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

Q 1 :
Let ${(a + b x + c x^{2})}^{10} = \sum_{i = 0}^{20} p_{i} x^{i}, a, b, c \in ℕ$ . If $p_{1} = 20$ and $p_{2} = 210,$ then $2 (a + b + c)$ is equal to [2023]

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(3)

We have, ${(a + b x + c x^{2})}^{10} = \sum_{i = 0}^{20} p_{i} x_{i}, a, b, c \in ℕ$ , $p_{1} = 20$ and $p_{2} = 210$

Now ${(a + b x + c x^{2})}^{10} = p_{0} + p_{1} x + p_{2} x^{2} + \dots$

${}^{10}C_{1} a^{9} \cdot b = p_{1} and {}^{10}C_{2} a^{8} \cdot b^{2} + {}^{10}C_{1} \cdot a^{9} \cdot c = p_{2}$

$\Rightarrow 10 b a^{9} = 20 and 45 b^{2} a^{8} + 10 c \cdot a^{9} = 210 \Rightarrow b a^{9} = 2$

$\Rightarrow$ $b = 2, a = 1$ $[∵ a, b \in ℕ]$

Now, $45 \times {(2)}^{2} + 10 \cdot c = 210$

$\Rightarrow 10 c = 210 - 180 \Rightarrow c = 3$

$∴$ $2 (a + b + c) = 2 (1 + 2 + 3) = 2 \times 6 = 12$

Q 2 :
The coefficient of $x^{7}$ in ${(1 - x + 2 x^{3})}^{10}$ is _________ . [2023]

0%

(960)

General term is given by

$\frac{10!}{r_{1}! r_{2}! r_{3}!} {(1)}^{r_{1}} {(- x)}^{r_{2}} {(2 x^{3})}^{r_{3}}$ i.e., $\frac{10!}{r_{1}! \cdot r_{2}! \cdot r_{3}!} {(- 1)}^{r_{2}} {\cdot (2)}^{r_{3}} x^{r_{2} + 3 r_{3}}$

where $r_{1} + r_{2} + r_{3} = 10$ and $r_{2} + 3 r_{3} = 7$

Possible values of $(r_{1}, r_{2}, r_{3})$ are $(3, 7, 0), (5, 4, 1)$ and $(7, 1, 2)$

Required coefficient

$= \frac{10!}{3! 7!} {(- 1)}^{7} + \frac{10!}{5! 4!} {(- 1)}^{4} (2) + \frac{10!}{7! 2!} {(- 1)}^{1} {(2)}^{2}$

$= - 120 + 2520 - 1440 = 960$

Q 3 :
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]

$B = A^{3}$
$3 A = B$
$A = 3 B$
$A = B^{3}$

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4

(4)

Putting $x = 1$ in ${(1 - 3 x + 10 x^{2})}^{n},$ we get

$A = {(1 - 3 + 10)}^{n} = 8^{n}$ ...(i)

Putting $x = 1$ in ${(1 + x^{2})}^{n},$ we get

$B = {(1 + 1)}^{n} = 2^{n}$ ...(ii)

From (i) and (ii), we get $A = B^{3}$

Q 4 :
The sum of all rational terms in the expansion of ${(1 + 2^{1 / 3} + 3^{1 / 2})}^{6}$ is equal to __________. [2025]

0%

(612)

Given, ${(1 + 2^{1 / 3} + 3^{1 / 2})}^{6}$

$∴$ The general term of multinomial expansion is $\frac{6!}{x! y! z!} {{(1)}^{x} {(2^{1 / 3})}^{y} (3^{1 / 2})}^{z}$

For terms to be rational 3 divides y and 2 divides z.

