The determinant $$\left| {\matrix{ {1 + b} & b & 1 \cr b & {1 + b} & 1 \cr 1 & {2b} & 1 \cr } } \right|$$ equals to

A long glass cylinder of inner diameter $$=0.03m$$ and outer diameter $$=0.05m$$ carries hot fluid inside. If the thermal conductivity of glass $$=1.05$$ $$W/mK,$$ the thermal resistance $$(K/W)$$ per unit length of the cylinder is

The $$DC$$ power source for arc welding has the characteristic $$3V + 1 = 240,$$ where $$V=$$ Voltage and $${\rm I}\,\, = $$ current in amp. For maximum arc power at the electrode, voltage should be set at

In a $$CAD$$ package, a point $$P(6, 3, 1)$$ is projected along a vector $$v(-2, 1, -1).$$ The projection of this point on $$X-Y$$ plane will be