If $n - {\;^1}{C_r} = \left( {{k^2} - 3} \right){\;^n}{C_{r + 1}},$ then $k \in $

$\left( { - \infty ,\; - 2} \right)$

$\left[ { - \sqrt 3 ,\;\sqrt 3 } \right]$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Medium

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 451-468 of 468 questions

451 Medium

451 of 468

Application +4 | -1

(1+x)^n=\sum\limits_{{r=0}}^n c_r x^r

, then

\sum\limits_{{r=0}}^{200} \frac{c_r}{r+1}=

View answer and explanation

Correct option: D

\frac{2^{201}-1}{201}

Option D (

\frac{{2^{201}-1}}{{201}}

) is correct because it is derived from integrating the binomial expansion and evaluating the sum.

452 Medium

452 of 468

Application +4 | -1

If '

n

' is positive integer and three consecutive coefficient in the expansion of

(1+x)^n

are in the ratio

6: 33: 110

, then

n

is equal to :

View answer and explanation

Correct option: C

Option C (12) is correct because solving the ratio of consecutive coefficients in the binomial expansion leads to n=12.

453 Medium

453 of 468

Analyzing +4 | -1

The value of

\sum\limits_{\mathop {r = 0}\limits_{\,\,\,\,\,\,\,r \le s} }^n {\sum\limits_{s = 1}^n {^n} {C_s}^s{C_r}}

View answer and explanation

Correct option: A

3^n-1

Option A (

3^n-1

) is correct because the double summation simplifies to a binomial expansion minus the first term.

454 Medium

454 of 468

Application +4 | -1

If in the expansion of

(1+x)^m(1-x)^n

, the coefficients of

x

and

x^2

are 3 and -6 respectively, then

m

View answer and explanation

Correct option: C

Option C (12) is correct because solving the system of equations derived from the coefficients of x and x² yields m=12.

455 Medium

455 of 468

Application +4 | -1

The greatest term in the expansion of

(2 x+5 y)^{13}

when

x=10

and

y=2

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the calculated greatest term does not match any of the provided options.

456 Medium

456 of 468

Application +4 | -1

Middle term in the expansion of

\left(x^2+\frac{1}{x^2}+2\right)^n

View answer and explanation

Correct option: B

\frac{(2 n) !}{(n !)^2}

Option B (

\frac{{(2 n) !}}{{(n !)^2}}

) is correct because the middle term of the simplified binomial expansion is found using the binomial coefficient

{}^{2n}C_n

457 Medium

457 of 468

Application +4 | -1

The greatest term (numerically) in the expansion of

(3-5 x)^{15}

, when

x=\frac{1}{5}

View answer and explanation

Correct option: C

{ }^{15} C_3 \times 3^{12}

Option C (

{ }^{15} C_3 \times 3^{12}

) is correct because it represents the numerically greatest term in the expansion, calculated by comparing consecutive terms and finding the largest.

458 Medium

458 of 468

Analyzing +4 | -1

10 th term of

\left(3-\sqrt{\frac{17}{4}+3 \sqrt{2}}\right)^{20}

View answer and explanation

Correct option: A

An irrational number

Option A (An irrational number) is correct because the 10th term of the binomial expansion contains a term with

\sqrt{2}

raised to an odd power, making it irrational.

459 Medium

459 of 468

Application +4 | -1

The coefficient of

a^8 b^6 c^4

in the expansion of

(a+b+c)^{18}

View answer and explanation

Correct option: A

{ }^{18} C_{14} \cdot{ }^{14} C_8

Option A (

{ }^{18} C_{14} \cdot{ }^{14} C_8

) is correct because it represents the multinomial coefficient for the given term in the expansion.

