Download SGBAU BSc 2019 Summer 2nd Sem Mathematics Vector Analysis n Solid Geometry Question Paper

Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 2nd Sem Mathematics Vector Analysis n Solid Geometry Previous Question Paper

AW?l 656
B. Sc. (Part?l) Semester?II Examination
MATHEMATICS
(Vector Analysis and Solid Geometry)
Paper?IV
Time : Three Hours] [Maximum Marks : 60
Note :? (1) Question No. l is compulsory; attempt it once only.
(2) Attempt one question from each unit.
I. Choose the correct alternative :
(i) If three vectors 5, B, a are coplanar, then for scalar triple product. which of the
following is correct ?
(a) Bxa is perpendicular to the vector 5
(b) 5X5 is parallel to the vector 5
(c) 5X5 is equal to the vector 5
(d) None of these. 1
(ii) The scalar triple product represents the volume of the ____
(a) rectangle (b) sphere
(c) parallelepiped (d) ellipse 1
(iii) The curvature k is determined
(a) only in magnitude (b) only in sign
(c) both in magnitude and sign (d) neither in magnitude nor sign 1
(iv) A plane determined by the tangent and binormal at PG) to the curve ? = F(S) is
a
(a) osculating plane (b) rectifying plane
(c) normal plane (d) none ofthesc 1
(v) Which of the following quantity is de?ned ?
(a) div (div?) (b) curl (div E)
(c) grad (curt?) (d) grad (div F) 1
(vi) A vector fis solenoidal if
(a) curl ?= 0 h (b) div 3:: 0
(c) grad ?= 0 (d) grad (div 1?") = 0 1
YBC?15230 I (Contd.)

(vii)If the radius 0fthc circle is equal to the radius of the sphere, the circle is called a _?
(a) srrall circle (b) imaginary circle
(c) great circle (d) none of these
(viii)Thc equations of the sphere and the plane taken together represent a
(a) sphere (b) plane
(c) straightlinc (d) circle 1
(ix) Every section 01' a right circular cone by a plane perpendicular to its axis is
(a) a sphere (b) a cone
(c) a circle (d) a cylinder 1
(x) The general equation of the cone passing through the coordinate axes is
(a) fyz + gzx + hxy = 0 (b) yz + 2x + xy = 0
(c) ax2+by2+czz=0 (d) X2+y2+22=0 1
UNIT?I
2. (a) Show that a' x (B x 516x (6 x Li). 6 x (5 x B) are coplanar. S
_ , . l - . -
(b) If a', b, E be three umt vectors such that a? x (b x E) = E b, ?nd the angles whtch a makes
with B and 6, B and E being non-parallel. 5
3. (p) If t: is a vector function of t and u is a scalar function of t, then prove that :
d ~ d? du ?.
? uf = u ? + ? t.
dt ( ) dt dt 5
(t) Fvaluatc ?Exdz?f dt- where
1 , I dt?
r'(t'_)=513t+ tj?t?i? 5
INIT?ll
4. (a) Prove that helices ZITC Iht? onlyv twisted curves whose Darboux's vector has a constant
direction. 5
(b) For the curve x = 31, y 2 W. 7 T 2l?at the point t = I. ?nd the equations for osculating
plane, normal plane and rectitying plane. 5
5. (p) For the curve x - a(3t ? t L _\? 32113, 1 a(31 + t?). show that the curvature and torsion
are equal. 5
(q) If ieax?aed ?3.626 .45. then ?nd the vector 3. 5
YBC?15230 3
(Contd.)

6 (a)
(b)
(C)
7- (p)
(q)
8 (a)
(b)
9. (p)
(q)
10. (a)
(b)
H. (p)
(q)
UNlT?IH
If f : x} + y} + 2E, then show that div(z? F): (n + 3)r". 4
Find the directional derivative of d) = xy2 + yz2 at the point (2, - l, l) in the direction
of the vector i' + 2] + 212. .3
If ct) = 3x7y ? y?zz, ?nd grad (p at the point (1, ? 2, ? 1). 3
If ?=(2x + yz)? + (3y ?4x)], evaluate IF'dF along the parabolic arc y = xzjoining (0, 0)
and (1, l). 5
Apply Green?s theorem to prove that the area enclosed by a simple plane curve C is'
l
3 ?my ' ydx). Hence ?nd the area of an ellipse whose semi-major and semi-minor axes
are of lengths a and b. 3+2
UNIT?IV
Find the equation of a sphere for which the circle x2 + y2 + z2 +7y ? 22 + 2 = 0,
2x + 3y + 42 = 8 is a great circle. 5
Find the equation of the sphere circumscribing the tetrahedron whose faces are :
x y z
= = = ? +? ? = l
x 0,y 0,2 0'21 b+c . 5
State and prove the condition for the orthogonality of two spheres. 1+4
Find the coordinates of the centre and radius of the circle x + 2y '+ 22 = 15 ;
x2+y3+23?2y?4z=11. 5
UNlT?V
Find the equation of the cone whose vertex is at the point (a, B, y) and whose generators
touch the sphere x2 + y2 + z2 = 33. 5
Find the equation of right circular cone whose vertical angle is 90? and its axis is along
thelinex=?2y=z. 5
Find the equation to the cylinder whose generators are parallel to the line ? = i = g
and the guiding curve is the ellipse x2 + 2y2 = 1, z = 3. 5
F ind the equation of the right circular cylinder of radius 2 whose axis passes through
(1, 2, 3) and has direction cosines proportional to 2, ? 3, 6. 5
YBC?l5230 3 625