Old Exam Papers June 2012 (Set 2)

1
2 2
x 2a{ t t 1 t }, y 2at , z 4at
sin c h
between the points where t t1
and t t2
is
4 2 abt 2 t1g.
2. (a) State and prove the Serret-Frenet Formulae.
(b) Show that the curve given by
x a sin u, y 0,
z a cos u
lies on a sphere.
3. (a) Find the evolute of the circular helix
x a cos , y a sin , z a tan
(b) Show that each characteristic touches the edge of
regression.
4. (a) Find the equation to the edge of regression of the
developable :
y xt t , z t y t
3 3 6
800 2 M.A./M.Sc.-M.T.-04
1z i . i
10. (a) Show that the integral f x , x dt has stationary
0
c h
value on the curve whose differential equations are
.
where x
d
i i
x
dt x
F HG I KJ f f
. 0
i
i
dx
dt
(b) State and prove Bianchi's identity of covariant
curvature tensor.
M.A./M.Sc.-M.T.-04 5 800