In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency 500Hz is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length 50.7cm and 83.9 cm. Which of the following statements is (are) true?

let $n_1$  harmonic is corresponding to 50.7 cm &  $n_2$ harmonic is corresponding 
83.9 cm.
$\therefore \frac{\lambda }{2}=(83.9-50.7)$ cm                                      

 $\Rightarrow \frac{\lambda }{2}=33.2$cm
$\Rightarrow \lambda =66.4$ cm $\therefore Option \left(3\right) is correct$                                     

 $\therefore Fundamental frequency=\frac{\lambda }{4}=16.6 cm$

$\therefore Option \left(4\right) is incorrect$
Length corresponding to fundamental mode must be close to  $\frac{\lambda }{4}$ & 50.7 cm must be an odd multiple of this length $16.6\times 3=49.8$cm. therefore 50.7 is  $3^{rd}$ harmonic 
If end correction is e, then
$e+50.7=\frac{3\lambda }{4}$

$\Rightarrow e=-0.9cm$

$\text{End correction can never be negative}$ 

$\therefore \mathrm{option}(2) \mathrm{is} \mathrm{incorrect}$

Speed of sound,  $v=f\lambda$
$\therefore v=500\times 66.4$  cm/sec $=332.000$ m/s

 $\therefore Option \left(1\right) is correct$

therefore options (1), (3) are correct