Binomial Expansion for Positive Integral Index

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JEE Main 2012

LEVELBoard

For natural numbers $m, n$ if $(1 - y)^{m} (1 + y)^{n} = 1 + a_{1} y + a_{2} y^{2} + ....$ and $a_{1} = a_{2} = 10$ , then $(m, n)$ is

(A)

(20, 45)

(B)

(35, 20)

(C)

(45, 35)

(D)

(35, 45)

JEE Main 2005

LEVELJEE Main

If the coefficient of $x^{7}$ in $[a x^{2} + (\frac{1}{b x})]^{11}$ equals the coefficient of $x^{- 7}$ in $[a x - (\frac{1}{b x ^{2}})]^{11}$ , then $a$ and $b$ satisfy the relation

(A)

a - b = 1

(B)

a + b = 1

(C)

a / b = 1

(D)

ab = 1

JEE Main 2004

LEVELBoard

The coefficient of $x^{n}$ in expansion of $(1 + x) (1 - x)^{n}$ is

(A)

(- 1)^{n - 1} n

(B)

(- 1)^{n} (1 - n)

(C)

(- 1)^{n - 1} (n - 1)^{2}

(D)

(n - 1)

JEE Main 2003

LEVELBoard

The number of integral terms in the expansion of $(3 + 85)^{256}$ is

The expression $[x + (x^{3} - 1)^{1/2}]^{5} + [x - (x^{3} - 1)^{1/2}]^{5}$ is a polynomial of degree

If in the expansion of $(1 + x)^{m} (1 - x)^{n}$ , the coefficients of $x$ and $x^{2}$ are 3 and $- 6$ respectively, then $m$ is

The coefficient of $x^{9}$ in the expansion of $(1 + x) (1 + x^{2}) (1 + x^{3}) \dots (1 + x^{100})$ is \dots.

JEE Advanced 1983

LEVELBoard

If $(1 + a x)^{n} = 1 + 8 x + 24 x^{2} + ....$ then $a = .........$ and $n = .........$

JEE Advanced 1982

LEVELBoard

The larger of $9 9^{50} + 10 0^{50}$ and $10 1^{50}$ is .........

JEE Advanced 2002

LEVELJEE Main

Use mathematical induction to show that $(25)^{n + 1} - 24 n + 5735$ is divisible by $(24)^{2}$ for all $n = 1, 2, .......$

JEE Advanced 1996

LEVELJEE Main

Using mathematical induction prove that for every integer $n \geq 1, (3^{2 n} - 1)$ is divisible by $2^{n + 2}$ but not by $2^{n + 3}$ .

JEE Advanced 1990

LEVELJEE Main

Prove that $\frac{n ^{7}}{7} + \frac{n ^{5}}{5} + \frac{2 n ^{3}}{3} - \frac{n}{105}$ is an integer for every positive integer $n$ .

JEE Advanced 1988

LEVELJEE Main

Let $R = (55 + 11)^{2 n + 1}$ and $f = R - [R]$ , where $[]$ denotes the greatest integer function. Prove that $R f = 4^{2 n + 1}$ .

JEE Advanced 1984

LEVELJEE Main

If $p$ be a natural number then prove that $p^{n + 1} + (p + 1)^{2 n - 1}$ is divisible by $p^{2} + p + 1$ for every positive integer $n$ .

JEE Advanced 1982

LEVELJEE Main

Prove that $7^{2 n} + 2^{3 n - 3} (3^{n - 1})$ is divisible by 25 for any natural number $n$ .

JEE Main 2018 (Paper 1)

LEVELBoard

The sum of the co-efficients of all odd degree terms in the expansion of $(x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, (x > 1)$ is :

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

If n is the degree of the polynomial, $[\frac{2}{5 x ^{3} + 1 - 5 x ^{3} - 1}]^{8} + [\frac{2}{5 x ^{3} + 1 + 5 x ^{3} - 1}]^{8}$ and m is the coefficient of $x^{n}$ in it, then the ordered pair (n,m) is equal to:

(A)

(24, (10)^{8})

(B)

(12, (20)^{4})

(C)

(12, 8 (10)^{4})

(D)

(8, 5 (10)^{4})

JEE Main 2018 (16 April Shift 1)

LEVELJEE Main

The coefficient of $x^{2}$ in the expansion of the product $(2 - x^{2}) \cdot ((1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6})$ is :

JEE Main 2019 (9 January)

LEVELJEE Main

If the fractional part of the number $\frac{2 ^{403}}{15}$ is $\frac{k}{15}$ , then k is equal to :

JEE Main 2019 (8 April Shift 1)

LEVELJEE Main

The sum of the co-efficients of all even degree terms in $x$ in the expansion of $(x + x^{3} - 1)^{6} + (x - x^{3} - 1)^{6}, (x > 1)$ is equal to :

(A)

32

(B)

26

(C)

29

(D)

24

JEE Main 2020 (7 January Shift 2)

LEVELJEE Main

The coefficient of $x^{7}$ in the expression $(1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10}$ is :

JEE Main 2020 (8 January Shift 2)

LEVELJEE Main

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $(x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6}$ , then:

(A)

α + β = - 30

(B)

α - β = - 132

(C)

α + β = 60

(D)

α - β = 60

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If n is a positive integer , then (3​+1)2n−(3​−1)2n is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x7 in the expansion of (1−x−x2+x3)6 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The remainder left out when 82n−(62)2n+1 is divided by 9 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For natural numbers m,n if (1−y)m(1+y)n=1+a1​y+a2​y2+.... and a1​=a2​=10, then (m,n) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the coefficient of x7 in [ax2+(bx1​)]11 equals the coefficient of x−7 in [ax−(bx21​)]11, then a and b satisfy the relation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of xn in expansion of (1+x)(1−x)n is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of integral terms in the expansion of (3​+85​)256 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The expression [x+(x3−1)1/2]5+[x−(x3−1)1/2]5 is a polynomial of degree

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in the expansion of (1+x)m(1−x)n, the coefficients of x and x2 are 3 and −6 respectively, then m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x9 in the expansion of (1+x)(1+x2)(1+x3)…(1+x100) is \dots.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (1+ax)n=1+8x+24x2+.... then a=......... and n=.........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The larger of 9950+10050 and 10150 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Use mathematical induction to show that (25)n+1−24n+5735 is divisible by (24)2 for all n=1,2,.......

