Binomial Theorem · JEE Main previous year questions

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2022 12 questions

Q232022 · 29 Jul 2022 · Shift 1· Numerical answer
Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $(42 + \frac{1}{4 3})^{n}$ , in the increasing powers of $\frac{1}{4 3}$ be $46 : 1$ . If the sixth term from the beginning is $\frac{α}{4 3}$ , then $α$ is equal to ____.
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Answer: 84
Q212022 · 28 Jul 2022 · Shift 2· Numerical answer
Let the coefficients of the middle terms in the expansion of $(\frac{1}{6} + β x)^{4}$ , $(1 - 3 β x)^{2}$ and $(1 - \frac{β}{2} x)^{6}$ , $β > 0$ , respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then $50 - \frac{2 d}{β ^{2}}$ is equal to _________.
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Answer: 57
Q222022 · 27 Jul 2022 · Shift 2· Numerical answer
Let for the $9^{th}$ term in the binomial expansion of $(3 + 6 x)^{n}$ , in the increasing powers of $6 x$ , to be the greatest for $x = \frac{3}{2}$ , the least value of $n$ is $n_{0}$ . If $k$ is the ratio of the coefficient of $x^{6}$ to the coefficient of $x^{3}$ , then $k + n_{0}$ is equal to :
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Answer: 24
Q292022 · 26 Jul 2022 · Shift 1· Numerical answer
If the coefficients of $x$ and $x^{2}$ in the expansion of $(1 + x)^{p} (1 - x)^{q}$ , $p, q \leq 15$ , are $- 3$ and $- 5$ respectively, then the coefficient of $x^{3}$ is equal to ____ .
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Answer: 23
Q52022 · 30 Jun 2022 · Shift 1
For two positive real numbers $a$ and $b$ such that $\frac{1}{a ^{2}} + \frac{1}{b ^{3}} = 4$ , then minimum value of the constant term in the expansion of $(a x^{\frac{1}{8}} + b x^{- \frac{1}{12}})^{10}$ is :
1. A. $\frac{105}{2}$
2. B. $\frac{105}{4}$
3. C. $\frac{105}{8}$
4. D. $\frac{105}{16}$
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Answer: C
Q192022 · 30 Jun 2022 · Shift 1
If a random variable X follows the Binomial distribution B(5, p) such that $P (X = 0) = P (X = 1)$ , then $\frac{P ( X = 2 )}{P ( X = 3 )}$ is equal to :
1. A.1
2. B.10
3. C.25
4. D.5
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Answer: D
Q42022 · 28 Jun 2022 · Shift 2
The term independent of x in the expansion of $(1 - x^{2} + 3 x^{3}) (\frac{5}{2} x^{3} - \frac{1}{5 x ^{2}})^{11}$ , $x \neq = 0$ is :
1. A. $\frac{7}{40}$
2. B. $\frac{33}{200}$
3. C. $\frac{39}{200}$
4. D. $\frac{11}{50}$
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Answer: B
Q242022 · 27 Jun 2022 · Shift 1· Numerical answer
If the coefficient of $x^{10}$ in the binomial expansion of $(\frac{x}{5 ^{\frac{1}{4}}} + \frac{5}{x ^{\frac{1}{3}}})^{60}$ is $5^{k} \cdot l$ , where $l, k \in N$ and $l$ is co-prime to 5, then $k$ is equal to ____.
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Answer: 5
Q292022 · 27 Jun 2022 · Shift 2· Numerical answer
If the sum of the coefficients of all the positive powers of $x$ , in the Binomial expansion of $(x^{n} + \frac{2}{x ^{5}})^{7}$ is $939$ , then the sum of all the possible integral values of $n$ is ________.
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Answer: 57
Q262022 · 25 Jun 2022 · Shift 1· Numerical answer
Let $C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1 + x)^{10}$ . If for $α, β \in R$ , $C_{1} + 3 \cdot 2 C_{2} + 5 \cdot 3 C_{3} + \dots upto 10 terms = \frac{α \times 2 ^{11}}{2 ^{β} - 1} (C_{0} + \frac{C _{1}}{2} + \frac{C _{2}}{3} + \dots upto 10 terms)$ then the value of $α + β$ is equal to .
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Answer: 286
Q72022 · 25 Jun 2022 · Shift 2
The coefficient of $x^{101}$ in the expression $(5 + x)^{500} + x (5 + x)^{499} + x^{2} (5 + x)^{498} + \dots + x^{500}$ , $x > 0$ , is
1. A. $^{501} C_{101} (5)^{399}$
2. B. $^{501} C_{101} (5)^{400}$
3. C. $^{501} C_{100} (5)^{400}$
4. D. $^{500} C_{101} (5)^{399}$
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Answer: A
Q242022 · 25 Jun 2022 · Shift 2· Numerical answer
If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of $(2 x^{3} + \frac{3}{x})^{10}$ is $5^{10} - β \cdot 3^{9}$ , then $β$ is equal to .
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Answer: 83