General Term and Middle Term

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JEE Main 2014

LEVELJEE Main

If the coefficients of $x^{3}$ and $x^{4}$ in the expansion of $(1 + a x + b x^{2}) (1 - 2 x)^{18}$ in powers of $x$ are both zero, then $(a, b)$ is equal to

The term independent of $x$ in expansion of $(\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}})^{10}$ is

In the binomial expansion of $(a - b)^{n}, n \geq 5$ , the sum of $5^{t h}$ and $6^{t h}$ terms is zero, then $a / b$ equals

(A)

\frac{n - 5}{6}

(B)

\frac{n - 4}{5}

(C)

\frac{5}{n - 4}

(D)

\frac{6}{n - 5}

JEE Main 2004

LEVELBoard

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + α x)^{4}$ and of $(1 - α x)^{6}$ is the same if $α$ equals

(A)

3/5

(B)

10/3

(C)

- 3/10

(D)

- 5/3

JEE Advanced 2003

LEVELBoard

Coefficient of $t^{24}$ in $(1 + t^{2})^{12} (1 + t^{12}) (1 + t^{24})$ is

(A)

^{12} C_{6} + 3

(B)

^{12} C_{6} + 1

(C)

^{12} C_{6}

(D)

^{12} C_{6} + 2

JEE Advanced 2001

LEVELBoard

In the binomial expansion of $(a - b)^{n}, n \geq 5$ , the sum of the $5^{t h}$ and $6^{t h}$ terms is zero. Then $a / b$ equals

(A)

(n - 5) /6

(B)

(n - 4) /5

(C)

5/ (n - 4)

(D)

6/ (n - 5)

JEE Advanced 1983

LEVELBoard

Given positive integers $r > 1, n > 2$ and that the coefficient of $(3 r)$ th and $(r + 2)$ th terms in the binomial expansion of $(1 + x)^{2 n}$ are equal. Then

(A)

n = 2 r

(B)

n = 2 r + 1

(C)

n = 3 r

The coefficient of $x^{4}$ in $(\frac{x}{2} - \frac{3}{x ^{2}})^{10}$ is

(A)

\frac{405}{256}

(B)

\frac{504}{259}

(C)

\frac{450}{263}

The sum of the rational terms in the expansion of $(2 + 3^{1/5})^{10}$ is .........

JEE Main 2018 (15 April Evening)

LEVELJEE Main

The coefficient of $x^{10}$ in the expansion of $(1 + x)^{2} (1 + x^{2})^{3} (1 + x^{3})^{4}$ is equal to :-

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

If the third term in the binomial expansion of $(1 + x^{l o g_{2} x})^{5}$ equals 2560, then a possible value of x is :

(A)

2\sqrt{2}

(B)

\frac{1}{8}

(C)

4\sqrt{2}

(D)

\frac{1}{4}

JEE Main 2019 (12 January)

LEVELJEE Main

The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is :

(A)

55

(B)

49

(C)

48

(D)

54

JEE Main 2019 (12 January Shift 1)

LEVELJEE Main

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $(2^{1/3} + \frac{1}{2 ( 3 ) ^{1/3}})^{10}$ is :

(A)

1 : 4 (16)^{1/3}

(B)

1 : 2 (6)^{1/3}

(C)

2 (36)^{1/3} : 1

(D)

4 (36)^{1/3} : 1

JEE Main 2019 (11 January)

LEVELJEE Main

The sum of the real values of $x$ for which the middle term in the binomial expansion of $(\frac{x ^{3}}{3} + \frac{3}{x})^{8}$ equals 5670 is :

JEE Main 2019 (10 January)

LEVELBoard

The positive value of $λ$ for which the co-efficient of $x^{2}$ in the expression $x^{2} (x + \frac{λ}{x ^{2}})^{10}$ is 720, is :

(A)

5

(B)

(C)

22

(D)

JEE Main 2019 (08 April Shift 2)

LEVELJEE Main

If the fourth term in the binomial expansion of $(1/ (1 0^{1 + l o g_{10} x}) + 12 x^{6})^{6}$ is equal to 200, and $x > 1$ , then the value of x is :

(A)

1 0^{3}

(B)

100

(C)

1 0^{4}

(D)

10

JEE Main 2019 (09 April Shift 1)

