The total revenue in Rupees received from the sale of $x$ units of a product is given by $R(x) = 3x^2 + 36x + 5$. The marginal revenue,when $x = 15$,is

For a crystal,the angle of diffraction $(2 \theta)$ is $90^{\circ}$ and the second order line has a $d$ value of $2.28 \ \mathring{A}$. The wavelength (in $\mathring{A}$) of $X$-rays used for Bragg's diffraction is

The value of $\int_0^1 \tan ^{-1}\left(\frac{2 x-1}{1+x-x^2}\right) d x$ is

An auditorium has a volume of $10^5 \ m^3$ and a total absorption area of $2 \times 10^4 \ m^2$. Its average absorption coefficient is $0.2$. The reverberation time of the auditorium in seconds is:

$A$ thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and is suddenly brought to rest. Assuming no heat is lost to the surroundings,its temperature increases by

The major product of the reaction is: