Basic Algebra

If the domain of the function $f(x)=\sqrt{\log_{(0.6)}\left(\left|\dfrac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty)$, then the value of $a+b+c+d+e$ is ________.

If the set of all solutions of $|x^2+x-9|=|x|+|x^2-9|$ is $[\alpha,\beta]\cup[\gamma,\infty)$, then $(\alpha^2+\beta^2+\gamma^2)$ is equal to:

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:

Let $\alpha=\dfrac14+\dfrac18+\dfrac1{16}+\dots\infty$ and $\beta=\dfrac13+\dfrac19+\dfrac1{27}+\dots\infty$. Then the value of $(0.2)^{\log_{\sqrt5}(\alpha)}+(0.04)^{\log_5(\beta)}$ is equal to:

The sum of squares of all the real solutions of the equation $\log_{(x+1)}\left(2x^2+5x+3\right)=4-\log_{(2x+3)}\left(x^2+2x+1\right)$ is equal to ________.

Let $f$ be a polynomial function such that
$$\log_2(f(x))=\left(\log_2\left(2+\frac23+\frac29+\dots\infty\right)\right)\cdot\log_3\left(1+\frac{f(x)}{f(1/x)}\right),$$
$x>0$ and $f(6)=37$. Then $\displaystyle\sum_{n=1}^{10}f(n)$ 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?

Let $a = 3\sqrt{2}$ and $b = \dfrac{1}{5^{1/6}\sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that

$$3x + 2y = \log_a (18)^{\frac{5}{4}} \quad\text{and}$$
$$2x - y = \log_b\left(\sqrt{1080}\right),$$

then $4x + 5y$ is equal to _____.

Let $\alpha$ be a positive real number. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : (\alpha, \infty) \to \mathbb{R}$ be the functions defined by
$$f(x) = \sin\left(\frac{\pi x}{12}\right) \quad\text{and}\quad g(x) = \frac{2\log_e\left(\sqrt x - \sqrt\alpha\right)}{\log_e\left(e^{\sqrt x} - e^{\sqrt\alpha}\right)}.$$
Then the value of $\displaystyle\lim_{x\to\alpha^+} f(g(x))$ is _____.

The greatest integer less than or equal to
$$\int_1^2 \log_2(x^3+1)\,dx + \int_1^{\log_2 9} (2^x - 1)^{\frac13}\,dx$$
is _____.

The product of all positive real values of $x$ satisfying the equation
$$x^{\left(16(\log_5 x)^3 - 68\log_5 x\right)} = 5^{-16}$$
is _____.

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

The sum of absolute maximum and absolute minimum values of the function $f(x)=|2x^2+3x-2|+\sin x\cos x$ in the interval $[0,1]$ is :

The domain of the function $f(x)=\dfrac{\cos^{-1}\left(\dfrac{x^2-5x+6}{x^2-9}\right)}{\log_e\left(x^2-3x+2\right)}$ is :

The sum of all the real roots of the equation $\left(e^{2x}-4\right)\left(6e^{2x}-5e^x+1\right)=0$ is

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is ______.

If $\dfrac1{2\cdot3^{10}}+\dfrac1{2^2\cdot3^9}+\dots+\dfrac1{2^{10}\cdot3}=\dfrac K{2^{10}\cdot3^{10}}$, then the remainder when K is divided by 6 is :

Let $f(x)$ be a polynomial function such that $f(x)+f'(x)+f''(x)=x^5+64$. Then, the value of $\displaystyle\lim_{x\to1}\frac{f(x)}{x-1}$ is equal to :

Let $A=\{x\in\mathbb R:|x+1|<2\}$ and $B=\{x\in\mathbb R:|x-1|\ge2\}$. Then which one of the following statements is NOT true ?