Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 48

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Consider the following statements
$S_{1}$ : The total number of terms in ${(x^{2} + 2 x + 1)}^{10}$ is $21$ .
$S_{2}$ : The coefficient of $x^{10}$ in ${(x^{2} + \frac{1}{x})}^{20}$ is ${}^{20}C_{10}$ .

$S_{3} :$ The middle term in the expansion of $(1 + x)^{12}$ is ${}^{12}C_{6} x^{6}$ .

$S_{4}$ : If the coefficients of the fifth and ninth term in the expansion of $(1 + x)^{n}$ are same then $n = 12$ .
Now identify the correct combination of true statements.

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Important Questions on Binomial Theorem

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 49

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of

{(2^{\frac{1}{4}} + \frac{1}{3^{\frac{1}{4}}})}^{n}

is

\sqrt{6} : 1

then value of

n

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 50

The value of the coefficient of

x^{4}

in the expansion of

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

is

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 51

The greatest value of

n

for which

9^{7} + 7^{9}

is divisible by

2^{n}

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 52

If the coefficient of

x^{n}

in the expansion of

{(1 - 9 x + 20 x^{2})}^{- 1}

is

a^{n + 1} - b^{n + 1}

(where

a, b

are coprime) then

\frac{a}{b}

is equal to

(a > b)

.

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 53

The coefficient of

x^{9}

in the expansion of

\prod_{r = 1}^{100} (1 + x^{r})

is equal to

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 1

If

{(1 - x^{3})}^{n} = \sum_{r = 0}^{n} a_{r} x^{r} (1 - x)^{3 n - 2 r}, x \neq 0

then the value of

a_{r}

where

n \in N

is

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 2

Let

{(1 + x^{2})}^{2} {(1 + x)}^{n} = \sum_{k = 0}^{n + 4} a_{k} \cdot x^{k}

, If

a_{1}, a_{2}

and

a_{3}

are in A.P., then the value of

n

is

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Gamma Question Bank for Engineering Mathematics>Chapter 6 - Binomial Theorem>Assignment>Q 3

\sum_{r = 0}^{n} {(- 1)}^{r} \cdot {}^{n}C_{r} [\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + \dots m terms] =