A rod of length $10\ meter$ at $0\,^oC$ having expansion coefficient $\alpha  = (2x^2 + 1) \times  10^{-6}\,C^{-1}$ where $x$ is the distance from one end of rod. The length of rod at $10\,^oC$ is

A copper rod of length $l_1$ and an iron rod of length $l_2$ are always maintained at the same common temperature $T$. If the difference $(l_2 -l_1)$ is $15\,cm$ and is independent of the value of $T,$ the $l_1$ and $l_2$ have the values (given the linear coefficient of expansion for copper and iron are $2.0 \times 10^{-6}\,C^{-1}$ and $1.0\times10^{-6}\,C^{ -1}$  respectively)

A surveyor's $30$-$m$ steel tape is correct at some temperutre. On a hot day the tape has expanded to $30.01$ $m$. On that day, the tape indicates a distance of $15.52$ $m$ between two points. The true distance between these points is :-

The diagram below shows the change in the length $X$ of a thin uniform wire caused by the application of stress $F$ at two different temperatures $T_1$ and $T_2$. The variation shown suggests that

A bakelite beaker has volume capacity of $500\, cc$ at $30^{\circ} C$. When it is partially filled with $V _{ m }$ volume (at $30^{\circ}$ ) of mercury, it is found that the unfilled volume of the beaker remains constant as temperature is varied. If $\gamma_{\text {(beaker) }}=6 \times 10^{-6}{ }^{\circ} C ^{-1}$ and $\gamma_{(\text {mercury })}=1.5 \times 10^{-4}{ }^{\circ} C ^{-1},$ where $\gamma$ is the coefficient of volume expansion, then $V _{ m }($in $cc )$ is close to

The coefficients of thermal expansion of steel and a metal $X$ are respectively $12 × 10^{-6}$ and $2 × 10^{-6} per^o C$. At $40^o C$, the side of a cube of metal $X$ was measured using a steel vernier callipers. The reading was $100 \,\,mm$.Assuming that the calibration of the vernier was done at $0^o C$, then the actual length of the side of the cube at $0^o C$ will be