4} = 7315$

$\Rightarrow$ $\frac{22 \times 21 \times 20 \times 19}{24} \cdot α^{4} = 7315$

$∴$ $| α |$ $= 1$

Q 34 :

The coefficient of $x^{18}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is _________.

(5005)

$Given,$ ${(x^{4} - \frac{1}{x^{3}})}^{15}$

$T_{r + 1} = {}^{15}C_{r} {(x^{4})}^{15 - r} {(- \frac{1}{x^{3}})}^{r}$

$= {}^{15}C_{r} {(x)}^{60 - 4 r} {\cdot (- 1)}^{r} x^{- 3 r} = {}^{15}C_{r} {(x)}^{60 - 7 r} \cdot {(- 1)}^{r}$

$For coefficient of x^{18}, put 60 - 7 r = 18 \Rightarrow r = \frac{42}{7} = 6$

$Hence, coefficient of x^{18} is {}^{15}C_{6} {(- 1)}^{6} = 5005$

Q 35 :

If the coefficient of $x^{9}$ in ${(α x^{3} + \frac{1}{β x})}^{11}$ and the coefficient of $x^{- 9}$ in ${(α x - \frac{1}{β x^{3}})}^{11}$ are equal, then ${(α β)}^{2}$ is equal to _______.

(1)

${(r + 1)}^{th} term in {(α x^{3} + \frac{1}{β x})}^{11} = {}^{11}C_{r} α^{11 - r} \cdot β^{- r} \cdot x^{33 - 4 r}$

$Here,$ $33 - 4 r = 9 \Rightarrow r = 6$

$∴$ $Coefficient of x^{9} = {}^{11}C_{6} α^{5} β^{- 6}$

${(r + 1)}^{th} term in {(α x - \frac{1}{β x^{3}})}^{11} = {}^{11}C_{r} α^{11 - r} {(- \frac{1}{β})}^{r} \cdot x^{11 - 4 r}$

$Here,$ $11 - 4 r = - 9 \Rightarrow r = 5$

$∴$ $Coefficient of x^{- 9} = - {}^{11}C_{5} α^{6} β^{- 5}$

$Now,$ ${}^{11}C_{6} \frac{α^{5}}{β^{6}} = - {}^{11}C_{5} \frac{α^{6}}{β^{5}}$ $[∵ Given]$

$\Rightarrow$ $\frac{1}{β} = - α \Rightarrow {(α β)}^{2} = 1$

Q 36 :

The coefficient of $x^{- 6}$ in the expansion of ${(\frac{4 x}{5} + \frac{5}{2 x^{2}})}^{9}$ is _________.

(5040)

$Here,$ $T_{r + 1} = {}^{9}C_{r} {(\frac{4 x}{5})}^{9 - r} {(\frac{5}{2 x^{2}})}^{r}$

$= {}^{9}C_{r} {(\frac{4}{5})}^{9 - r} {(\frac{5}{2})}^{r} x^{9 - r - 2 r}$

$For coefficient of x^{- 6}, we have 9 - r - 2 r = - 6 \Rightarrow 3 r = 15$

$\Rightarrow$ $r = 5$

$So, coefficient of x^{- 6} = {}^{9}C_{5} {(\frac{4}{5})}^{4} {(\frac{5}{2})}^{5}$

$= \frac{9!}{5! 4!} (\frac{4 \times 4 \times 4 \times 4}{5 \times 5 \times 5 \times 5}) (\frac{5 \times 5 \times 5 \times 5 \times 5}{2 \times 2 \times 2 \times 2 \times 2})$

$= \frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2} (8) (5) = 126 \times 40$

$∴$ $126 \times 40 \cdot x^{- 6} = 5040$

Q 37 :

If $k$ is the coefficient of $x^{4}$ in the expansion of ${(1 + 2 x + \frac{3}{x^{2}})}^{6}$ , then the sum of the digits of $k$ is _______.

(6)

$The general term is \frac{6!}{α! β! γ!} 1^{α} {(2 x)}^{β} \cdot {(\frac{3}{x^{2}})}^{γ}$

$β - 2 γ = 4,$ $α + β + γ = 6 \Rightarrow α = 2 - 3 γ$

$β = 2 γ + 4$ $∴$ $(α, β, γ) = (2, 4, 0)$

$The desired coefficient is$

$\frac{6!}{2! 4!} \cdot 2^{4} = 15 \times 16 = 240$

Q 38 :

If the term independent of $x$ in the expansion of ${(\sqrt{x} - \frac{λ}{x^{2}})}^{10}$ is 405, then the value of $| 4 λ |$ is _________.