$∴$

x	y	z
6	0	0
4	0	2
2	0	4
0	0	6
3	3	0
1	3	2
0	6	0

$∴$ Required sum = $\frac{6!}{6! 0! 0!} + \frac{6!}{4! 0! 2!} \times 3 + \frac{6!}{2! 0! 4!} \times {(3)}^{2} + \frac{6!}{0! 0! 6!} {(3)}^{3} + \frac{6!}{3! 3! 0!} (2) + \frac{6!}{1! 3! 2!} (2) \times (3) + \frac{6!}{0! 6! 0!} {(2)}^{2}$

= 1 + 45 + 135 + 27 + 40 + 360 + 4 = 612

Q 5 :
Let ${(1 + x + x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20}$ . If $(a_{1} + a_{3} + a_{5} + . . . + a_{19}) - 11 a_{2} = 121 k$ , then k is equal to __________. [2025]

0%

(239)

${(1 + x + x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20}$

Put x = 1, we have

$3^{10} = a_{0} + a_{1} + a_{2} + . . . . . + a_{20}$ ... (i)

Put x = –1, we have

$1 = a_{0} - a_{1} + a_{2} + . . . . . + a_{20}$ ... (ii)

Subtracting equation (ii) from (i),

$a_{1} + a_{3} + . . . + a_{19} = \frac{3^{10} - 1}{2} = 29524$

Also, ${1 + x (1 + x)}^{10} = 1 + {}^{10}C_{1} x (1 + x) + {}^{10}C_{2} x^{2} {(1 + x)}^{2} + . . .$

$∴ a_{2} = {}^{10}C_{1} + {}^{10}C_{2} = 55$

$∴ k = \frac{(a_{1} + a_{3} + . . . + a_{19}) - 11 a_{2}}{121} = \frac{29524 - 605}{121} = 239$ .

Topic Question Set

Q 1 :
Let ${(a + b x + c x^{2})}^{10} = \sum_{i = 0}^{20} p_{i} x^{i}, a, b, c \in ℕ$ . If $p_{1} = 20$ and $p_{2} = 210,$ then $2 (a + b + c)$ is equal to [2023]

Q 2 :
The coefficient of $x^{7}$ in ${(1 - x + 2 x^{3})}^{10}$ is _________ . [2023]

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Q 3 :
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]

Q 4 :
The sum of all rational terms in the expansion of ${(1 + 2^{1 / 3} + 3^{1 / 2})}^{6}$ is equal to __________. [2025]

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Q 5 :
Let ${(1 + x + x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20}$ . If $(a_{1} + a_{3} + a_{5} + . . . + a_{19}) - 11 a_{2} = 121 k$ , then k is equal to __________. [2025]

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Topic Question Set

Q 1 : Let (a+bx+cx2)10=∑i=020pixi,a,b,c∈ℕ. If p1=20 and p2=210, then 2(a+b+c) is equal to [2023]

Q 2 : The coefficient of x7 in (1-x+2x3)10 is _________ . [2023]

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Q 3 : If A denotes the sum of all the coefficients in the expansion of (1-3x+10x2)n and B denotes the sum of all the coefficients in the expansion of (1+x2)n, then [2024]

Q 4 : The sum of all rational terms in the expansion of (1+21/3+31/2)6 is equal to __________. [2025]

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Q 5 : Let (1+x+x2)10=a0+a1x+a2x2+...+a20x20. If (a1+a3+a5+...+a19)–11a2=121k, then k is equal to __________. [2025]

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Q 1 :
Let ${(a + b x + c x^{2})}^{10} = \sum_{i = 0}^{20} p_{i} x^{i}, a, b, c \in ℕ$ . If $p_{1} = 20$ and $p_{2} = 210,$ then $2 (a + b + c)$ is equal to [2023]

Q 2 :
The coefficient of $x^{7}$ in ${(1 - x + 2 x^{3})}^{10}$ is _________ . [2023]

Q 3 :
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]

Q 4 :
The sum of all rational terms in the expansion of ${(1 + 2^{1 / 3} + 3^{1 / 2})}^{6}$ is equal to __________. [2025]

Q 5 :
Let ${(1 + x + x^{2})}^{10} = a_{0} + a_{1} x + a_{2} x^{2} + . . . + a_{20} x^{20}$ . If $(a_{1} + a_{3} + a_{5} + . . . + a_{19}) - 11 a_{2} = 121 k$ , then k is equal to __________. [2025]