LEVELJEE Main

If the fourth term in the binomial expansion of $(\frac{2}{x} + x^{l o g_{8} x})^{6} (x > 0)$ is $20 \times 8^{7}$ , then a value of $x$ is :

(A)

(B)

8^{2}

(C)

8^{- 2}

(D)

8^{3}

JEE Main 2019 (10 April Shift 2)

LEVELBoard

The smallest natural number $n$ , such that the coefficient of $x$ in the expansion of $(x^{2} + \frac{1}{x ^{3}})^{n}$ is $^{n} C_{23}$ , is :

JEE Main 2019 (12 April)

LEVELJEE Main

The term independent of x in the expansion of $(\frac{1}{60} - \frac{x ^{8}}{81}) (2 x^{2} - \frac{3}{x ^{2}})^{6}$ is equal to :

JEE Main 2020 (9 January Shift 2)

LEVELJEE Main

In the expansion of $(\frac{x}{c o s θ} + \frac{1}{x s i n θ})^{16}$ , if $l_{1}$ is the least value of the term independent of $x$ when $\frac{π}{8} \leq θ \leq \frac{π}{4}$ and $l_{2}$ is the least value of the term independent of $x$ when $\frac{π}{16} \leq θ \leq \frac{π}{8}$ , then the ratio $l_{2} : l_{1}$ is equal to :

(A)

16 : 1

(B)

8 : 1

(C)

1 : 8

(D)

1 : 16

JEE Main 2020 - 9 Jan (Evening)

LEVELJEE Main

In the expansion of $(x cos θ + \frac{1}{x s i n θ})^{16}$ , if $l_{1}$ is the least value of the term independent of $x$ when $\frac{π}{8} \leq θ \leq \frac{π}{4}$ and $l_{2}$ is the least value of the term independent of $x$ when $\frac{π}{16} < θ < \frac{π}{8}$ , then the ratio $l_{2} : l_{1}$ is equal to:

JEE Main 2020 - 2 Sep (Morning)

LEVELJEE Main

Let $α > 0, β > 0$ be such that $α^{3} + β^{2} = 4$ . If the maximum value of the term independent of $x$ in the binomial expansion of $(α x^{1/9} + β x^{- 1/6})^{10}$ is $10 k$ , then $k$ is equal to :

JEE Main 2020 - 3 Sep (Morning)

LEVELJEE Main

If the number of integral terms in the expansion of $(33 + 55)^{n}$ is exactly 33, then the least value of $n$ is:

JEE Main 2020 - 3 Sep (Evening)

LEVELJEE Main

If the term independent of $x$ in the expansion of $(\frac{3}{2} x^{2} - \frac{1}{3 x})^{9}$ is $k$ , then $18 k$ is equal to :

JEE Main 2020 - 5 Sep (Morning)

LEVELBoard

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the coefficients of x3 and x4 in the expansion of (1+ax+bx2)(1−2x)18 in powers of x are both zero, then (a,b) is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The term independent of x in expansion of (x2/3−x1/3+1x+1​−x−x1/2x−1​)10 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In the binomial expansion of (a−b)n,n≥5, the sum of 5th and 6th terms is zero, then a/b equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of the middle term in the binomial expansion in powers of x of (1+αx)4 and of (1−αx)6 is the same if α equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Coefficient of t24 in (1+t2)12(1+t12)(1+t24) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In the binomial expansion of (a−b)n,n≥5, the sum of the 5th and 6th terms is zero. Then a/b equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Given positive integers r>1,n>2 and that the coefficient of (3r)th and (r+2)th terms in the binomial expansion of (1+x)2n are equal. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x4 in (2x​−x23​)10 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the rational terms in the expansion of (2​+31/5)10 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x10 in the expansion of (1+x)2(1+x2)3(1+x3)4 is equal to :-