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Using mathematical induction prove that for every integer n≥1,(32n−1) is divisible by 2n+2 but not by 2n+3.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that 7n7​+5n5​+32n3​−105n​ is an integer for every positive integer n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let R=(55​+11)2n+1 and f=R−[R], where [ ] denotes the greatest integer function. Prove that Rf=42n+1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p be a natural number then prove that pn+1+(p+1)2n−1 is divisible by p2+p+1 for every positive integer n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove that 72n+23n−3(3n−1) is divisible by 25 for any natural number n.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the co-efficients of all odd degree terms in the expansion of (x+x3−1​)5+(x−x3−1​)5,(x>1) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If n is the degree of the polynomial, [5x3+1​−5x3−1​2​]8+[5x3+1​+5x3−1​2​]8 and m is the coefficient of xn in it, then the ordered pair (n,m) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x2 in the expansion of the product (2−x2)⋅((1+2x+3x2)6+(1−4x2)6) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the fractional part of the number 152403​ is 15k​, then k is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the co-efficients of all even degree terms in x in the expansion of (x+x3−1​)6+(x−x3−1​)6,(x>1) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x7 in the expression (1+x)10+x(1+x)9+x2(1+x)8+⋯+x10 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α and β be the coefficients of x4 and x2 respectively in the expansion of (x+x2−1​)6+(x−x2−1​)6, then:

If $n$ is a positive integer , then $(3 + 1)^{2 n} - (3 - 1)^{2 n}$ is

The coefficient of $x^{7}$ in the expansion of $(1 - x - x^{2} + x^{3})^{6}$ is

The remainder left out when $8^{2 n} - (62)^{2 n + 1}$ is divided by 9 is

For natural numbers $m, n$ if $(1 - y)^{m} (1 + y)^{n} = 1 + a_{1} y + a_{2} y^{2} + ....$ and $a_{1} = a_{2} = 10$ , then $(m, n)$ is

If the coefficient of $x^{7}$ in $[a x^{2} + (\frac{1}{b x})]^{11}$ equals the coefficient of $x^{- 7}$ in $[a x - (\frac{1}{b x ^{2}})]^{11}$ , then $a$ and $b$ satisfy the relation

The coefficient of $x^{n}$ in expansion of $(1 + x) (1 - x)^{n}$ is

The number of integral terms in the expansion of $(3 + 85)^{256}$ is

The expression $[x + (x^{3} - 1)^{1/2}]^{5} + [x - (x^{3} - 1)^{1/2}]^{5}$ is a polynomial of degree

If in the expansion of $(1 + x)^{m} (1 - x)^{n}$ , the coefficients of $x$ and $x^{2}$ are 3 and $- 6$ respectively, then $m$ is

The coefficient of $x^{9}$ in the expansion of $(1 + x) (1 + x^{2}) (1 + x^{3}) \dots (1 + x^{100})$ is \dots.

If $(1 + a x)^{n} = 1 + 8 x + 24 x^{2} + ....$ then $a = .........$ and $n = .........$

The larger of $9 9^{50} + 10 0^{50}$ and $10 1^{50}$ is .........

Use mathematical induction to show that $(25)^{n + 1} - 24 n + 5735$ is divisible by $(24)^{2}$ for all $n = 1, 2, .......$

Using mathematical induction prove that for every integer $n \geq 1, (3^{2 n} - 1)$ is divisible by $2^{n + 2}$ but not by $2^{n + 3}$ .

Prove that $\frac{n ^{7}}{7} + \frac{n ^{5}}{5} + \frac{2 n ^{3}}{3} - \frac{n}{105}$ is an integer for every positive integer $n$ .

Let $R = (55 + 11)^{2 n + 1}$ and $f = R - [R]$ , where $[]$ denotes the greatest integer function. Prove that $R f = 4^{2 n + 1}$ .

If $p$ be a natural number then prove that $p^{n + 1} + (p + 1)^{2 n - 1}$ is divisible by $p^{2} + p + 1$ for every positive integer $n$ .

Prove that $7^{2 n} + 2^{3 n - 3} (3^{n - 1})$ is divisible by 25 for any natural number $n$ .

The sum of the co-efficients of all odd degree terms in the expansion of $(x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, (x > 1)$ is :

If n is the degree of the polynomial, $[\frac{2}{5 x ^{3} + 1 - 5 x ^{3} - 1}]^{8} + [\frac{2}{5 x ^{3} + 1 + 5 x ^{3} - 1}]^{8}$ and m is the coefficient of $x^{n}$ in it, then the ordered pair (n,m) is equal to:

The coefficient of $x^{2}$ in the expansion of the product $(2 - x^{2}) \cdot ((1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6})$ is :

If the fractional part of the number $\frac{2 ^{403}}{15}$ is $\frac{k}{15}$ , then k is equal to :

The sum of the co-efficients of all even degree terms in $x$ in the expansion of $(x + x^{3} - 1)^{6} + (x - x^{3} - 1)^{6}, (x > 1)$ is equal to :

The coefficient of $x^{7}$ in the expression $(1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10}$ is :

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $(x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6}$ , then: