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

Q 11 :
The coefficient of $x^{301}$ in ${(1 + x)}^{500} + x {(1 + x)}^{499} + x^{2} {(1 + x)}^{498} + . . . + x^{500}$ is [2023]

${}^{500}{C_{301}}$
${}^{501}{C_{200}}$
${}^{500}{C_{300}}$
${}^{501}{C_{302}}$

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2

3

4

(2)

$The coefficient of x^{301} in$

${(1 + x)}^{500} + x {(1 + x)}^{499} + x^{2} {(1 + x)}^{498} + \dots + x^{500}$

$= {}^{500}C_{301} + {}^{499}C_{300} + {}^{498}C_{299} + \dots + {}^{199}C_{0}$

$= {}^{500}C_{199} + {}^{499}C_{199} + {}^{498}C_{199} + \dots + {}^{199}C_{199} = {}^{501}C_{200}$

Q 12 :
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 ${(a x^{1 / 3} - \frac{1}{b x^{3}})}^{15}$ , where $a$ and $b$ are positive real numbers, then for each such ordered pair (a, b) [2023]

a = 3b
a = b
ab = 1
ab = 3

1

2

3

4

(3)

The coefficient of $x^{15}$ in the expansion of

${(a x^{3} + \frac{1}{b x^{\frac{1}{3}}})}^{15}$ is $T_{r + 1} = {}^{15}{C_{r}} {(a x^{3})}^{15 - r} \times {(b^{- 1} x^{- \frac{1}{3}})}^{r}$

$= {}^{15}{C_{r}} a^{15 - r} \cdot x^{45 - 3 r} \cdot b^{- r} \cdot x^{- \frac{r}{3}} = {}^{15}{C_{r}} a^{15 - r} \cdot b^{- r} x^{45 - 3 r - \frac{r}{3}}$

We have to find the coefficient of $x^{15}$ , so $x^{45 - 3 r - \frac{r}{3}} = x^{15}$

$\Rightarrow 45 - 3 r - \frac{r}{3} = 15 \Rightarrow \frac{10 r}{3} = 30 \Rightarrow r = 9$

So, $T_{9 + 1} = {}^{15}{C_{9}} a^{15 - 9} \cdot b^{- 9} \cdot x^{15};$ $T_{10} = {}^{15}{C_{9}} a^{6} \cdot b^{- 9} \cdot x^{15}$

$∴$ $The coefficient of x^{15} = {}^{15}{C_{9}} a^{6} \cdot b^{- 9}$

Now, the coefficient of $x^{- 15}$ in the expansion of

${({a x}^{\frac{1}{3}} - \frac{1}{b x^{3}})}^{15}$ is, $T_{r + 1} = {{}^{15}C}_{r} {(a x^{1 / 3})}^{15 - r} \cdot {(\frac{- 1}{b x^{3}})}^{r}$

$= {{}^{15}C}_{r} a^{15 - r} x^{\frac{15 - r}{3}} \cdot b^{- r} \cdot x^{- 3 r} {(- 1)}^{r}$

$= {}^{15}{C_{r}} a^{15 - r} \cdot b^{- r} \cdot x^{\frac{15 - r}{3} - 3 r} {(- 1)}^{r}$

We need to find the coefficient of $x^{- 15}$ , so

$x^{\frac{15 - r}{3} - 3 r} = x^{- 15} \Rightarrow \frac{15 - r}{3} - 3 r = - 15 \Rightarrow r = 6$

$∴$ $The coefficient of x^{- 15} = {}^{15}{C_{6}} a^{9} b^{- 6}$

Now, according to the question, ${}^{15}{C_{9}} a^{6} b^{- 9} = {}^{15}{C_{6}} a^{9} b^{- 6}$

$\Rightarrow a^{3} b^{3} = 1 ∴ a b = 1$

Q 13 :
The coefficient of $x^{18}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is __________ . [2023]

0%

(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 the coefficient of $x^{18}$

$60 - 7 r = 18 \Rightarrow r = \frac{42}{7} = 6$

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

Q 14 :
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]

0%

(1275)

Let the $r^{t h}$ term be the constant term of the given expansion.

$∴$ $T_{r + 1} = {}^{7}{C_{r}} {(3 x^{2})}^{7 - r} \cdot {(\frac{- 1}{2 x^{5}})}^{r} = {}^{7}{C_{r}} 3^{7 - r} \cdot {(\frac{- 1}{2})}^{r} \cdot x^{14 - 2 r - 5 r}$

For constant term, put $14 - 2 r - 5 r = 0 \Rightarrow 7 r = 14 \Rightarrow r = 2$

$∴$ $T_{2 + 1} = {}^{7}{C_{2}} 3^{7 - 2} {(\frac{- 1}{2})}^{2} = 21 \times 3^{5} \times \frac{1}{4}$

$\Rightarrow α = 1275.75 \Rightarrow [α] = 1275$

Q 15 :
The number of integral terms in the expansion of ${(3^{1 / 2} + 5^{1 / 4})}^{680}$ is equal to ___________ . [2023]

0%

(171)

The expansion of the general term is

$= {}^{680}{C_{r}} {(3^{1 / 2})}^{680 - r} {(5^{1 / 4})}^{r} = {}^{680}{C_{r}} 3^{\frac{680 - r}{2}} 5^{\frac{r}{4}}$

Possible values of $r$ , where $\frac{r}{4}$ is an integer

$r = 0, 4, 8, 12, \dots, 680$

All these values of $r$ are accepted as well.

