A cup of coffee cools from $90^{\circ}\text{C}$ $\mathrm{to}$ $80^{\circ}\text{C}$ in $t$ minutes, when the room temperature is $20^{\circ}\text{C}.$ The time taken by a similar cup of coffee to cool from $80^{\circ}\text{C}$ $\mathrm{to}$ $60^{\circ}\text{C}$ at room temperature same at $20^{\circ}\text{C}$ is:
1. $\frac{10}{13}t$ 
2. $\frac{5}{13}t$
3. $\frac{13}{10}t$
4. $\frac{13}{5}t$

The quantities of heat required to raise the temperature of two solid copper spheres of radii $r_1$ and $r_2$ $(r_1=1.5~r_2)$ through $1~\text{K}$ are in the ratio:
1. $\dfrac{9}{4}$

2. $\dfrac{3}{2}$

3. $\dfrac{5}{3}$

4. $\dfrac{27}{8}$

A copper rod of $88$ cm and an aluminium rod of an unknown length have an equal increase in their lengths independent of an increase in temperature. The length of the aluminium rod is:
$\left(\alpha_{Cu}= 1.7\times10^{-5}~\text{K}^{-1}~\text{and}~\alpha_{Al}= 2.2\times10^{-5}~\text{K}^{-1}\right)$
1. $68~\text{cm}$
2. $6.8~\text{cm}$
3. $113.9~\text{cm}$
4. $88~\text{cm}$

An object kept in a large room having an air temperature of $25^\circ \text{C}$ takes $12$ minutes to cool from $80^\circ \text{C}$ to $70^\circ \text{C}.$ The time taken to cool for the same object from $70^\circ \text{C}$ to $60^\circ \text{C}$ would be nearly:
1. $10$ min
2. $12$ min
3. $20$ min
4. $15$ min

A deep rectangular pond of surface area $A$, containing water (density = $\rho,$ specific heat capacity = $s$), is located in a region where the outside air temperature is at a steady value of $-26^{\circ}\text{C}$. The thickness of the ice layer in this pond at a certain instant is $x$. Taking the thermal conductivity of ice as $k$, and its specific latent heat of fusion as $L$, the rate of increase of the thickness of the ice layer, at this instant, would be given by:
1. $\dfrac{26k}{x\rho L-4s}$
2. $\dfrac{26k}{x^2\rho L}$
3. $\dfrac{26k}{x\rho L}$
4. $\dfrac{26k}{x\rho L+4s}$

The power radiated by a black body is $P$ and it radiates maximum energy at wavelength $\lambda_0.$ Temperature of the black body is now changed so that it radiates maximum energy at the wavelength $\dfrac{3}{4}\lambda_0.$ The power radiated by it now becomes $nP.$ The value of $n$ is: