Binomial expansion

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 ?

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 ________.

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

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 ______.

The remainder when $(2021)^{2023}$ is divided by 7 is :

Let $S=2+\dfrac67+\dfrac{12}{7^2}+\dfrac{20}{7^3}+\dfrac{30}{7^4}+\dots$. Then $4S$ is equal to

Let $A=\begin{pmatrix}2&-1\\0&2\end{pmatrix}$. If $B=I-{}^5C_1(\operatorname{adj}A)+{}^5C_2(\operatorname{adj}A)^2-\dots-{}^5C_5(\operatorname{adj}A)^5$, then the sum of all elements of the matrix B is

Let $n\ge5$ be an integer. If $9^n-8n-1=64\alpha$ and $6^n-5n-1=25\beta$, then $\alpha-\beta$ is equal to

If the coefficients of $x$ and $x^2$ in the expansion of $(1+x)^p(1-x)^q$, $p,q\le15$, are $-3$ and $-5$ respectively, then the coefficient of $x^3$ is equal to ______.

The remainder when $(2021)^{2022}+(2022)^{2021}$ is divided by 7 is

Let for the 9$^{\text{th}}$ term in the binomial expansion of $(3+6x)^n$, in the increasing powers of $6x$, to be the greatest for $x=\dfrac32$, the least value of n is $n_0$. If k is the ratio of the coefficient of $x^6$ to the coefficient of $x^3$, then $k+n_0$ is equal to :

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :

If $z=2+3i$, then $z^5+(\bar z)^5$ is equal to :

If $(27)^{999}$ is divided by 7, then the remainder is :

If the number of terms in the expansion of $\left(1-\dfrac2x+\dfrac4{x^2}\right)^n$, $x\ne0$, is 28, then the sum of the coefficients of all the terms in this expansion, is :

Coefficient of $x^{11}$ in the expansion of $(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$ is

The number of terms in the expansion of $(1+x)^{101}\left(1+x^2-x\right)^{100}$ in powers of $x$ is :