B.Sc-B.Sc Mathematics 1st Year (Aligarh Muslim University (AMU), Aligarh, Uttar Pradesh-2010)
2009-2010
B.A./B.Sc. (lions.)(PART-I) EXAMINATION (MATHEMATICS)
VECTOR ANALYSIS AND GEOMETRY (MM-105)
Maximum Marks: 40 Duration: Three Hours
Note: Answer all questions.
All questions carry equal marks.
(a) Chords of the parabola y2 =4ax subtend a right angle at the vertex. Find the locus of their middle points.
Define diameters, conjugate diameters and find the condition that the lines Ax2 + 2Hxy+ By2 = 0,may the conjugate diameters of the conic ax1 + 2hxy + by7 = 1,
OR
Pair of tangents are drawn to the conic ax2 + Py2 = l,so as to be always parallel to conjugate diameters of the conic ax2 + 2hxy + by2 = l.Find the locus of their point of intersection.
Trace the conic
16x2 24xy + 9yJ + 77x - 64y + 95 = 0,
OR
Find the foci and the eccentricity of the conic x2 +4xy + y22x + 2y-6 = 0,
Find the equation of the asymptotes of the conic = 1 + e cos 0.
r
3 (a) Show that the equation of the right circular cylinder described on the circle throughhe points A: (1,0,0), B: (0,1,0) and C: (0,0,1) as the guiding curve is
x2 + y2 +z2 -yz-zx -xy = 1
Find the equation to the cone whose vertex is (a, p,y) and the base the parabola rz2 = 4ax, y = 0
OR
Find the semi-vertical angle of a right circular cone which has three mutually perpendicular tangent planes.
(b*)
Contd.
(a) A section of surface is obtained by cutting it through a plane. Find the locus of the centres of the section of the surface ax2 + by2 + cz2=1 if the plane touches ax2 + 0y2 + yz2 = 1
OR
(aO Prove that the pole of the plane through the extremities of three conjug9emi-diarndersoftbeeflipeok] x2/a2+y2/b2+z2/c2=1 lies on the ellipsoid x2/a2 + y2/b2 +z2/c2 =3
(b) Find the locus of the point of intersection of three mutually perpendicular tangent planes to the surface ax2 + by2 = 2z
(a) If a,b,c are non zero vectors and (axb)xc=ax(bxo), then show that (axc)x b = 0. Moreover for four vectors a, d, c, d 'show that (axb)x(cxd) = |cxd)-a)b-|cxd) b)a
(b) Define scaler field, Vector field and gradient of a scaler field and
df" dr
show that = V--and deduce from the above relation that the
ds ds
vector Vpoints in the direction in which has maximum value,
ds
also this maximum value is equal to |Vf|
OR
(b') Define the divergence and curl of a vector field and show that Vx(Vf)=0 and V-(Vxa)=0
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Earning: Approval pending. |