The population $p(t)$ at time $t$ of a certain mouse species follows the differential equation $\frac{d p(t)}{d t}=0.5 p(t)-450$. If $p(0)=850$, then the time at which the population becomes zero is

If $(-2, 6)$ is the image of the point $(4, 2)$ with respect to the line $L = 0$,then $L$ is equal to

$\sec ^2(\tan ^{-1} 2) + \operatorname{cosec}^2(\cot ^{-1} 3)$ is equal to

The ratio of the specific heat capacities of a gas is $1.5$. When the gas undergoes an adiabatic process, its volume is doubled and pressure becomes $P_1$. When the gas undergoes an isothermal process, its volume is doubled and pressure becomes $P_2$. If $P_1 = P_2$, the ratio of the initial pressures of the gas when it undergoes adiabatic and isothermal processes is

If the kinetic energy of a body is directly proportional to time $t$,the magnitude of force acting on the body is
$(i)$ directly proportional to $\sqrt{t}$
$(ii)$ inversely proportional to $\sqrt{t}$
$(iii)$ directly proportional to the speed of the body
$(iv)$ inversely proportional to the speed of the body

If $\frac{1+\sqrt{3}i}{2}$ is a root of the equation $x^4-x^2+x-1=0$,then its real roots are: