The minimum value of the function \(\mathrm{f}(x)=2 x^3-15 x^2+36 x-48 \quad\) on the set \(\mathrm{A}=\left\{x \mid x^2+20 \leqslant 9 x\right\}\) is

A vector \(\overline{\mathrm{a}}\) has components 1 and \(2 \mathrm{p}\) with respect to a rectangular Cartesian system. This system is rotated through a certain angle about origin in the counter clock wise sense. If, with respect to the new system, \(\bar{a}\) has components 1 and \((p+1)\), then

The equation of the circle which passes through the centre of the circle \(x^2+y^2+8 x+10 y-7=0\) and concentric which the circle \(2 x^2+2 y^2-8 x-12 y-9=0\) is

Let \(\quad \overline{\mathrm{a}}=\alpha \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}, \quad \overline{\mathrm{b}}=3 \hat{\mathrm{i}}-\beta \hat{\mathrm{j}}+4 \hat{\mathrm{k}} \quad\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}\), where \(\alpha, \beta \in \mathbb{R}\), be three vectors. If the projection at \(\overline{\mathrm{a}}\) on \(\overline{\mathrm{c}}\) is \(\frac{10}{3}\) and \(\overline{\mathrm{b}} \times \overline{\mathrm{c}}=-6 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+7 \hat{\mathrm{k}}\), then the value of \(\alpha^2+\beta^2-\alpha \beta\) is equal to

\(\int_0^{\frac{\pi}{2}} \frac{\mathrm{~d} x}{1+(\cot x)^{101}}=\)

Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm ? ( 1 decimeter \(=10 \mathrm{~cm}\) )

The value of \(\int_{0}^{\pi} x \sin ^{3} x d x\) is

Mixture of sodium chloride and ammonium chloride is separated by

The ________ is a primary constriction.

A vessel completely filled with water has two holes 'P' and 'Q' at depths ' 2 h ' and ' 8 h ' from the top respectively. Hole ' P ' is square of side ' a ' and hole ' \(Q\) ' is a circle of radius ' \(r\) '. The water flowing out per second from both the holes is same, then side ' \(a\) ' of hole ' \(P\) ' is

Paul Ehrlich by Rivet Popper hypothesis has explained

500 gram of a diatomic gas is enclosed at a pressure of \(10^5 \mathrm{Nm}^{-2}\). The density of the gas is \(5 \mathrm{kgm}^{-3}\). The energy of one mole of the gas due to its thermal motion is [consider the gas molecule as a rigid rotator]

A metal ball of radius \(9 \times 10^{-4} \mathrm{~m}\) and density \(10^4 \mathrm{~kg} / \mathrm{m}^3\) falls freely under gravity through a distance 'h' and erfters a tank of water. Considering that the metal ball has constant velocity, the value of h is [coefficient of viscosity of water \(=8.1 \times 10^{-4} \mathrm{pa}-\mathrm{s}, \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) density of water \(\left.=10^3 \mathrm{~kg} / \mathrm{m}^3\right]\)