The number of roots of the equation $\log ( - 2x)$ $ = 2\log (x + 1)$ are

Let $a, b, c, d$ be real numbers such that $|a-b|=2$, $|b-c|=3,|c-d|=4$. Then, the sum of all possible values of $|a-d|$ is

Let $p(x)=a_0+a_1 x+\ldots+a_n x^n$ be a non-zero polynomial with integer coefficients. If $p(\sqrt{2}+\sqrt{3}+\sqrt{6})=0$, then the smallest possible value of $n$ is

If the product of roots of the equation ${x^2} - 3kx + 2{e^{2\log k}} - 1 = 0$ is $7$, then its roots will real when

Exact set of values of $a$ for which ${x^3}(x + 1) = 2(x + a)(x + 2a)$ is having four real solutions is

The maximum value $M$ of $3^x+5^x-9^x+15^x-25^x$, as $x$ varies over reals, satisfies