Basic Algebra

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ______.

Let $f,g:\mathbf N-\{1\}\to\mathbf N$ be functions defined by $f(a)=\alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^\alpha$ divides $a$, and $g(a)=a+1$, for all $a\in\mathbf N-\{1\}$. Then, the function $f+g$ is

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

Let $f(x)=2x^2-x-1$ and $S=\{n\in\mathbf Z:|f(n)|\le800\}$. Then, the value of $\displaystyle\sum_{n\in S}f(n)$ is equal to ______.

The domain of the function $f(x)=\sin^{-1}\left[2x^2-3\right]+\log_2\left(\log_{\frac12}\left(x^2-5x+5\right)\right)$, where $[t]$ is the greatest integer function, is :

If $\alpha$, $\beta$ are the roots of the equation

$$x^2-\left(5+3^{\sqrt{\log_35}}-5^{\sqrt{\log_53}}\right)x+3\left(3^{(\log_35)^{\frac13}}-5^{(\log_53)^{\frac23}}-1\right)=0,$$

then the equation, whose roots are $\alpha+\dfrac1\beta$ and $\beta+\dfrac1\alpha$, is :

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

Let $S=\left\{x\in[-6,3]-\{-2,2\}:\dfrac{|x+3|-1}{|x|-2}\ge0\right\}$ and $T=\left\{x\in\mathbf Z:x^2-7|x|+9\le0\right\}$. Then the number of elements in $S\cap T$ is :

Let S be the set of all $a\in\mathbf R$ for which the angle between the vectors $\vec u=a(\log_eb)\hat i-6\hat j+3\hat k$ and $\vec v=(\log_eb)\hat i+2\hat j+2a(\log_eb)\hat k$, $(b>1)$ is acute. Then S is equal to :

Let S = {1, 2, 3, ..., 2022}. Then the probability, that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is :

Let p and p + 2 be prime numbers and let

$$\Delta=\begin{vmatrix}p! & (p+1)! & (p+2)!\\ (p+1)! & (p+2)! & (p+3)!\\ (p+2)! & (p+3)! & (p+4)!\end{vmatrix}$$

Then the sum of the maximum values of α and β, such that $p^\alpha$ and $(p+2)^\beta$ divide Δ, is _______.

Let S = {4, 6, 9} and T = {9, 10, 11, ..., 1000}. If A = {$a_1+a_2+\dots+a_k$ : k ∈ N, $a_1,a_2,a_3,\dots,a_k$ ∈ S}, then the sum of all the elements in the set T − A is equal to _______.

If [t] denotes the greatest integer ≤ t, then the value of $\displaystyle\int_0^1\left[2x-\left|3x^2-5x+2\right|+1\right]dx$ is :

For |α| ≥ 1, let $5^{4-2\alpha}$, 63, $5^{2\alpha-1}$ be in A.P. with the common difference d. Let S$_n$ denote the sum of the first n terms of an A.P., whose first term is α$^2$ and the common difference is d. If S$_{30}$ − S$_{15}$ = 30k, then k is equal to

The number of solutions of the equation $|x^2+3x+2|+|x+5|-2=0$, x ∈ ℝ, is _______.

If $p_1$ and $p_2$, respectively, are the second largest and the fourth largest prime numbers in the factorization of the coefficient of $x^{30}$ in the expression of $\left(x^2+2+\dfrac1{x^2}\right)^{-5}(1+x^2)^{40}$, then $p_1+p_2$ is equal to _______

For $x \in \mathbb R$, the number of real roots of the equation
$$3x^2 - 4|x^2 - 1| + x - 1 = 0$$
is ____.

For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. If
$$I = \int_0^{10}\left[\sqrt{\frac{10x}{x + 1}}\right]dx,$$
then the value of $9I$ is ____.

Let $m$ be the minimum possible value of $\log_3\left(3^{y_1} + 3^{y_2} + 3^{y_3}\right)$, where $y_1, y_2, y_3$ are real numbers for which $y_1 + y_2 + y_3 = 9$. Let $M$ be the maximum possible value of $(\log_3 x_1 + \log_3 x_2 + \log_3 x_3)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1 + x_2 + x_3 = 9$. Then the value of $\log_2(m^3) + \log_3(M^2)$ is ______

Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If

$$AP(1;3)\cap AP(2;5)\cap AP(3;7) = AP(a;d)$$

then $a + d$ equals ______