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

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

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

Back to BINOMIAL THEOREM

Your Progress

0 / 951 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

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.

Take a chapter quiz →

Showing 931-951 of 951 questions

931 Easy

931 of 951

Application +4 | -1

The coefficient of

{x^4}

in the expansion of

{\left( {1 + x + {x^2} + {x^3}} \right)^{11}},

View answer and explanation

Correct option: C

990

Option C (990) is correct because it is the sum of the products of binomial coefficients that contribute to the x⁴ term in the expansion.

932 Medium

932 of 951

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

933 Easy

933 of 951

Application +4 | -1

If the fourth term in the expansion of

{\left\{ {\sqrt {{x^{\left( {\frac{1}{{\log x + 1}}} \right)}}} + {x^{1/12}}} \right\}^6}

is equal to 200 and

x > 1,

, then

x

is equal to

View answer and explanation

Correct option: B

Option B (10) is correct because the value of x is found by setting the fourth term in the given expansion equal to 200 and solving the resulting equation for x, given that x > 1.

934 Easy

934 of 951

Application +4 | -1

{\;^n}{C_{12}} = {\;^n}{C_6}

, then

{\;^n}{C_2}

is equal to

View answer and explanation

Correct option: B

153

Option B (153) is correct. Using the property that if

^nC_x = ^nC_y

, then either x=y or x+y=n, we can find n and then calculate

^nC_2

935 Medium

935 of 951

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.

936 Medium

936 of 951

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.

937 Easy

937 of 951

Analyzing +4 | -1

The total number of terms in the expansion of

(x+a)^{200}+(x-a)^{200}

after simplification is

View answer and explanation

Correct option: A

101

Option A (101) is correct because after simplification, only the even-powered terms remain, resulting in 101 terms.

938 Easy

938 of 951

Application +4 | -1

If the fourth term in the expansion of

{(x + {x^{{{\log }_2}x}})^7}

is 4480, then the value of x, Where

x \in \mathrm{N}

is equal to

View answer and explanation

Correct option: A

Option A (2) is correct because substituting x = 2 into the binomial expansion and solving the resulting equation yields a consistent result.

939 Medium

939 of 951

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.

940 Medium

940 of 951

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}

941 Medium

941 of 951

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.

942 Easy

942 of 951

Application +4 | -1

The sum of the coefficients of all the integral powers of

x

in the expansion of

(1+2 \sqrt{x})^{40}

View answer and explanation

Correct option: D

\frac{1}{2}\left(3^{40}+1\right)

Option D (

\frac{1}{2}\left(3^{40}+1\right)

) is correct because it represents the sum of the coefficients of all integral powers of x in the expansion of

(1+2 \sqrt{x})^{40}

943 Easy

943 of 951

Application +4 | -1

\left| x \right| < \frac{1}{2}

, then the coefficient of

{x^r}

in the expansion of

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

, is

View answer and explanation

Correct option: D

\left( {2r + 1} \right){2^r}

Option D (

(2r + 1){2^r}

) is correct because it represents the coefficient of

{x^r}

after expanding and simplifying the given expression.

944 Hard

944 of 951

Application +4 | -1

If the sum of the coefficients in the expansion of

{\left( {\alpha \;{x^2} - 2x + 1} \right)^{35}}

is equal to the sum of the coefficient in the expansion of

{\left( {x - \alpha \;y} \right)^{35}},

then

\alpha =

View answer and explanation

Correct option: B

Option B (1) is correct because setting the sum of coefficients equal in both expressions leads to the equation (

\alpha

- 1)³⁵ = -(

\alpha

- 1)³⁵, which is only true when

\alpha

= 1.

945 Medium

945 of 951

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.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

946 Medium

946 of 951

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.

947 Medium

947 of 951

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.

948 Medium

948 of 951

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.

949 Medium

949 of 951

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.

950 Medium

950 of 951

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).

951 Medium

951 of 951

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

Review BINOMIAL THEOREM in your textbook or class notes, then use this set to check your understanding.

Showing 931-951 of 951 questions

Continue Your Practice

📚 Switch Set (ALL)

Chapter Sets

ALL 951 MCQs EASY 434 MCQs HARD 49 MCQs MEDIUM 468 MCQs

0 /951

0✓ 0✗

0% Accuracy