If $y=a \cos (\log x)+b \sin (\log x)$,where $a, b$ are parameters,then $x^2 y^{\prime \prime}+x y^{\prime}$ is equal to

The value of $\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}$ is :

If $\alpha, \beta, \gamma$ are the roots of $2x^3 - 2x - 1 = 0$,then $(\Sigma \alpha \beta)^2$ is equal to

Read the following statements carefully and select the correct statements.
$(1)$ The wheat variety $Atlas-66$ has high protein content.
$(2)$ $SCP$ is an alternative source of protein for human and animal nutrition.
$(3)$ Plants developed through micropropagation are genetically different from the parent plant.
$(4)$ Semi-dwarf rice varieties were developed from $IR-8$ and $Taichung Native-1$.

If the slope of one of the lines in the pair of lines $8x^2 + axy + y^2 = 0$ is thrice the slope of the other line,then $a =$

The angular momentum of an electron depends on