460 Medium

460 of 468

Application +4 | -1

m,n,r

are positive integers such that

r < m,n,

then

{\;^m}{C_r} + {\;^m}{C_{r - 1}}{\;^n}{C_1} + {\;^m}{C_{r - 2}}{\;^n}{C_2} + \ldots + {\;^m}{C_1}{\;^n}{C_{r - 1}} + {\;^n}{C_r}

equals

View answer and explanation

Correct option: B

{\;^{m + n}}{C_r}

Option B (

{\;^{m + n}}{C_r}

) is correct because the given expression represents the coefficient of

x^r

in the expansion of

(1+x)^m(1+x)^n

, which simplifies to

(1+x)^{m+n}

461 Medium

461 of 468

Application +4 | -1

n - {\;^1}{C_r} = \left( {{k^2} - 3} \right){\;^n}{C_{r + 1}},

then

k \in

View answer and explanation

Correct option: D

\left( {\sqrt 3 ,2} \right]

Option D (

\left( {\sqrt 3 ,2} \right]

) is correct because it satisfies the condition

0 < k^2 - 3 \le 1

derived from the properties of binomial coefficients.

462 Medium

462 of 468

Analyzing +4 | -1

The value of

\sum\limits_{{r=0}}^n(-2)^r\left(\frac{{ }^n C_r}{{ }^{r+2} C_r}\right)

View answer and explanation

Correct option: C

\frac{1}{n+1}, n

is even

Option C (

\frac{{1}}{{(n+1)}}

, n is even) is correct because after simplifying the summation, the expression reduces to this form when n is even.

463 Medium

463 of 468

Application +4 | -1

In the expansion of

{(1 + 3x + 2{x^2})^6}

the coefficient of

{x^{11}}

View answer and explanation

Correct option: D

576

The correct answer is D (576) because it is the coefficient of

{x^{11}}

obtained from the multinomial expansion of the given expression.

464 Medium

464 of 468

Application +4 | -1

The coefficient of

x^4

in the expansion of

\left(1+5 x+9 x^2+13 x^3+17 x^4+\ldots ..\right)\left(1+x^2\right)^{11}

is equal to

View answer and explanation

Correct option: D

171

Option D (171) is correct because it calculates the coefficient of x⁴ by multiplying the series and considering terms that result in x⁴.

465 Medium

465 of 468

Application +4 | -1

The coefficient of

x^2

in the expansion of

\left(1+x+x^2\right)^{10}

View answer and explanation

Correct option: C

{ }^{11} C_2

Option C (

{ }^{11} C_2

) is correct because the coefficient of x² is found by simplifying the expression and applying the binomial theorem.

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466 Medium

466 of 468

Application +4 | -1

The 10 th term in the expansion of

(x-1)^{11}

(in decreasing powers of

x

) is

View answer and explanation

Correct option: D

-{ }^{11} C_9 x^2

Option D (-{ }^{11} C_9 x²) is correct because it is the 10th term in the expansion of (x-1)¹¹ using the binomial theorem.

467 Medium

467 of 468

Application +4 | -1

(1+x)^n=C_0+C_1 x+C_2 x^2+\ldots+C_n x^n, n \in N

, then for

2 \leq m \leq n

C_0-C_1+C_2-C_3+\ldots+(-1)^{m-1} C_{m-1}

is equal to

View answer and explanation

Correct option: A

(-1)^{m-1} \cdot{ }^{n-1} C_{m-1}

Option A ((-1)^(m-1) * { }^{n-1} C_{m-1}) is correct because it is derived by equating coefficients of x^(m-1) in the expansion of (1-x)^n(1-x)^(-1).

468 Medium

468 of 468

Application +4 | -1

The value of

{ }^{50} C_4+\sum\limits_{{r=1}}^6{ }^{56-r} C_3

View answer and explanation

Correct option: A

{ }^{56} \mathrm{C}_4

Option A (

{ }^{56} C_4

) is correct because the summation can be simplified using the identity

{ }^n C_r + { }^n C_{r-1} = { }^{n+1} C_r

repeatedly.

Revision notes and key takeaways

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