New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Binomial Theorem question

2023 · 11 Apr · Shift 1 · Q46

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Binomial Theorem
2023 · 11 Apr · Shift 1 · Q46

Binomial Theorem question

2023 · 11 Apr · Shift 1 · Q46

JEE MainMathematicsBinomial TheoremNumerical+4 / −1

The mean of the coefficients of

x, x^{2}, \ldots, x^{7}

in the binomial expansion of

(2+x)^{9}

is

\underline{\hspace{2cm}}

.

Numerical answer

View written solutionFree

Correct answer: 2736

In the expansion of

(2+x)^9= \sum_{k=0}^{9} \binom{9}{k}2^{9-k}x^k,

the coefficient of $x^k$ is

\binom{9}{k}2^{9-k}.

We need the mean of the coefficients of $x, x^2, \ldots, x^7,$ so we need the coefficients for $k=1$ to $k=7$ .

Their sum is

\sum_{k=1}^{7} \binom{9}{k}2^{9-k}.

Use the total sum of all coefficients:

\sum_{k=0}^{9} \binom{9}{k}2^{9-k}=(2+1)^9=3^9=19683.

Now subtract the coefficients of $x^0, x^8, x^9$ :

Coefficient of $x^0$ :

\binom{9}{0}2^9=512

Coefficient of $x^8$ :

\binom{9}{8}2^1=9\cdot 2=18

Coefficient of $x^9$ :

\binom{9}{9}2^0=1

Hence,

\sum_{k=1}^{7} \binom{9}{k}2^{9-k}=19683-512-18-1=19152.

There are $7$ coefficients from $x^1$ to $x^7$ , so the mean is

\frac{19152}{7}=2736.

Therefore, the required integer is

\boxed{2736}.

Comparison with stored answer: the derived answer matches the stored correct answer.

More from Binomial Theorem

The number of integral terms in the expansion of $(3^{\frac{1}{2}} + 5^{\frac{1}{4}})^{680}$ is equal to .2023 · Numerical
The sum of the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n + 2}$ , which are in the ratio $1 : 3 : 5$ , is equal to :2023 · MCQ
If the $101 1^{th}$ term from the end in the binominal expansion of $(\frac{4 x}{5} - \frac{5}{2 x})^{2022}$ is 1024 times $101 1^{th}$ R term from the beginning, then $∣ x ∣$ is equal to2023 · MCQ
Fractional part of the number $\frac{4 ^{2022}}{15}$ is equal to2023 · MCQ
Let $α$ be the constant term in the binomial expansion of $(x - \frac{6}{x ^{\frac{3}{2}}})^{n}, n \leq 15$ . If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of…2023 · Numerical
The coefficient of $x^{5}$ in the expansion of $(2 x^{3} - \frac{1}{3 x ^{2}})^{5}$ is :2023 · MCQ
The remainder, when $7^{103}$ is divided by 17, is 2023 · Numerical
Let $(a + b x + c x^{2})^{10} = i = 0 \sum 20 p_{i} x^{i}, a, b, c \in N$ . If $p_{1} = 20$ and $p_{2} = 210$ , then $2 (a + b + c)$ is equal to :2023 · MCQ