List of top Mathematics Questions on Binomial theorem asked in TS EAMCET

Let \([t]\) represent the greatest integer less than or equal to \(t\). If \[ x = (7\sqrt{5} + 15)^9 \quad \text{and} \quad y = (5\sqrt{7} + 13)^{11}, \] then \([x]\) and \([y]\) are:

Evaluate binomial sum involving $\sum C_r \frac{2^r}{r+1}$ for $n=10$.}

Approximate $(0.98)^{0.2}$ using binomial expansion.}

The term independent of \(x\) in expansion of \[ \left(\frac{\sqrt{x}}{2}-\frac{3}{x}\right)^{12} \] is

Coefficient of \(x^3\) in the expansion of \[ \frac{(1-2x^2)^{\frac13}}{(2+x)^{\frac12}} \] is

The term independent of \(x\) in expansion of \[ \left(\frac{\sqrt{x}}{2}-\frac{3}{x}\right)^{12} \] is

Let K be the number of rational terms in the expansion of $(\sqrt{2}+\sqrt[6]{3})^{6144}$. If the coefficient of $x^P (P \in N)$ in the expansion of $\frac{1}{(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})}$ is $a_P$, then $a_K - a_{K+1} - a_{K-1} =$

Numerically greatest term in the expansion of $(3x-4y)^{23}$ when $x=\frac{1}{6}$ and $y=\frac{1}{8}$ is

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

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

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

If the expression \( 5^{2n} - 48n + k \) is divisible by 24 for all \( n \in \mathbb{N} \), then the least positive integral value of k is

If $C_0, C_1, C_2, \dots, C_n$ are the binomial coefficients in the expansion of $(1+x)^n$ then $\sum_{r=1}^{n} \frac{r C_r}{C_{r-1}} =$

Numerically greatest term in the expansion of $(2x-3y)^n$ when $x=\frac{7}{5}, y=\frac{3}{7}$ and $n=13$ is