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33
V Semester B.Sc. Examination, February/March 2A24
(NEP) (Freshers)
MATHEMATICS (Major)
Paper - 5.2 : Vector Calculus and Analytical Geometry
Time ;2Yz Hours Max. Marks : 60
lnstruction : Answer all questians.
PAHT _ A
L Answer any ten questions. (10x2=20)
{\t) tf ;: f -t21+sintt, rino [t, qlat r:0.
+\ ,zs
ldt dr') ie,--Vg
2) lf 6(x, Y, z) = 3x2 + 2y - 32, find lv$l at (2, '1, -3).
3) Find the unit normal vector to the surface yz + Zx + XY = C at (-'1, 2, 3).
4) State Gauss-Divergence theorem.
5) Evaluate by Stoke's theorem, { Vrd* + zxdy + xydz where 'c' is the
curve, *'*y'=1,2=y2.
6) Show that area of the ellipse
'azb' *.*: 1 is nab by using Green's theorem.
7) Find the equation of the line passing through the points t2, 4, 8) and
(-1, 6, 3).
8) Find the angle between the planes 3x - 6y + 2z + 5 = 0 and
4x-12y+32-3=0.
9) Findthecentreand radiusof thesphere *'*y'*z'-Gx-8y -22- 1 =0.
10) Find the equation of a cone whose veftex is origin, semivertical angle is
60'and axis along z-axis.
11) Write the general second order degree equation of conicoid.
12) Write the condition for a plane to touch the conicoid at a point (f, g, h).
P.T.O.
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PART _ B
ll. Answer any two questions. (2x5=10)
13) Find the directional derivative of Q(x, y, Z) = x2 - 2y2 + 422 at the point
(1, 1, -1) in the direction of z?+ j-ri.
14) Find the equation of the tangent plane and the normal line to the surface
xo * y" + 3xyz = 3 at the point (1,2, -1).
15) lf the vector F=(ax+3y +42)i+(x-2y+32)j+(3x+ 2y-z)i is solenoidal,
find'a'.
16) For any scalar field 0 and any vector field F, prove that
curl ((r r) = S cuil r+ (grad o) x r.
PART _ C
lll. Answer any two questions. (2x5=10)
17) State and prove Green's theorem,
18) verify Green's theorem in the plane $"(s*' - gy2ydx + {4y - 6xy)dy, where
'c' is the boundary of the rectangular area enclosed by the lines x = 0,
x=1,y=0andy=2.
19) Evaluate the Gauss-Divergence theorem for F = 4xi- zy2j+z2k taken over
the region bounded by *' + y2 = 1, z =0 and z= 3.
20) Evaluate $" r'or using Stoke's theorem for the function F: xi+ z2iny2k
overthe plane surface x + y * z= 1 that lies in the first octant.
'Pr
- D
lV. Answer any two questions.'ART (2x5=10)
21) Find the equation of the plane which bisects the angle between the planes
3x - 4y + z-3 = 0 and 5x + 3y - 4z - g = 0.
22) Find the reflection of the point {2, *1, 0) in the line * ^' -Y;' = '-:,3
211
23) Find the angle between the line '^' = y *,' = '*1 andthe plane
2x+3y-z-4=0. 2 -1 3
24) Find the equation of the sphere which passes through the points (7, 9, 1),
(-2, -3, 2), {1, 5, 5) and (-6, 2,5).
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PAHT _ E
V. Answer any two questions. {2x5=10)
25) Derive the equation of the right circular cone in its standard form
x'*Y'=z2tan26-.
26) Find the equation of right circular cone whose axis is * : y -2 -z-3
-1-' 33
and generator is *"'
234=
Y -3 :z-3
.
27) Explain the equation of hyperboloid of one sheet with properties.
28) showthatthe plane x+2y + 3z =2touchesthe conicoidx2- zy'*Jzz =2.
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