If $a = \log_{24} 12$,$b = \log_{36} 24$,and $c = \log_{48} 36$,then $1 + abc$ is equal to:

$A$ satellite $S$ moves around a planet $P$ in an elliptical orbit as shown in the figure. The ratio of the speed of the satellite at point $a$ to that at point $b$ is

$P$ is a point on the segment joining the feet of two vertical poles of heights $a$ and $b$. The angles of elevation of the tops of the poles from $P$ are $45^{\circ}$ each. Then,the square of the distance between the tops of the poles is

The sum of two numbers is $20$. If the product of the square of one number and the cube of the other is maximum,then the numbers are:

$A$ radioactive material decays by simultaneous emission of two particles with respective half-lives $1620 \ years$ and $810 \ years$. The time (in years) after which one-fourth of the material remains is

The hydride ion $H^-$ is a stronger base than the hydroxide ion $OH^-$. If sodium hydride $(NaH)$ is dissolved in water,which of the following reactions will occur?