University : University of Madras
Degree : B.Sc
Department :Mathematics
Subject :PAS Vector Analysis & Analytical Solid
Document type : Question Paper
Website : ideunom.ac.in
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OCT 2012 : https://www.pdfquestion.in/uploads/ideunom.ac.in/6859-._DEC12_uid4720%20PAS.pdf
PAS Vector Analysis & Analytical Solid Model Paper :
OCTOBER 2012 U/ID 4720/PAS
Time : Three hours
Maximum : 100 marks
Related / Similar Question Paper : IDEUNOM B.Sc UCMH Graph Theory Question Paper
SECTION A — (10 ´ 2 = 20 marks)
Answer ALL questions.
Each question carries 2 marks.
1. Find the directional derivative of f where ( 2 2 2 ) log
2 y = 2x in the xy plane from (0,0) to (1,2). A xyi y j
4. Evaluate ??( + + ) S xdydz ydzdx zdxdy over the
5. Find the equation of the line passing through the point (3,2,-8)
6. Find the equation of the sphere with centre(-1,2,- 3) and radius
7. Find the equation to the cone with vertex at the origin
8. Find the equation of the right circular cone whose vertex
9. Test whether the sequence ¥ = ? ?
10. State the comparison test for absolute convergence.
SECTION B — (5 × 16 = 80 marks)
Answer ALL questions.
Each question carries 16 marks.
11. (a) Determine f (r) so that the vector f (r) both solenoida
Or
(c) Find the angle between surfaces 3 z = x2 + y2 – and 9 x2 + y2 + z2
(d) If f and f differentiable scalar point functions, prove that Ñf ´Ñf
12. (a) Verify Gauss’s divergence theorem for F x i y j z k r r r r
(b) Verify Stokes theorem for F y i xyj xzk r r r r
Or
13. (a) Prove that the origin lies in the acute angle between the planes
(b) Show that the lines x – = y – = z +
Or
(c) Find the equation of the sphere which touches the plane 3x + 2y – z + 2
(d) Find the equation of the sphere having the circle 9 x2 + y2 + z2
(C) 3x + 2y – z + 2 = 0 GßÓ uÍzvøÚ P(1,- 2,1)
14. (a) Find the equation of the right circular cone which passes
(b) Find the condition for the equation 2 2 2 0 ax2 + by2 + cz2 + fyz
Or
(c) Find the equation to the right circular cylinder whose guiding curve is
(d) Find the equation to the right circularcylinder of radius 3
15. (a) Find the limit of the sequence
(b) Show that a non decreasing sequence
Or
which is bounded above is convergent.
(c) Show that the series S is divergent. (d) If S
OCTOBER 2011 U/ID 4720/PAS
Time : Three hours
Maximum : 100 marks
SECTION A : (10 × 2 = 20 marks)
Answer ALL questions.
Each question carries 2 marks :
1. Show that F ( xy z )i x j xz k 3 2 2 = 4 – + 2 – 3 is irrotational.
F ( xy z )i x j xz k 3 2 2 = 4 – + 2 – 3 Gߣx _Ç»ØÓx GÚ {ÖÄP.
2. Find the directional derivative of 2 3 x + xy + yz at (0, 1, 1) along 2i + 2 j – k .
3. Evaluate f dr C ∫ × where f x i y j = 2 + 2 and C is the straight line y = x joining the points (0, 0) and (1, 1)
4. Using Stoke’s theorem show that ∫r × dr = 0 C where r = xi + yj + zk .
5. Find the point where the line x – = y + z meets the plane 2x – 3y + 2z + 3 = 0 .
6. Find the equation of the sphere with centre (–1, 2, –3) and radius 3 units.
7. Write the condition for the equation 2 2 2 0 ax2 + by2 + cz2 + fyz + gzx + hxy = to represent a right circular cone.
8. Find the equation of the cone of second degree which passes through the axes.
9. Examine the convergence of {n-1/2n+3}
10. Give two example for oscillating sequence.
SECTION B : (5 × 16 = 80 marks)
Answer ALL questions.
Each question carries 16 marks.
11. (a) Prove that (r r ) n(n )r r
Or
(c) If F x yi xzj 2yzk = 2 + + then prove that div (curl F ) = 0.
(d) Find the equation of the tangent plane to the surface 4 6 x2yz + xz2 = at the point (1, –2, –1).
12. (a) Verify Gauss divergence theorem for F yi xj z k = + + 2 over the cylindrical region 9 x2 + y2 = , z = 0 , z = 2 .
Or
(b) Verify Stoke’s theorem for xyi + yzj + zxk taken over the surface in the plane x + y + z = 1 bounded by x = 0 , y = 0 , z = 0 .
13. (a) Find the equation of the sphere that passes through the circle 2 4 0 x2 + y2 + z2 – x – y = , x + 2y + 3z = 8 and touches the plane 4x + 3y = 25 .
Or
(b) Find the image of the point (1, –2, 3) in the plane 2x – 3y + 2z + 3 = 0 .
(c) Find the shortest distance and equation of the shortest distance between the lines
14. (a) Prove that the equation 2 3 14 5 6 8 19 2 20 0 x2 – y2 + z2 – xy + yz – zx + x – y – z – =
represents a cone. Find the vertex of the cone.
(b) Prove that the equation 2 2 2 0