Binomial Theorem Questions

MathematicsBinomial Theorem69 Questions Found

MathematicsProperties of Binomial CoefficientsJEE Main 2008Easy

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Statement-1: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}$ . Statement-2: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}$ .

(A)

Statement-1 is false, Statement-2 is true

(B)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

(C)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

(D)

Statement-1 is true, Statement-2 is false

Answer:B

MathematicsBinomial Expansion for Positive Integral IndexJEE Main 2009Easy

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

(A)

(B)

(C)

(D)

Answer:A

MathematicsProperties of Binomial CoefficientsJEE Main 2010Moderate

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Let $S_{1} = \sum_{j = 1}^{10} j (j - 1)^{10} C_{j}, S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}$ and $S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j$ . Statement-1: $S_{3} = 55 \times 2^{9}$ . Statement-2: $S_{1} = 90 \times 2^{8}$ and $S_{2} = 10 \times 2^{8}$ .

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

Answer:B

MathematicsBinomial Expansion for Positive Integral IndexJEE Main 2011Easy

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

(A)

-132

(B)

-144

(C)

132

(D)

144

Answer:B

MathematicsBinomial Expansion for Positive Integral IndexJEE Main 2012Easy

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

(A)

an irrational number

(B)

an odd positive integer

(C)

an even positive integer

(D)

a rational number other than positive integers

Answer:A

MathematicsGeneral Term and Middle TermJEE Main 2013Easy

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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

(A)

(B)

120

(C)

210

(D)

310

Answer:C

MathematicsGeneral Term and Middle TermJEE Main 2014Moderate

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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

(A)

(14, 272/3)

(B)

(16, 272/3)

(C)

(16, 251/3)

(D)

(14, 251/3)

Answer:B

MathematicsProperties of Binomial CoefficientsJEE Main 2015Easy

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The sum of coefficients of integral power of $x$ in the binomial expansion $(1 - 2 x)^{50}$ is

(A)

\frac{1}{2} (3^{50} - 1)

(B)

\frac{1}{2} (2^{50} + 1)

(C)

\frac{1}{2} (3^{50} + 1)

(D)

\frac{1}{2} (3^{50})

Answer:C

MathematicsMultinomial TheoremJEE Main 2016Easy

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If the number of terms in the expansion of $(1 - \frac{2}{x} + \frac{4}{x ^{2}})^{n}, x \neq = 0$ , is 28, then the sum of the coefficients of all the terms in this expansion, is

(A)

243

(B)

729

(C)

(D)

2187

Answer:B

...6

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Statement-1: ∑r=0n​(r+1)nCr​=(n+2)2n−1. Statement-2: ∑r=0n​(r+1)nCr​xr=(1+x)n+nx(1+x)n−1.

.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; } Let S1​=∑j=110​j(j−1)10Cj​,S2​=∑j=110​j10Cj​ and S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j. Statement-1: S3​=55×29. Statement-2: S1​=90×28 and S2​=10×28.

.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; } 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 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; } 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 sum of coefficients of integral power of x in the binomial expansion (1−2x​)50 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the number of terms in the expansion of (1−x2​+x24​)n,x=0, is 28, then the sum of the coefficients of all the terms in this expansion, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Statement-1: ∑r=0n​(r+1)nCr​=(n+2)2n−1. Statement-2: ∑r=0n​(r+1)nCr​xr=(1+x)n+nx(1+x)n−1.

.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; } Let S1​=∑j=110​j(j−1)10Cj​,S2​=∑j=110​j10Cj​ and S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j. Statement-1: S3​=55×29. Statement-2: S1​=90×28 and S2​=10×28.

.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; } 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 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; } 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 sum of coefficients of integral power of x in the binomial expansion (1−2x​)50 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the number of terms in the expansion of (1−x2​+x24​)n,x=0, is 28, then the sum of the coefficients of all the terms in this expansion, is

Statement-1: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}$ . Statement-2: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}$ .

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

Let $S_{1} = \sum_{j = 1}^{10} j (j - 1)^{10} C_{j}, S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}$ and $S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j$ . Statement-1: $S_{3} = 55 \times 2^{9}$ . Statement-2: $S_{1} = 90 \times 2^{8}$ and $S_{2} = 10 \times 2^{8}$ .

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

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

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

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 sum of coefficients of integral power of $x$ in the binomial expansion $(1 - 2 x)^{50}$ is

If the number of terms in the expansion of $(1 - \frac{2}{x} + \frac{4}{x ^{2}})^{n}, x \neq = 0$ , is 28, then the sum of the coefficients of all the terms in this expansion, is

Statement-1: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}$ . Statement-2: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}$ .

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

Let $S_{1} = \sum_{j = 1}^{10} j (j - 1)^{10} C_{j}, S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}$ and $S_3 = \sum_{j=1}^{10} j^2 ^{10}C_j$ . Statement-1: $S_{3} = 55 \times 2^{9}$ . Statement-2: $S_{1} = 90 \times 2^{8}$ and $S_{2} = 10 \times 2^{8}$ .

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

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

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

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 sum of coefficients of integral power of $x$ in the binomial expansion $(1 - 2 x)^{50}$ is

If the number of terms in the expansion of $(1 - \frac{2}{x} + \frac{4}{x ^{2}})^{n}, x \neq = 0$ , is 28, then the sum of the coefficients of all the terms in this expansion, is