Let $K$ be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[3]{3})^{6144}$. If the coefficient of $x^P(P \in N)$ in the expansion of $\frac{1}{(1+x)\left(1+x^2\right)\left(1+x^4\right)\left(1+x^8\right)\left(1+x^{16}\right)}$ is $\alpha_p$, then $\alpha_k-\alpha_{k+1}-\alpha_{k-1}=$

If $C_0, C_1, C_2, \ldots, C_{10}$ represent the binomial coefficients in the expansion of $(1+x)^{10}$, then

$ C_0 C_6+C_1 C_7+C_2 C_8+C_3 C_9+C_4 C_{10}= $

When $|x|<\frac{1}{2}$ the coefficient of $x^6$ in the expansion of $\left(\frac{2-x}{1+2 x}\right)^2$ is

If $C_0, C_1, C_2, \ldots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$ then the value of $\Sigma r^3 \cdot C_r$ when $n=5$ is

The coefficient of $x^{12}$ in the expansion of $\left(x^2+2 x+2\right)^8$ is

Numerically greatest term in the expansion of $(2 x-3 y)^n$ when $x=\frac{7}{2}, y=\frac{3}{7}$ and $n=13$ is

If $C_0, C_1, C_2, \ldots, C_8$ are the binomial coefficients in the expansion of $(1+x)^8$, then $\sum\limits_{r = 1}^8 {} r^3 \frac{C_r}{C_{r-1}}=$

The constant term in the expansion of $\left(1+\frac{1}{x}\right)^{20}\left(30 x(1+x)^{29}+(1+x)^{30}\right)$ is

When $|x|>3$, then coefficient of $\frac{1}{x^n}$ in the expansion of $x^{3 / 2}(3+x)^{1 / 2}$ is

If the coefficient of 3rd term from the beginning in the expansion of $\left(a x^2-\frac{8}{b x}\right)^9$ is equal to the coefficient of 3rd term from the end in the expansion of $\left(a x-\frac{2}{b x^2}\right)^9$, then the relation between $a$ and $b$ is

If $X \sim B(7, P)$ is a binomial variate and $P(X=3)=P(X=5)$, then $P=$

The term independent of $x$ in the expansion of $\left(1-3 x+2 x^3\right)\left(\frac{3 x^2}{2}-\frac{1}{3 x}\right)^9$ is

If $\sum_{r=0}^{20}{ }^{20+r} C_r=\frac{p}{q}{ }^{40} C_{20}$ and GCD of $(p, q)=1$, then $p^2-q^2=$

If $x=\frac{2 \cdot 5}{2!3}+\frac{2 \cdot 5 \cdot 7}{3!3^2}+\frac{2 \cdot 5 \cdot 7 \cdot 9}{4!3^3}+\ldots$, then $x^2+8 x+8=$

If the coefficient of $x^4$ in the expansion of $\frac{x}{(x-1)^2(x-2)}$ is $\frac{m}{n}$ and $|m|,|n|$ are coprimes, then $\sqrt{|m+n|}=$

If $(-c, c)$ is the set of all values of $x$ for which the expansion of $(7-5 x)^{\frac{-2}{3}}$ is valid, then $5 c+7=$

If $n$ is a positive integer and $f(n)$ is the coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$, then $f(2023)=$

If $y=\frac{3}{4}+\frac{3 \cdot 5}{4 \cdot 8}+\frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12}+\ldots$ to $\infty$, then

The numerically greatest term in the binomial expansion of $(2 x-3 y)^5$, when $x=\frac{3}{2}$ and $y=\frac{2}{3}$ is

If $\frac{2 x^3+3 x^2+3 x+5}{\left(x^2+1\right)\left(x^2+2\right)}$ is expanded in terms of the powers of $x$, then the coefficient of $x^5$ is

In the expansion of $(x-2 y+3 z)^5$, if the total number of terms is $p$ and the coefficient of $x^2 y z^2$ is $q$, then $\frac{q}{p}=$

