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 + 9y^J + 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.

(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

Attachment: aligarh-muslim-university--amu--2010-b-sc-mathematics-338-1.pdf

Earning:   Approval pending.