(12)

$Let {(r + 1)}^{th} term is independent of x .$

$We have,$ $T_{r + 1} = {}^{10}C_{r} {(\sqrt{x})}^{10 - r} \cdot {(- \frac{λ}{x^{2}})}^{r}$

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

$The term will be independent of x if 5 - \frac{r}{2} - 2 r = 0$

$\Rightarrow$ $r = 2$

$Then,$ $T_{2 + 1} = {}^{10}C_{2} \cdot {(- λ)}^{2} = 45 λ^{2} = 405$ $(Given)$

$∴$ $λ^{2} = 9 \Rightarrow λ = \pm 3 \Rightarrow$ $| 4 λ |$ $= 12$

Q 39 :

If the magnitude of the coefficient of $x^{7}$ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{8}$ , where $a, b$ are positive numbers, is equal to the magnitude of the coefficient of $x^{- 7}$ in the expansion of ${(a x + \frac{1}{b x^{2}})}^{8}$ , then $| a b |$ is equal to _________.

(1)

$In the expansion {(a x^{2} + \frac{1}{b x})}^{8},$ ${(r + 1)}^{th} term is$

$T_{r + 1} = {}^{8}C_{r} {(a x^{2})}^{8 - r} {(\frac{1}{b x})}^{r}$

$Let the {(r + 1)}^{th} term contain x^{7} .$

$∴$ $x^{16 - 3 r} = x^{7} \Rightarrow r = 3$

$∴$ $Coefficient of x^{7} = {}^{8}C_{3} a^{5} b^{- 3}$ ...(i)

$In the expansion {(a x + \frac{1}{b x^{2}})}^{8},$ ${(r + 1)}^{th} term is$

$T_{r + 1} = {}^{8}C_{r} {(a x)}^{8 - r} {(\frac{1}{b x^{2}})}^{r} = {}^{8}C_{r} a^{8 - r} {(b)}^{- r} x^{8 - 3 r}$

$Let the {(r + 1)}^{th} term contain x^{- 7} .$

$∴$ $x^{8 - 3 r} = x^{- 7} \Rightarrow r = 5$

$∴$ $Coefficient of x^{- 7} = {}^{8}C_{5} a^{3} b^{- 5}$ ...(ii)

$From (i) and (ii), we have$ $| {}^{8}C_{3} a^{5} b^{- 3} |$ $=$ $| {}^{8}C_{5} a^{3} b^{- 5} |$

$\Rightarrow$ $a^{2} b^{2} = 1 \Rightarrow a b = 1$

Q 40 :

$Coefficient of x^{99} in x^{100} + 2 x^{99} (1 + x) + 3 {(1 + x)}^{2} x^{98} + \dots + 101 {(1 + x)}^{100} is____$

${}^{101}C_{99} - {}^{101}C_{98}$
$100 ({}^{101}C_{99}) - {}^{101}C_{98}$
${}^{- 101}C_{98} + 101 ({}^{101}C_{99})$
$101 ({}^{101}C_{98}) - {}^{101}C_{99}$

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(3)

$Required = coefficient of x^{99} in x^{100} (1 + 2 (\frac{1 + x}{x}) + . . . . . 101 {(\frac{1 + x}{x})}^{100})$

Q 41 :

Statement-1: The sum of rational terms in ${(\sqrt{2} + \sqrt[3]{3} + \sqrt[6]{5})}^{10}$ is 1262

Statement-2: The number of irrational terms in the expansion of ${(5^{1 / 6} + 2^{1 / 8})}^{100}$ is 97

The correct statements among the following are

Statement-1 is true, statement-2 is true
Statement-1 is true, statement-2 is false
Statement-1 is false, statement-2 is true
Statement-1 is false, statement-2 is false

1

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3

4

(3)

$Statement-1 General term = \frac{10!}{α! β! γ!} 2^{α / 2} 3^{β / 3} 5^{γ / 6} for rational terms$

$α = 0, 2, 4, 6, 8, 10, β = 0, 3, 6, γ = 0, 6$

$Hence possible sets = (4, 6, 0), (4, 0, 6), (10, 0, 0)$

$Hence, there are 5 rational terms.$

$∴$ $required = \frac{10!}{4! 6!} 2^{24} \frac{10!}{10!} 2^{5} = 12632$

$Statement-3 t_{r + 1}, the (r + 1) th in the expansion of$ ${(5^{1 / 6} + 2^{1 / 8})}^{10} is$ $t_{r + 1} = {}^{10}C_{r} {(5)}^{\frac{100 - r}{6}} {(2^{1 / 8})}^{r}$

$As 5 and 2 are relatively prime, t_{r + 1} will be rational if \frac{100 - r}{6} and \frac{r}{8} are both integers.$

$i.e. If 100 - r is a multiple of 6 and r is a multiple of 8.$

$As 0 \leq r \leq 100, multiple of 8 upto 100 and corresponding value of 100 - r$

