Binomial Theorem

Let $\mathbb R$ denote the set of all real numbers. Consider the polynomial function $f:\mathbb R\to\mathbb R$ defined by
$$f(x)=\frac{d^{10}}{dx^{10}}\left(\left(x^2-1\right)^{10}\right),\qquad\text{for all }x\in\mathbb R.$$
Here $\dfrac{d^{10}}{dx^{10}}\left(\left(x^2-1\right)^{10}\right)$ is the $10^{\text{th}}$ order derivative of the function $\left(x^2-1\right)^{10}$.

Then which of the following statements is (are) TRUE ?

The number of elements in the set $S=\left\{(r,k):k\in\mathbb Z\ \text{and}\ {}^{36}C_{r+1}=\dfrac{6\left({}^{35}C_r\right)}{\left(k^2-3\right)}\right\}$, is :

If for $3\le r\le30$, $\left({}^{30}C_{30-r}\right)+3\left({}^{30}C_{31-r}\right)+3\left({}^{30}C_{32-r}\right)+\left({}^{30}C_{33-r}\right)={}^mC_r$, then $m$ equals :

Let the smallest value of $k\in\mathbb N$, for which the coefficient of $x^3$ in $(1+x)^3+(1+x)^4+(1+x)^5+\dots+(1+x)^{99}+(1+kx)^{100}$, $x\ne0$, is $\left(43n+\dfrac{101}4\right)\left({}^{100}C_3\right)$ for some $n\in\mathbb N$, be $p$. Then the value of $p+n$ is:

In the expansion of $\left(9x-\dfrac1{3\sqrt x}\right)^{18}$, $x>0$, if the term independent of $x$ is $(221)k$, then $k$ is equal to:

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of $\left(\dfrac1{x^3}-x^4\right)^n$, $x\ne0$, is zero, then the value of $n$ is ________.

The coefficient of $x^2$ in the expansion of $\left(2x^2+\dfrac1x\right)^{10}$, $x\ne0$, is :

A variable X takes values $0,0,2,6,12,20,\dots,n(n-1)$ with frequencies ${}^nC_0,{}^nC_1,{}^nC_2,{}^nC_3,{}^nC_4,{}^nC_5,\dots,{}^nC_n$, respectively. If the mean of this data is 60, then its median is :

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}$, $\alpha\ne0$, are equal, then the value of $\alpha$ is:

Let $A=\begin{bmatrix}1&0&0\\3&1&0\\9&3&1\end{bmatrix}$ and $B=[b_{ij}]$, $1\le i,j\le3$. If $B=A^{99}-I$, then the value of $\dfrac{b_{31}-b_{21}}{b_{32}}$ is :

If $\left(1-x^3\right)^{10}=\displaystyle\sum_{r=0}^{10}a_rx^r(1-x)^{30-2r}$, then $\dfrac{9a_9}{a_{10}}$ is equal to ________.

If $26\left(\dfrac{2^3}3\left({}^{12}C_2\right)+\dfrac{2^5}5\left({}^{12}C_4\right)+\dfrac{2^7}7\left({}^{12}C_6\right)+\dots+\dfrac{2^{13}}{13}\left({}^{12}C_{12}\right)\right)=3^{13}-\alpha$, then $\alpha$ is equal to :

Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1, 2, 3\}$. Define the functions $f : \mathbb{R} \to \mathbb{R}$, $g : \mathbb{R} \to \mathbb{R}$, and $h : \mathbb{R} \to \mathbb{R}$ by

$$\begin{aligned} f(x) &= a_1 + 10x + a_2x^2 + a_3x^3 + x^4,\\ g(x) &= b_1 + 3x + b_2x^2 + b_3x^3 + x^4,\\ h(x) &= f(x+1) - g(x+2). \end{aligned}$$

If $f(x) \neq g(x)$ for every $x \in \mathbb{R}$, then the coefficient of $x^3$ in $h(x)$ is

Let $a_0, a_1, \ldots, a_{23}$ be real numbers such that

$$\left(1 + \frac25x\right)^{23} = \sum_{i=0}^{23}a_ix^i$$

for every real number $x$. Let $a_r$ be the largest among the numbers $a_j$ for $0 \le j \le 23$. Then the value of $r$ is _____.

Let $S = \left\{a + b\sqrt{2} : a, b \in \mathbb{Z}\right\}$, $T_1 = \left\{\left(-1+\sqrt{2}\right)^n : n \in \mathbb{N}\right\}$, and $T_2 = \left\{\left(1+\sqrt{2}\right)^n : n \in \mathbb{N}\right\}$. Then which of the following statements is (are) TRUE?

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^5$ in the expansion of $\left(ax^2 + \dfrac{70}{27bx}\right)^4$ is equal to the coefficient of $x^{-5}$ in the expansion of $\left(ax - \dfrac{1}{bx^2}\right)^7$, then the value of $2b$ is _____.

The remainder when $3^{2022}$ is divided by 5 is :

The remainder on dividing $1+3+3^2+3^3+\dots+3^{2021}$ by 50 is ______.

Let $C_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x)^{10}$. If for $\alpha,\beta\in\mathbb R$,
$$C_1+3\cdot2\,C_2+5\cdot3\,C_3+\dots\text{ upto 10 terms}=\frac{\alpha\times2^{11}}{2^\beta-1}\left(C_0+\frac{C_1}2+\frac{C_2}3+\dots\text{ upto 10 terms}\right)$$
then the value of $\alpha+\beta$ is equal to ______.

The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+\dots+x^{500}$, $x>0$, is