IAS PYQs 4
2008
1) Show that
∮Sds√a2x2+b2y2+c2z2=4π√abc, where s is the surface of the ellipsoid ax2+by2+cz2=1.
2) Find the unit vector along the normal to the surface z=x2+y2 at the point (−1,−2,5).
[10M]
3) Prove that the necessary and sufficient condition for the vector function →V of the scalar variable t to have constant magnitude is →V⋅d→vdt=0.
[10M]
4) Prove that the shortest distances between a diagonal of a rectangular parallelopiped whose sides are of lengths a, b and c and the edges not meeting it are
bc√b2+c2,ca√c2+a2,ab√a2+b2[10M]
5) If →F=2x2ˆi−4yzˆj+zxˆk, evaluate
∬5→F⋅ˆnds,
where s is the surface of the cube bounded by the planes x=0, x=1, y=0, y=1, z=0, z=1.
[10M]
2007
1) Evaluate
∫0ˉF¯drwhere ˉF=c[−3asin2θcosθˉi+a(2sinθ−3sin2θ)ˆj+bsin2θ→k] and the curve C is given by →r=acosθ→i+asinθj+bθˉk θ varying from π4 to π2.
2) Show that
Curl(→a×→rr3)=−˙ar3+3rr3(→a,→r)Where →a is a constant vector and
→r=x→i+y→j+z→k[10M]
3) Find the curvature and torsion at any point of the curve
x=acos2t,y=asin2t,z=2asint
[10M]
4) Evaluate the surface integral ∫S(yzˉi+zxˉj+xyˉk).d→a Where S is the surface of the sphere x2+y2+z2=1 in the first octant.
[10M]
5) Apply Stokes’ theorem to prove that
∫n(ydx+zdy+xdz)=−2√2πa2Where C is curve given by
x2+y2+z2−2ax+2ay=0,x+y=2a[10M]
2006
1) If →f=3xyi−y2j, determine the value of ∫C→f.d→r, where C is the curve y=2x2 in the xy-plane from (0,0) to (1,2).
[10M]
2) If u→f=→∇v, where u, y are scalar fields and →f is a vector field, find the value of →f⋅ curt →f.
[10M]
3) If O be the origin, A,B, two fixed points and P(x,y,z) a vaniable point, show that curl(→PA×→PB)=2→AB
[10M]
4) Using Stokes’ theorem, determine the value of the integral
∫Γ(ydx+zdy+xdz)where Γ is the curve defined by
x2+y2+z2=a2y+z=a[10M]
5) Prove that the cylindrical co-ordinate system is orthogonal.
[10M]
2005
1) For the curve \vec{r}=a\left(3 t-t^{3}\right) \vec{i}+3 a t^{2}\vec{j}+a\left(3 t+b^{3}\right) \vec{k} being a constant. Show that the radius of curvature is equal to its radius of torsion.
2) Find f(r) if f(x)=6 both solenoidal and irrotational.
[10M]
3) Evaluate ∬SˉF⋅dˉs, where ˉF=yzˉi+zxˆj+xyˉk and S is the part of the sphere x2+y2+z2=1 that lies in the first octant.
[10M]
4) Verify the divergence theorem for ˉF=4xˉi−2y2ˉj+z2ˉk taken over the region bounded by x2+y2=4, z=0 and z=3.
[10M]
5) By using vector methods, find an equation for the tangent plane to the surface z=x2+y2 at the point (1,−1,2).
[10M]