What is $DDT$?

$A$ speaker emits a sound wave of frequency $f_0$. When it moves towards a stationary observer with speed $u$,the observer measures a frequency $f_1$. If the speaker is stationary,and the observer moves towards it with speed $u$,the measured frequency is $f_2$. Then:

$\cos \left[\cos ^{-1}\left(-\frac{1}{7}\right)+\sin ^{-1}\left(-\frac{1}{7}\right)\right]$ is equal to

One and a half mole of oxygen combine with aluminium to form $Al_2O_3$,then the weight of aluminium metal used in this reaction is ............. $g$ (Atomic weight of $Al = 27$)

Find the values of $x$,if: $(i)$ $\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|$ (ii) $\left|\begin{array}{ll}2 & 3 \\ 4 & 5\end{array}\right|=\left|\begin{array}{ll}x & 3 \\ 2 x & 5\end{array}\right|$

$A$ beam of light $AO$ is incident on a glass slab $(\mu = 1.54)$ in a direction as shown in the figure. The reflected ray $OB$ is passed through a Nicol prism. On viewing through the Nicol prism,we find on rotating the prism that: