JEE Advanced Maths – Binomial Theorem PYQ with Solutions PDF

Q 1 :

Coefficient of $x^{11}$ in the expansion of ${(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$ is [2014]

1051
1106
1113
1120

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2

3

4

(3)

$Coeff. of x^{11} in expansion of {(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$

$= [Coeff. of x^{a} in {(1 + x^{2})}^{4}] \times [Coeff. of x^{b} in {(1 + x^{3})}^{7}] \times [Coeff. of x^{c} in {(1 + x^{4})}^{12}]$

$Such that a + b + c = 11$

$Here a = 2 m, b = 3 n, c = 4 p$

$∴$ $2 m + 3 n + 4 p = 11$

$Case I: m = 0, n = 1, p = 2$

$Case II: m = 1, n = 3, p = 0$

$Case III: m = 2, n = 1, p = 1$

$Case IV: m = 4, n = 1, p = 0$

$∴$ $Required coefficient$

$= {}^{4}C_{0} \times {}^{7}C_{1} \times {}^{12}C_{2} + {}^{4}C_{1} \times {}^{7}C_{3} \times {}^{12}C_{0} + {}^{4}C_{2} \times {}^{7}C_{1} \times {}^{12}C_{1} + {}^{4}C_{4} \times {}^{7}C_{1} \times {}^{12}C_{0}$

$= 462 + 140 + 504 + 7 = 1113$

Q 2 :

The value of $(\binom{30}{0}) (\binom{30}{10}) - (\binom{30}{1}) (\binom{30}{11}) + (\binom{30}{2}) (\binom{30}{12}) - \dots + (\binom{30}{20}) (\binom{30}{30})$ is

where $(\binom{n}{r}) = {}^{n}C_{r}$ [2005]

$(\binom{30}{10})$
$(\binom{30}{15})$
$(\binom{60}{30})$
$(\binom{31}{10})$

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4

(1)

$To find {}^{30}C_{0} {}^{30}C_{10} - {}^{30}C_{1} {}^{30}C_{11} + {}^{30}C_{2} {}^{30}C_{12} - \dots + {}^{30}C_{20} {}^{30}C_{30}$

$∵$ ${(1 + x)}^{30} = {}^{30}C_{0} + {}^{30}C_{1} x + {}^{30}C_{2} x^{2} + \dots + {}^{30}C_{20} x^{20} + . . . + {}^{30}C_{30} x^{30} . . . (i)$

$and {(x - 1)}^{30} = {}^{30}C_{0} x^{30} - {}^{30}C_{1} x^{29} + \dots + {}^{30}C_{10} x^{20} - {}^{30}C_{11} x^{19} + {}^{30}C_{12} x^{18} + \dots + {}^{30}C_{30} x^{0} . . . (i i)$

$On multiplying (i) and (ii), we get$

${(x^{2} - 1)}^{30} = {(1 + x)}^{30} {(x - 1)}^{30}$

$Equating the coefficients of x^{20} on both sides, we get$

${}^{30}C_{10} = {}^{30}C_{0} {}^{30}C_{10} - {}^{30}C_{1} {}^{30}C_{11} + {}^{30}C_{2} {}^{30}C_{12} - \dots + {}^{30}C_{20} {}^{30}C_{30}$

$∴$ $Required value = {}^{30}C_{10}$

Q 3 :

The sum $\sum_{i = 0}^{m} (\binom{10}{i}) (\binom{20}{m - i}),$ (where $(\binom{p}{q}) = 0$ if $p > q$ ) is maximum when $m$ is [2002]

5
10
15
20

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(3)

$\sum_{i = 0}^{m} {}^{10}C_{i} {}^{20}C_{m - i} = {}^{10}C_{0} {}^{20}C_{m} + {}^{10}C_{1} {}^{20}C_{m - 1} + {}^{10}C_{2} {}^{20}C_{m - 2} + \dots + {}^{10}C_{m} {}^{20}C_{0}$

$= Coeff. of x^{m} in the expansion of product {(1 + x)}^{10} {(1 + x)}^{20}$

$= Coeff. of x^{m} in the expansion of {(1 + x)}^{30} = {}^{30}C_{m}$

$\sum_{i = 0}^{m} {}^{10}C_{i} {}^{20}C_{m - i} will be maximum, if {}^{30}C_{m} will be maximum.$

$Clearly, {}^{30}C_{m} will be maximum when m = \frac{30}{2} = 15$ $[∵ \max \cdot ({}^{n}C_{r}) = {\begin{matrix} {}^{n}C_{\frac{n}{2}} if n is even \\ {}^{n}C_{\frac{n + 1}{2}} if n is odd \end{matrix}]$

