1. A system with unity negative feedback has the following characteristic equation 1+k s(s+4) s2+28+2 (1) Sketch the root locus, (2) Find the gain when the roots are equal, and (3) Find the value of the roots when they are equal. - 2. A system is in closed loop feedback Ge(s) = K, H(s) = 1 and G(s) = 1/[s(s = 1)]. Show that the closed-loop control system is stable regardless of the value of K, when Ge(s) = K(s+2) 8+20 Sketch the root locus and determine the range of K for which the system is stable. Determine the value of K and the complex roots when two roots lie on the jw-axis. 3. A unity feedback system has a loop transfer function L(s) G(s)Ge(s) = 4(s+1) s(s+a) Sketch the root locus for 0 < a < . 4. A unity feedback system has the loop transfer function K L(s) G(s)Ge(s) = s(s+2)(s+5) Find (a) the breakaway point on the real axis and the gain K for this point, (b) the gain and the roots when two roots lie on the imaginary axis, and (c) the roots when K 6. (d) Sketch the root locus.