Let $C_0, C_1, C_2, \ldots, C_n$ be the binomial coefficients in the expansion of $(1+x)^n$. If $S_{n+1}=5 \cdot C_0+8 \cdot C_1+11 \cdot C_2+\ldots(n+1)$ terms, then $S_{11}=$

If $|x|$ is so small that $x^3$ and higher powers of $x$ can be neglected, then an approximate value of $\frac{1}{\sqrt{4-x}(2+x)^3}$ is

The number of integral terms in the expansion of $(\sqrt{3}+\sqrt[8]{5})^{256}$ is

The expansion of $\left(1+x+x^2\right)^{-3 / 2}$ in powers of $x$ is valid, if

If $(1+x)^n=c_0+c_1 x+c_2 x^2+\ldots \ldots+c_n x^n$ for $n \in N$, then $c_0+\frac{c_1}{2}+\frac{c_2}{3}+\ldots \ldots+\frac{c_n}{n+1}=$

Numerically greatest term in the expansion of $(2 x-3 y)^{11}$ when $x=\frac{1}{3}$ and $y=\frac{1}{2}$ is

$\frac{1}{8}-\frac{7}{8 \cdot 12}+\frac{7 \cdot 10}{8 \cdot 12 \cdot 16}-\ldots=$

The expansion of $(a+x)^n$ contains 15 terms. When $x=1$ the ratio of the neighbouring terms to the middle term in this expansion is 16 . Then, the positive integral value of ' $a$ ' is

If $k$ is the coefficient of $x^5$ in the expansion of $\left(2 x^2-\frac{1}{3 x^3}\right)^5$, then $\frac{3 k}{2}=$

If the 4 th term in the expansion of $\left(\frac{x}{2}-\frac{2 y}{3}\right)^6$ is -20, then $x y=$

18. If $L$ and $M$ are respectively the coefficient of $x^{-7}$ in $\left(a x+\frac{b}{x^2}\right)^{11}$ and the coefficient of $x^7$ in $\left(b x^2+\frac{a}{x^2}\right)^{11}$, then $L+M=$

If ${ }^n C_0,{ }^n C_1,{ }^n C_2, \ldots,{ }^n C_n$ respectively are the binomial coefficients in the expansion of $(1+x)^n$, then when $n=10, \sum_{r=1}^{10}{ }^n C_r \cdot r(r-4)=$

If sum of the coefficients of $x^r(r=0,1,2, \ldots, 2 n)$ in the expansion of $\left(1+3 x-2 x^2\right)^n$ is 128 , then $\sum_{r=1}^{2 n} r \frac{(2 n)_{C_r}}{(2 n)_{C_{r-1}}}=$

The approximate value of $\left(3 \sqrt{126}+\sin 61^{\circ}\right)$ correct to three decimal places, obtained by taking $1^{\circ}=0.0174$ radians, is

If $x$ is so small that all terms containing $x^2$ and higher powers of $x$ can be neglected, then the approximate value of $\frac{\left(1+\frac{2 x}{3}\right)^{-4}(4+5 x)^{1 / 2}}{(9+x)^{3 / 2}}$, when $x=\frac{6}{371}$, is

The sum of the coefficients of $x^{-3 / 2}$ and $x^3$ in the expansion of $\sqrt{3+x}+\sqrt{5+x}$ when $3 < x< 5$, is

If the 9th and 10th terms are the numerically greatest terms in the expansion of $(5 x-6 y)^n$ when $x=2 / 5$ and $y=1 / 2$, then the absolute value of the middle terms of that expansion is

$ 1-\frac{3}{16}+\frac{1 \cdot 4}{1 \cdot 2}\left(\frac{3}{16}\right)^2-\frac{1 \cdot 4 \cdot 7}{1 \cdot 2 \cdot 3}\left(\frac{3}{16}\right)^3+\ldots $

Let $x \in \mathbf{R}$ be so small that the powers of $x$ beyond two are insignificant and negligibly small. For such $x$, if $(1-x)^3(2+x)^6$ is approximated by $a+b x+c x^2$, then $a+b+c=$

For $0 < x < 1$, the expansion of $\left(1+\frac{1}{x}\right)^{\frac{1}{2}}$ is