Engineering Sciences (XE) 2008

If the characteristic equation of a 3×3 matrix is λ3−λ2+λ−1=0, then the matrix should be

lim (x,y)→(0,0) (x4+xy)/(x2−y2) is

If f(z)=u+iv is an analytic function and u−v=(x−y)3+kxy(x−y), then k is

The directional derivative at the point P(1,2,3) to the surface x2+4/9 y2−z2/9=3 in the direction of the vector →OP, where O denotes the origin, is

If the solution of the differential equation dy/dx + P(x)y = xy3 is y3(1+cex)=1, c being an arbitrary constant, then P(x) is

The system of equations ax + by + a2 = 0, bx + ay - b2 = 0, x + y + a - b = 0 admits unique solution if