For a narrow base PNP BJT, the excess minority carrier concentration ($$\bigtriangleup n_E$$ for emitter, $$\bigtriangleup p_B$$ for base, $$\bigtriangleup n_E$$ for collector) normalized to equilibrium minority carrier concentration ($$\bigtriangleup n_{E0}$$ for emitter, $$\bigtriangleup p_{B0}$$ for base, $$\bigtriangleup n_{C0}$$ for collector) in the quasi-neutral emitter, base and collector regions are shown below. Which one of the following biasing modes is the transistor operating in?

For the circuit shown, assume that the NMOS transistor is in saturation. Its threshold voltage V_tn = 1 V and its trans-conductance parameter $${\mu _n}{C_{ox}}\left( {{W \over L}} \right) = 1m{\rm A}/{V^2}.$$ Neglect channel length modulation and body bias effects. Under these conditions the drain current I_D in mA is______.

The rank of the matrix $$M = \left[ {\matrix{ 5 & {10} & {10} \cr 1 & 0 & 2 \cr 3 & 6 & 6 \cr } } \right]$$ is

Consider the $$5 \times 5$$ matrix $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 & 5 \cr 5 & 1 & 2 & 3 & 4 \cr 4 & 5 & 1 & 2 & 3 \cr 3 & 4 & 5 & 1 & 2 \cr 2 & 3 & 4 & 5 & 1 \cr } } \right]$$
It is given that $$A$$ has only one real eigen value. Then the real eigen value of $$A$$ is