If \(\mathrm{A} \equiv(1,-1,0), \mathrm{B} \equiv(0,1,-1)\) and \(\mathrm{C} \equiv(-1,0,1)\), then the unit vector \(\overline{\mathrm{d}}\) such that \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{d}}\) are perpendiculars and \(\overline{\mathrm{b}}, \overline{\mathrm{c}}, \overline{\mathrm{d}}\) are coplanar is

If the vectors \(2 \hat{i}-3 \hat{j}+6 \hat{k}\) and \(\bar{b}\) are collinear and \(|\bar{b}|=14\), then \(\bar{b}\) has the value

The value of \(\tan \left(2 \tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)\) is

If \(\mathrm{f}(x)=\mathrm{e}^x, \mathrm{~g}(x)=\sin ^{-1} x\) and \(\mathrm{h}(x)=\mathrm{f}(\mathrm{g}(x))\), then \(\frac{\mathrm{h}^{\prime}(x)}{\mathrm{h}(x)}\) is

There are 6 periods in each working day of a school. The number of ways one can arrange 5 subjects such that each is allowed at least one period, is

If the line \(6 x-y-4=0\) touches the curve \(y^{2}=a x^{3}+b\) at the point \((1,2)\) then
\(a+b=\)

Mrs. Rajni deposited Rs. 10,0000 in a bank that pays \(4 \%\) interest compounded continuously then the amount she gets after 10 years is Rs. approximately.
(given \(e^{(0.4)}=1.49182\) )

What is the the coordination number of a particle in simple cubic close packed structure?

\(\int_{-1}^1 \log \left(\frac{2-x}{2+x}\right) d x=\)

If \(\bar{x}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \bar{y}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) and \(\overline{\mathrm{z}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) where \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors, then value of \(\bar{x} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})+\bar{y} \cdot(\overline{\mathrm{~b}}+\overline{\mathrm{c}})+\overline{\mathrm{z}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})\) is

A vessel is filled with an ideal gas at a pressure of 10 atm and temperature \(27^{\circ} \mathrm{C} .\) Half of the mass is removed from the vessel and temperature of the remaining gas is increased to \(87^{\circ} \mathrm{C}\). Then the pressure of the gas in the vessel will be

Out of total oxygen transported, the plasma carries ________ in dissolved state.

A progressive wave of frequency 400 Hz is travelling with a velocity \(336 \mathrm{~m} / \mathrm{s}\). How far apart are the two points which are \(60^{\circ}\) out of phase?