Q 4 :

For $2 \leq r \leq n, (\binom{n}{r}) + 2 (\binom{n}{r - 1}) + (\binom{n}{r - 2}) =$ [2000]

$(\binom{n + 1}{r - 1})$
$2 (\binom{n + 1}{r + 1})$
$2 (\binom{n + 2}{r})$
$(\binom{n + 2}{r})$

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3

4

(4)

$(\binom{n}{r}) + 2 (\binom{n}{r - 1}) + (\binom{n}{r - 2})$

$= [(\binom{n}{r}) + (\binom{n}{r - 1})] + [(\binom{n}{r - 1}) + (\binom{n}{r - 2})]$

$[Here [\begin{matrix} n \\ r \end{matrix}], [\begin{matrix} n \\ r - 1 \end{matrix}] and [\begin{matrix} n \\ r - 2 \end{matrix}] represent {}^{n}C_{r}, {}^{n}C_{r - 1} and {}^{n}C_{r - 2}]$

$= (\binom{n + 1}{r}) + (\binom{n + 1}{r - 1})$ $= (\binom{n + 2}{r})$ $[∵ {}^{n}{C_{r}} + {}^{n}{C_{r - 1}} = {}^{n + 1}{C_{r}}]$

Q 5 :

Let $a_{0}, a_{1}, \dots, a_{23}$ be real numbers such that ${(1 + \frac{2}{5} x)}^{23} = \sum_{i = 0}^{23} a_{i} x^{i}$ for every real number $x$ . Let $a_{r}$ be the largest among the numbers $a_{j}$ for $0 \leq j \leq 23$ . Then the value of $r$ is ________. [2025]

(6)

$Put x = 1$

${(1 + \frac{2}{5})}^{23} = a_{0} + a_{1} + a_{2} + \dots + a_{23}$

$For numerically greatest term$

$\frac{n + 1}{1 + | \frac{a}{b} |} = \frac{23 + 1}{1 + \frac{5}{2}} = \frac{48}{7}$

$\Rightarrow [\frac{48}{7}] = 6$ (where $[.]$ denotes greatest integer function)

So, $a_{7}$ is numerically greatest term

Hence, $r = 6$

Q 6 :

Suppose det $[\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} k^{2} \\ \sum_{k = 0}^{n} {}^{n}{C_{k}} k & \sum_{k = 0}^{n} {}^{n}{C_{k}} 3^{k} \end{matrix}] = 0$ holds for some positive integer $n$ . The $\sum_{k = 0}^{n} \frac{{}^{n}{C_{k}}}{k + 1}$ equals _________ . [2019]

(6.20)

$Here \sum_{k = 0}^{n} k = \frac{n (n + 1)}{2}$

$\sum_{k = 0}^{n} {}^{n}C_{k} k^{2} = \sum_{k = 1}^{n} \frac{n}{k} . {}^{n - 1}C_{k - 1} . k^{2} = \sum_{k = 1}^{n} n . {}^{n - 1}C_{k - 1} . k$

$= n \sum_{k = 1}^{n} {}^{n - 1}C_{k - 1} (k - 1 + 1)$ $= n [\sum_{k = 2}^{n} \frac{n - 1}{k - 1} {}^{n - 2}C_{k - 2} (k - 1) + \sum_{k = 1}^{n} {}^{n - 1}C_{k - 1}]$

$= n (n - 1) 2^{n - 2} + n \times 2^{n - 1}$

$\sum_{k = 0}^{n} {}^{n}C_{k} \times k = \sum_{k = 1}^{n} \frac{n}{k} {}^{n - 1}C_{k - 1} \times k = n \times 2^{n - 1}$

$and \sum_{k = 0}^{n} {}^{n}C_{k} 3^{k} = 4^{n}$

$∴$ det $[\begin{matrix} \sum_{k = 0}^{n} k & \sum_{k = 0}^{n} {}^{n}C_{k} k^{2} \\ \sum_{k = 0}^{n} {}^{n}C_{k} k & \sum_{k = 0}^{n} {}^{n}C_{k} 3^{k} \end{matrix}] = 0$

$\Rightarrow$ $| \begin{matrix} \frac{n (n + 1)}{2} & n (n - 1) 2^{n - 2} + n \times 2^{n - 1} \\ n \times 2^{n - 1} & 4^{n} \end{matrix} |$ $= 0$