$Hence, number of integral terms = 171$

Q 16 :
Let $α$ be the constant term in the binomial expansion of ${(\sqrt{x} - \frac{6}{x^{\frac{3}{2}}})}^{n}, n \leq 15$ . If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{- n}$ is $λ α$ , then $λ$ is equal to _________ . [2023]

0%

(36)

General term in the expansion of ${(\sqrt{x} - \frac{6}{x^{3 / 2}})}^{n}$ is given by
$T_{r + 1} = {}^{n}{C_{r}} {(x)}^{\frac{n - r}{2}} {(- 6)}^{r} {(x)}^{\frac{- 3}{2} r}$

For the constant term, the power of $x$ should be zero.

$\Rightarrow \frac{n - r}{2} - \frac{3}{2} r = 0 \Rightarrow n - 4 r = 0 \Rightarrow r = \frac{n}{4}$

For the constant term, put $x = 1$ in ${(\sqrt{x} - \frac{6}{x^{3 / 2}})}^{n}$

Constant term, ${(- 5)}^{n}$

$∴$ $Sum of coefficients except constant term is {(- 5)}^{n} - {}^{n}{C_{n / 4}} {(- 6)}^{n / 4} = 649$

Put $n = 4,$ $625 + 24 = 649$

Thus, $n = 4$ satisfies the above equation.

Now, $r = \frac{n}{4} = \frac{4}{4} = 1$

Required coefficient of $x^{- 4}$ = ${}^{4}{C_{3}} {(- 6)}^{3} = 4 (- 216)$

and constant term $a = {}^{4}{C_{1}} {(- 6)}^{1} = 4 (- 6)$

$∴$ $4 (- 216) = λ [4 (- 6)] \Rightarrow λ = 36$

Q 17 :
Let the sixth term in the binomial expansion of ${(\sqrt{2^{\log_{2} (10 - 3^{x})}} + \sqrt[5]{2^{(x - 2) \log_{2} 3}})}^{m}$ , in the increasing powers of $2^{(x - 2) \log_{2} 3}$ , be 21. If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P., then the sum of the squares of all possible values of $x$ is _______ . [2023]

0%

(4)

Sixth term $T_{6} = {}^{m}{C_{5}} {(10 - 3^{x})}^{\frac{m - 5}{2}} \cdot (3^{x - 2}) = 21 ...(i)$

${}^{m}{C_{1}}, {}^{m}{C_{2}}, {}^{m}{C_{3}} are in A.P. \Rightarrow 2 {}^{m}{C_{2}} = {}^{m}{C_{1}} + {}^{m}{C_{3}}$

$\Rightarrow \frac{2 \cdot m!}{(m - 2)! 2!} = \frac{m!}{(m - 1)! 1!} + \frac{m!}{(m - 3)! 3!}$

$∴$ $m = 7, 2 (Rejected)$

Put $m = 7$ in equation (i):

${}^{7}{C_{5}} {(10 - 3^{x})}^{\frac{7 - 5}{2}} \cdot (3^{x - 2}) = 21$ $\Rightarrow 21 \cdot (10 - 3^{x}) \cdot \frac{3^{x}}{9} = 21$

$\Rightarrow x = 0, 2$

Sum of the squares of all possible values of $x$ is $0^{2} + 2^{2} = 4$

Q 18 :
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]

0%

(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 19 :
The constant term in the expansion of ${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}$ is _______. [2023]

0%

(1080)

$General term = \frac{5!}{r_{1}! r_{2}! r_{3}!} {(2 x)}^{r_{1}} {(x^{- 7})}^{r_{2}} {(3 x^{2})}^{r_{3}}$

$= \frac{5!}{r_{1}! r_{2}! r_{3}!} 2^{r_{1}} \cdot 3^{r_{3}} \cdot x^{r_{1} - 7 r_{2} + 2 r_{3}}$

For the constant term, $r_{1} - 7 r_{2} + 2 r_{3} = 0$ and $r_{1} + r_{2} + r_{3} = 5$

By hit and trial, $r_{1} = 1$ , $r_{2} = 1$ and $r_{3} = 3$ .

