Differentia geometry past/Previous question paper

Q1:

(a) Define the rectifying plane?

Prove that the normal plane at any point to the locus of center of curvature any cur bisect curvature at the corresponding the original

Q2:

(a). Define involutes and evolutes ? (b). Prove that there exist an infinite family of evolutes for space curve (involutes)? (10)

Q3:

a). Define characteristics and envelope of a family? (b). Prove that the envelope of one parameter family of the surfaces touch each member the family each point of chracteristics? (10)

04:

(a). Find the envelope of the plane ix+my+nz=p: Where p²= a21²+ b²m² + c²n² and a, b, c are constant? (b). Find whether xyz = a³ is developable (10)

Q5:

curvature of indicatrix of normal? (b). Prove that the edge of regression of the tangent developable the curve itself? (10)