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the third term in the binomial expansion of (1+xlog2​x)5 equals 2560, then a possible value of x is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The total number of irrational terms in the binomial expansion of (71/5−31/10)60 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of (21/3+2(3)1/31​)10 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the real values of x for which the middle term in the binomial expansion of (3x3​+x3​)8 equals 5670 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The positive value of λ for which the co-efficient of x2 in the expression x2(x​+x2λ​)10 is 720, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the fourth term in the binomial expansion of (1/(101+log10​x)​+12x6​)6 is equal to 200, and x>1, then the value of x is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the fourth term in the binomial expansion of (x2​+xlog8​x)6(x>0) is 20×87, then a value of x is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The smallest natural number n, such that the coefficient of x in the expansion of (x2+x31​)n is nC23​, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The term independent of x in the expansion of (601​−81x8​)(2x2−x23​)6 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α>0,β>0 be such that α3+β2=4. If the maximum value of the term independent of x in the binomial expansion of (αx1/9+βx−1/6)10 is 10k, then k is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the number of integral terms in the expansion of (33​+55​)n is exactly 33, then the least value of n is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the term independent of x in the expansion of (23​x2−3x1​)9 is k, then 18k is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The natural number m, for which the coefficient of x in the binomial expansion of (xm+x21​)22 is 1540, is

If the coefficients of $x^{3}$ and $x^{4}$ in the expansion of $(1 + a x + b x^{2}) (1 - 2 x)^{18}$ in powers of $x$ are both zero, then $(a, b)$ is equal to

The term independent of $x$ in expansion of $(\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}})^{10}$ is

In the binomial expansion of $(a - b)^{n}, n \geq 5$ , the sum of $5^{t h}$ and $6^{t h}$ terms is zero, then $a / b$ equals

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + α x)^{4}$ and of $(1 - α x)^{6}$ is the same if $α$ equals

Coefficient of $t^{24}$ in $(1 + t^{2})^{12} (1 + t^{12}) (1 + t^{24})$ is

In the binomial expansion of $(a - b)^{n}, n \geq 5$ , the sum of the $5^{t h}$ and $6^{t h}$ terms is zero. Then $a / b$ equals

Given positive integers $r > 1, n > 2$ and that the coefficient of $(3 r)$ th and $(r + 2)$ th terms in the binomial expansion of $(1 + x)^{2 n}$ are equal. Then

The coefficient of $x^{4}$ in $(\frac{x}{2} - \frac{3}{x ^{2}})^{10}$ is

The sum of the rational terms in the expansion of $(2 + 3^{1/5})^{10}$ is .........

The coefficient of $x^{10}$ in the expansion of $(1 + x)^{2} (1 + x^{2})^{3} (1 + x^{3})^{4}$ is equal to :-

If the third term in the binomial expansion of $(1 + x^{l o g_{2} x})^{5}$ equals 2560, then a possible value of x is :

The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is :

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $(2^{1/3} + \frac{1}{2 ( 3 ) ^{1/3}})^{10}$ is :

The sum of the real values of $x$ for which the middle term in the binomial expansion of $(\frac{x ^{3}}{3} + \frac{3}{x})^{8}$ equals 5670 is :

The positive value of $λ$ for which the co-efficient of $x^{2}$ in the expression $x^{2} (x + \frac{λ}{x ^{2}})^{10}$ is 720, is :

If the fourth term in the binomial expansion of $(1/ (1 0^{1 + l o g_{10} x}) + 12 x^{6})^{6}$ is equal to 200, and $x > 1$ , then the value of x is :

If the fourth term in the binomial expansion of $(\frac{2}{x} + x^{l o g_{8} x})^{6} (x > 0)$ is $20 \times 8^{7}$ , then a value of $x$ is :

The smallest natural number $n$ , such that the coefficient of $x$ in the expansion of $(x^{2} + \frac{1}{x ^{3}})^{n}$ is $^{n} C_{23}$ , is :

The term independent of x in the expansion of $(\frac{1}{60} - \frac{x ^{8}}{81}) (2 x^{2} - \frac{3}{x ^{2}})^{6}$ is equal to :

Let $α > 0, β > 0$ be such that $α^{3} + β^{2} = 4$ . If the maximum value of the term independent of $x$ in the binomial expansion of $(α x^{1/9} + β x^{- 1/6})^{10}$ is $10 k$ , then $k$ is equal to :

If the number of integral terms in the expansion of $(33 + 55)^{n}$ is exactly 33, then the least value of $n$ is:

If the term independent of $x$ in the expansion of $(\frac{3}{2} x^{2} - \frac{1}{3 x})^{9}$ is $k$ , then $18 k$ is equal to :

The natural number $m$ , for which the coefficient of $x$ in the binomial expansion of $(x^{m} + \frac{1}{x ^{2}})^{22}$ is 1540, is