If the middle term of $\left(\dfrac 1x +x\sin x\right)^{10}$ is equal to $7\dfrac 78$, then the principal value of $x$ is: ;

Solution:

Middle term of the expansion $\left(\cfrac1x+x\sin x\right)^{10}$ is $T_6.$

$T_6={}^{10}C_5\times\left(\cfrac1x\right)^5\times(x\sin x)^5=7\cfrac78$

or, ${}^{10}C_5\times \sin^5x=\cfrac{63}8$

or, $252\times \sin^5x=\cfrac{63}8$

or, $\sin^5x=\cfrac{1}{32}$

or, $\sin x=\cfrac 12$

or, $x=30^\circ$

Hence, B is the correct option.