$r = 0, 8, 16, 24, \dots, 88, 96$

$100 - r = 100, 92, 84, 76, \dots, 12, 4$

$Out of 100 - r, multiple of 6 are 84, 60, 36, 12 .$

$∴$ $There are just four rational terms$

$\Rightarrow Number of irrational terms = 101 - 4 = 97$

Q 42 :

The remainder when the number $3^{2^{2^{3}}} - {(3^{2^{2}})}^{3}$ is divide by 8, is ______

(0)

$3^{256} - 3^{12} = 3^{12} \times (3^{256 - 12} - 1) = {(1 + 8)}^{6} ({(1 + 8)}^{112} - 1) = (1 + 8 λ) (1 + 8 μ - 1)$

$= 8 μ (1 + 8 λ), which is divisible by 8 . Hence, remainder is zero.$

Q 43 :

$Coefficient of x^{4} in 5 x^{2} {(1 + x^{2})}^{11} is$

22
33
44
55

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4

(4)

$5 x^{2} (1 + {}^{11}C_{1} x^{2} + \dots)$

Q 44 :

If $26 (\frac{2^{3}}{3} ({}^{12}C_{2}) + \frac{2^{5}}{5} ({}^{12}C_{4}) + \frac{2^{7}}{7} ({}^{12}C_{6}) + . . . + \frac{2^{13}}{13} ({}^{12}C_{12})) = 3^{13} - α$ , then $α$ is equal to: [2026]

45
48
51
54

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3

4

(3)

$26 [\frac{2^{3}}{3} ({}^{12}C_{2}) + \frac{2^{5}}{5} ({}^{12}C_{4}) + \frac{2^{7}}{7} ({}^{12}C_{6}) + . . . + \frac{2^{13}}{13} ({}^{12}C_{12})] = 3^{13} - α$

$= 2 [2^{3} \cdot {}^{13}C_{3} + 2^{5} \cdot {}^{13}C_{5} + 2^{7} \cdot {}^{13}C_{7} + . . . + 2^{13} \cdot {}^{13}C_{13}]$

$= 2 [\frac{{(1 + 2)}^{13} - {(1 - 2)}^{13}}{2} - 26]$

$= 3^{13} + 1 - 52 = 3^{13} - 51 ∴ α = 51$

Topic Question Set

Q 31 :

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}})}^{n}$ is $\sqrt{6} : 1$ , then the third term from the beginning is

Q 32 :

If the coefficients of three consecutive terms in the expansion of ${(1 + x)}^{n}$ are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

Q 33 :

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 ________.

Q 34 :

The coefficient of $x^{18}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is _________.

Q 35 :

If the coefficient of $x^{9}$ in ${(α x^{3} + \frac{1}{β x})}^{11}$ and the coefficient of $x^{- 9}$ in ${(α x - \frac{1}{β x^{3}})}^{11}$ are equal, then ${(α β)}^{2}$ is equal to _______.

Q 36 :

The coefficient of $x^{- 6}$ in the expansion of ${(\frac{4 x}{5} + \frac{5}{2 x^{2}})}^{9}$ is _________.

Q 37 :

If $k$ is the coefficient of $x^{4}$ in the expansion of ${(1 + 2 x + \frac{3}{x^{2}})}^{6}$ , then the sum of the digits of $k$ is _______.

Q 38 :

If the term independent of $x$ in the expansion of ${(\sqrt{x} - \frac{λ}{x^{2}})}^{10}$ is 405, then the value of $| 4 λ |$ is _________.

Q 39 :

If the magnitude of the coefficient of $x^{7}$ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{8}$ , where $a, b$ are positive numbers, is equal to the magnitude of the coefficient of $x^{- 7}$ in the expansion of ${(a x + \frac{1}{b x^{2}})}^{8}$ , then $| a b |$ is equal to _________.

Q 40 :

$Coefficient of x^{99} in x^{100} + 2 x^{99} (1 + x) + 3 {(1 + x)}^{2} x^{98} + \dots + 101 {(1 + x)}^{100} is____$

Q 41 :

Statement-1: The sum of rational terms in ${(\sqrt{2} + \sqrt[3]{3} + \sqrt[6]{5})}^{10}$ is 1262

Statement-2: The number of irrational terms in the expansion of ${(5^{1 / 6} + 2^{1 / 8})}^{100}$ is 97

The correct statements among the following are

Q 42 :

The remainder when the number $3^{2^{2^{3}}} - {(3^{2^{2}})}^{3}$ is divide by 8, is ______

Q 43 :

$Coefficient of x^{4} in 5 x^{2} {(1 + x^{2})}^{11} is$

Q 44 :

If $26 (\frac{2^{3}}{3} ({}^{12}C_{2}) + \frac{2^{5}}{5} ({}^{12}C_{4}) + \frac{2^{7}}{7} ({}^{12}C_{6}) + . . . + \frac{2^{13}}{13} ({}^{12}C_{12})) = 3^{13} - α$ , then $α$ is equal to: [2026]

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Topic Question Set

Q 31 : If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of (24+134)n is 6:1, then the third term from the beginning is

Q 32 : If the coefficients of three consecutive terms in the expansion of (1+x)n are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

Q 33 : If the term without x in the expansion of (x23+αx3)22 is 7315, then |α| is equal to ________.

Q 34 : The coefficient of x18 in the expansion of (x4-1x3)15 is _________.

Q 35 : If the coefficient of x9 in (αx3+1βx)11 and the coefficient of x-9 in (αx-1βx3)11 are equal, then (αβ)2 is equal to _______.

Q 36 : The coefficient of x-6 in the expansion of (4x5+52x2)9 is _________.

Q 37 : If k is the coefficient of x4 in the expansion of (1+2x+3x2)6, then the sum of the digits of k is _______.

Q 38 : If the term independent of x in the expansion of (x-λx2)10 is 405, then the value of |4λ| is _________.

Q 39 : If the magnitude of the coefficient of x7 in the expansion of (ax2+1bx)8, where a,b are positive numbers, is equal to the magnitude of the coefficient of x-7 in the expansion of (ax+1bx2)8, then |ab| is equal to _________.

Q 40 : Coefficient of x99 in x100+2x99(1+x)+3(1+x)2x98+⋯+101(1+x)100 is ____

Q 41 : Statement-1: The sum of rational terms in (2+33+56)10 is 1262 Statement-2: The number of irrational terms in the expansion of (51/6+21/8)100 is 97 The correct statements among the following are

Q 42 : The remainder when the number 3223-(322)3 is divide by 8, is ______

Q 43 : Coefficient of x4 in 5x2(1+x2)11 is

Q 44 : If 26(233(C212)+255(C412)+277(C612)+...+21313(C1212))=313–α, then α is equal to: [2026]

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Q 31 :

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of ${(\sqrt[4]{2} + \frac{1}{\sqrt[4]{3}})}^{n}$ is $\sqrt{6} : 1$ , then the third term from the beginning is

Q 32 :

If the coefficients of three consecutive terms in the expansion of ${(1 + x)}^{n}$ are in the ratio 1 : 5 : 20, then the coefficient of the fourth term is

Q 33 :

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 ________.

Q 34 :

The coefficient of $x^{18}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is _________.

Q 35 :

If the coefficient of $x^{9}$ in ${(α x^{3} + \frac{1}{β x})}^{11}$ and the coefficient of $x^{- 9}$ in ${(α x - \frac{1}{β x^{3}})}^{11}$ are equal, then ${(α β)}^{2}$ is equal to _______.

Q 36 :

The coefficient of $x^{- 6}$ in the expansion of ${(\frac{4 x}{5} + \frac{5}{2 x^{2}})}^{9}$ is _________.

Q 37 :

If $k$ is the coefficient of $x^{4}$ in the expansion of ${(1 + 2 x + \frac{3}{x^{2}})}^{6}$ , then the sum of the digits of $k$ is _______.

Q 38 :

If the term independent of $x$ in the expansion of ${(\sqrt{x} - \frac{λ}{x^{2}})}^{10}$ is 405, then the value of $| 4 λ |$ is _________.

Q 39 :

If the magnitude of the coefficient of $x^{7}$ in the expansion of ${(a x^{2} + \frac{1}{b x})}^{8}$ , where $a, b$ are positive numbers, is equal to the magnitude of the coefficient of $x^{- 7}$ in the expansion of ${(a x + \frac{1}{b x^{2}})}^{8}$ , then $| a b |$ is equal to _________.

Q 40 :

$Coefficient of x^{99} in x^{100} + 2 x^{99} (1 + x) + 3 {(1 + x)}^{2} x^{98} + \dots + 101 {(1 + x)}^{100} is____$

Q 41 :

Statement-1: The sum of rational terms in ${(\sqrt{2} + \sqrt[3]{3} + \sqrt[6]{5})}^{10}$ is 1262

Statement-2: The number of irrational terms in the expansion of ${(5^{1 / 6} + 2^{1 / 8})}^{100}$ is 97

The correct statements among the following are

Q 42 :

The remainder when the number $3^{2^{2^{3}}} - {(3^{2^{2}})}^{3}$ is divide by 8, is ______

Q 43 :

$Coefficient of x^{4} in 5 x^{2} {(1 + x^{2})}^{11} is$

Q 44 :

If $26 (\frac{2^{3}}{3} ({}^{12}C_{2}) + \frac{2^{5}}{5} ({}^{12}C_{4}) + \frac{2^{7}}{7} ({}^{12}C_{6}) + . . . + \frac{2^{13}}{13} ({}^{12}C_{12})) = 3^{13} - α$ , then $α$ is equal to: [2026]