Binomial Theorem · Mathematics · AP EAPCET

In the binomial expansion of $(p-q)^{14}$, if the sum of 7th term and 8 th term is zero, then $\frac{p+q}{p-q}=$

The numerically greatest term in the expansion of $(x+3 y)^{13}$, when $x=\frac{1}{2}$ and $y=\frac{1}{3}$ is

The remainder obtained when $(2 m+1)^{2 n}(m, n \in N)$ is divided by 8 is

$$ \sum_{r=1}^{15} r^2\left(\frac{{ }^{15} C_r}{{ }^{15} C_{r-1}}\right)= $$

$$ \frac{1}{81^n}-{ }^{2 n} C_1 \frac{10}{81^n}+{ }^{2 n} C_2 \frac{10^2}{81^n}-\ldots+\frac{10^{2 n}}{81^n}= $$

If $x$ is positive real number and the first negative term in the expansion of $(1+x)^{\frac{27}{5}}$ is $t_k$, then $k=$

The coefficient of $x^{10}$ in the expansion of $\left(x+\frac{2}{x}-5\right)^{12}$ is

Let $S_1=\sum\limits_{j=1}^{10} j(j-1) \cdot{ }^{10} C_j, S_2=\sum\limits_{j=1}^{10} j \cdot{ }^{10} C_j$ and

$$ S_3=\sum\limits_{j=1}^{10} j^2 \cdot{ }^{10} C_j $$

Assertion (A) $S_3=55 \times 2^9$

Reason (R) $S_1=90 \times 2^8$ and $S_2=10 \times 2^8$

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

Sum of the coefficients of $x^4$ and $x^6$ in the expansion of $\left(1+x-x^2\right)^6$ is

If $11^{12}-11^2=k\left(5 \times 10^9+6 \times 10^9+33 \times 10^8\right. \left.+110 \times 10^7+\ldots+33\right)$, then $k=$

If $C_0, C_2, \ldots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$, then

$$ \left(C_0+C_1\right)-\left(C_2+C_3\right)+\left(C_4+C_5\right)-\left(C_6+C_7\right)+\ldots= $$

The mean and variance of a binomial distribution are $x$ and 5 respectively. If $x$ is an integer, then the possible values for $x$ are

If the coefficients of $x^{10}$ and $x^{11}$ in the expansion of $\left(1+\alpha x+\beta x^2\right)(1+x)^{11}$ are 396 and 144 respectively, then $\alpha^2+\beta^2=$

If $-\frac{2}{3} < x < \frac{2}{3}$, then the value of the 5 th term in the expansion of $\frac{1}{\sqrt[3]{2-3 x}}$ when $x=\frac{1}{2}$ is

The terms containing $x^r y^s$ (for certain $r$ and $s$ ) are present in both the expansions of $\left(x+y^2\right)^{13}$ and $\left(x^2+y\right)^{14}$. If $\alpha$ is the number of such terms, then the $\operatorname{sum} \alpha \sum_{r, s}(r+s)=$

The coefficient of $x^3$ in the power series expansion of $\frac{1+4 x-3 x^2}{(1+3 x)^3}$ is

If $k$ is a positive integer and $10^k$ is a divisor of the number $9^{11}+11^9$, then the greatest value of $k$ is

Coefficient of $x^2$ in the expansion of $\left(x^2+x-2\right)^5$ is

If $P_n$ denotes the product of the binomial coefficients in the expansion of $(1+x)^n$, then $\frac{P_{n+1}}{P_n}=$

The coefficient of $x^3$ in the expansion of $\frac{x^4+1}{\left(x^2+1\right)(x-1)}$ when it is expressed in terms of positive integral powers of $x$, is

If $(1+x)^n=\sum_{r=0}^n C, x^r$, then the value of $C_0+\left(C_0+C_1\right)+\left(C_0+C_1+C_2\right)+\ldots+ \left(C_0+C_1+C_2+\ldots+C_n\right)$ is

If $x$ is so large that terms containing $x^{-3}, x^{-4}, x^{-5}, \ldots$ can be neglected, then the approximate value of $\left(\frac{3 x-5}{4 x^2+3}\right)^{-1 / 5}$ is

If the coefficients of $r$ th, $(r+1)$ th and $(r+2)$ th terms in the expansion of $(1+x)^n$ are in the ratio of $4: 15: 42$, then $n-r$ is equal to

If the coefficients of $(2 r+6)$ th and $(r-1)$ th terms in the expansion of $(1+x)^{21}$ are equal, then the value of $r$ is equal to

The least value of $$n$$ so that $${ }^{(n-1)} C_3+{ }^{(n-1)} C_4>{ }^n C_3$$