$A$ student records the initial length $l$,change in temperature $\Delta T$ and change in length $\Delta l$ of a rod as follows:

$S$.No.	$l (m)$	$\Delta T (^{\circ}C)$	$\Delta l (m)$
$(1)$	$2$	$10$	$4 \times 10^{-4}$
$(2)$	$1$	$10$	$4 \times 10^{-4}$
$(3)$	$2$	$20$	$2 \times 10^{-4}$
$(4)$	$3$	$10$	$6 \times 10^{-4}$

If the first observation is correct,what can you say about observations $(2)$,$(3)$ and $(4)$?

(N/A) The formula for linear thermal expansion is $\Delta l = \alpha l \Delta T$,where $\alpha$ is the coefficient of linear expansion.
From the first observation:
$\alpha = \frac{\Delta l}{l \Delta T} = \frac{4 \times 10^{-4}}{2 \times 10} = 2 \times 10^{-5} \, ^{\circ}C^{-1}$.
Now,we check the other observations using $\alpha = 2 \times 10^{-5} \, ^{\circ}C^{-1}$:
For observation $(2)$:
$\Delta l = \alpha l \Delta T = (2 \times 10^{-5}) \times 1 \times 10 = 2 \times 10^{-4} \, m$.
Since the recorded value is $4 \times 10^{-4} \, m$,observation $(2)$ is incorrect.
For observation $(3)$:
$\Delta l = \alpha l \Delta T = (2 \times 10^{-5}) \times 2 \times 20 = 8 \times 10^{-4} \, m$.
Since the recorded value is $2 \times 10^{-4} \, m$,observation $(3)$ is incorrect.
For observation $(4)$:
$\Delta l = \alpha l \Delta T = (2 \times 10^{-5}) \times 3 \times 10 = 6 \times 10^{-4} \, m$.
Since the recorded value is $6 \times 10^{-4} \, m$,observation $(4)$ is correct.