$\Rightarrow n (n + 1) \times 2^{2 n - 1} - n^{2} [(n - 1) 2^{2 n - 3} + 2^{2 n - 2}] = 0$

$\Rightarrow 2^{2 n - 3} \times n [4 (n + 1) - n (n - 1 + 2)] = 0$

$\Rightarrow 2^{2 n - 3} \times n [4 n + 4 - n^{2} - n] = 0$

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

$∴$ $\sum_{k = 0}^{n} \frac{{}^{n}C_{k}}{k + 1} = \sum_{k = 0}^{4} \frac{{}^{4}C_{k}}{k + 1} = \frac{{}^{4}C_{0}}{1} + \frac{{}^{4}C_{1}}{2} + \frac{{}^{4}C_{2}}{3} + \frac{{}^{4}C_{3}}{4} + \frac{{}^{4}C_{4}}{5}$

$= 1 + 2 + 2 + 1 + \frac{1}{5} = 6.20$

Topic Question Set

Q 1 :

Coefficient of $x^{11}$ in the expansion of ${(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$ is [2014]

Q 2 :

The value of $(\binom{30}{0}) (\binom{30}{10}) - (\binom{30}{1}) (\binom{30}{11}) + (\binom{30}{2}) (\binom{30}{12}) - \dots + (\binom{30}{20}) (\binom{30}{30})$ is

where $(\binom{n}{r}) = {}^{n}C_{r}$ [2005]

Q 3 :

The sum $\sum_{i = 0}^{m} (\binom{10}{i}) (\binom{20}{m - i}),$ (where $(\binom{p}{q}) = 0$ if $p > q$ ) is maximum when $m$ is [2002]

Q 4 :

For $2 \leq r \leq n, (\binom{n}{r}) + 2 (\binom{n}{r - 1}) + (\binom{n}{r - 2}) =$ [2000]

Q 5 :

Let $a_{0}, a_{1}, \dots, a_{23}$ be real numbers such that ${(1 + \frac{2}{5} x)}^{23} = \sum_{i = 0}^{23} a_{i} x^{i}$ for every real number $x$ . Let $a_{r}$ be the largest among the numbers $a_{j}$ for $0 \leq j \leq 23$ . Then the value of $r$ is ________. [2025]

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Topic Question Set

Q 1 : Coefficient of x11 in the expansion of (1+x2)4(1+x3)7(1+x4)12 is [2014]

Q 2 : The value of (300)(3010)-(301)(3011)+(302)(3012)-⋯+(3020)(3030) is where (nr)=Crn [2005]

Q 3 : The sum ∑i=0m(10i)(20m-i), (where (pq)=0 if p>q) is maximum when m is [2002]

Q 4 : For 2≤r≤n,(nr)+2(nr-1)+(nr-2)= [2000]

Q 5 : Let a0,a1,…,a23 be real numbers such that (1+25x)23=∑i=023aixi for every real number x. Let ar be the largest among the numbers aj for 0≤j≤23. Then the value of r is ________. [2025]

Q 6 : Suppose det [∑k=0nk∑k=0nCknk2∑k=0nCknk∑k=0nCkn3k]=0 holds for some positive integer n. The ∑k=0nCknk+1 equals _________ . [2019]

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Q 1 :

Coefficient of $x^{11}$ in the expansion of ${(1 + x^{2})}^{4} {(1 + x^{3})}^{7} {(1 + x^{4})}^{12}$ is [2014]

Q 2 :

The value of $(\binom{30}{0}) (\binom{30}{10}) - (\binom{30}{1}) (\binom{30}{11}) + (\binom{30}{2}) (\binom{30}{12}) - \dots + (\binom{30}{20}) (\binom{30}{30})$ is

where $(\binom{n}{r}) = {}^{n}C_{r}$ [2005]

Q 3 :

The sum $\sum_{i = 0}^{m} (\binom{10}{i}) (\binom{20}{m - i}),$ (where $(\binom{p}{q}) = 0$ if $p > q$ ) is maximum when $m$ is [2002]

Q 4 :

For $2 \leq r \leq n, (\binom{n}{r}) + 2 (\binom{n}{r - 1}) + (\binom{n}{r - 2}) =$ [2000]

Q 5 :

Let $a_{0}, a_{1}, \dots, a_{23}$ be real numbers such that ${(1 + \frac{2}{5} x)}^{23} = \sum_{i = 0}^{23} a_{i} x^{i}$ for every real number $x$ . Let $a_{r}$ be the largest among the numbers $a_{j}$ for $0 \leq j \leq 23$ . Then the value of $r$ is ________. [2025]