$∴$ $Constant term = \frac{5!}{1! 1! 3!} 2^{1} \cdot 3^{3}$

$= \frac{120}{6} \times 2 \times 27 = 20 \times 2 \times 27 = 1080$

Q 20 :
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 these three terms, is _________ . [2023]

0%

(1120)

$Coefficient of {(r + 1)}^{th} term is,$

$t_{r + 1} = {}^{n}C_{r} {(2 x)}^{r}$

Given consecutive term ratio as,

$\frac{{}^{n}C_{r - 1} {(2)}^{r - 1}}{{}^{n}C_{r} {(2)}^{r}} = \frac{2}{5} \Rightarrow \frac{n! / (r - 1)! (n - r + 1)!}{n! (2) / r! (n - r)!} = \frac{2}{5}$

$\Rightarrow \frac{r}{n - r + 1} = \frac{4}{5} \Rightarrow 5 r = 4 n - 4 r + 4 \Rightarrow 9 r = 4 (n + 1) ...(i)$

Again, consecutive term ratio are,

$\frac{{}^{n}C_{r} {(2)}^{r}}{{}^{n}C_{r + 1} {(2)}^{r + 1}} = \frac{5}{8} \Rightarrow \frac{n! / r! (n - r)!}{n! / (r + 1)! (n - r - 1)!} = \frac{5}{4}$

$\Rightarrow \frac{r + 1}{n - r} = \frac{5}{4} \Rightarrow 5 n - 4 = 9 r ...(ii)$

$From (i) and (ii): n = 8, r = 4$

$So, coefficient of middle term is,$

${}^{8}C_{4} 2^{4} = 16 \times \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 1120$

Topic Question Set

Q 11 :
The coefficient of $x^{301}$ in ${(1 + x)}^{500} + x {(1 + x)}^{499} + x^{2} {(1 + x)}^{498} + . . . + x^{500}$ is [2023]

Q 12 :
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 ${(a x^{1 / 3} - \frac{1}{b x^{3}})}^{15}$ , where $a$ and $b$ are positive real numbers, then for each such ordered pair (a, b) [2023]

Q 13 :
The coefficient of $x^{18}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is __________ . [2023]

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Q 14 :
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]

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Q 15 :
The number of integral terms in the expansion of ${(3^{1 / 2} + 5^{1 / 4})}^{680}$ is equal to ___________ . [2023]

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Q 16 :
Let $α$ be the constant term in the binomial expansion of ${(\sqrt{x} - \frac{6}{x^{\frac{3}{2}}})}^{n}, n \leq 15$ . If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{- n}$ is $λ α$ , then $λ$ is equal to _________ . [2023]

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Q 18 :
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]

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Q 19 :
The constant term in the expansion of ${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}$ is _______. [2023]

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Q 20 :
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 these three terms, is _________ . [2023]

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Topic Question Set

Q 11 : The coefficient of x301 in (1+x)500+x(1+x)499+x2(1+x)498+...+ x500 is [2023]

Q 12 : If the coefficient of x15 in the expansion of (ax3+1bx1/3)15 is equal to the coefficient of x-15 in the expansion of (ax1/3-1bx3)15, where a and b are positive real numbers, then for each such ordered pair (a, b) [2023]

Q 13 : The coefficient of x18 in the expansion of (x4-1x3)15 is __________ . [2023]

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Q 14 : If the constant term in the expansion of (3x2-12x5)7 is α, then [α] is equal to _______ . [2023]

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Q 15 : The number of integral terms in the expansion of (31/2+51/4)680 is equal to ___________ . [2023]

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Q 16 : Let α be the constant term in the binomial expansion of (x-6x32)n,n≤15. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of x-n is λα, then λ is equal to _________ . [2023]

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Q 18 : If the term without x in the expansion of (x23+αx3)22 is 7315, then |α| is equal to ________ . [2023]

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Q 19 : The constant term in the expansion of (2x+1x7+3x2)5 is _______. [2023]

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Q 20 : Let the coefficients of three consecutive terms in the binomial expansion of (1+2x)n be in the ratio 2:5:8.Then the coefficient of the term, which is in the middle of these three terms, is _________ . [2023]

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Q 11 :
The coefficient of $x^{301}$ in ${(1 + x)}^{500} + x {(1 + x)}^{499} + x^{2} {(1 + x)}^{498} + . . . + x^{500}$ is [2023]

Q 12 :
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 ${(a x^{1 / 3} - \frac{1}{b x^{3}})}^{15}$ , where $a$ and $b$ are positive real numbers, then for each such ordered pair (a, b) [2023]

Q 13 :
The coefficient of $x^{18}$ in the expansion of ${(x^{4} - \frac{1}{x^{3}})}^{15}$ is __________ . [2023]

Q 14 :
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]

Q 15 :
The number of integral terms in the expansion of ${(3^{1 / 2} + 5^{1 / 4})}^{680}$ is equal to ___________ . [2023]

Q 16 :
Let $α$ be the constant term in the binomial expansion of ${(\sqrt{x} - \frac{6}{x^{\frac{3}{2}}})}^{n}, n \leq 15$ . If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{- n}$ is $λ α$ , then $λ$ is equal to _________ . [2023]

Q 18 :
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]

Q 19 :
The constant term in the expansion of ${(2 x + \frac{1}{x^{7}} + 3 x^{2})}^{5}$ is _______. [2023]

Q 20 :
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 these three terms, is _________ . [2023]