A Firstranker's choice
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B. Sc. (Part-I) Semester—II Examination
MATHEMATICS
(Vector Analysis and Solid Geometry)
Paper—iV
Time : Three Hours] [Maximum Marks : 60
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Note :— (1) Question No. 1 is compulsory; attempt it once only.
(2) Attempt one question from each unit.
1. Choose the correct alternative :
(i) If three vectors 3, p, ¢ are coplanar, then for scalar triple product, which of the following is correct ?
(a) px¢g is perpendicular to the vector 3z
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(b) px¢g is parallel to the vector 3
(c) px¢ is equal to the vector 3
(d) None of these. 1
(ii) The scalar triple product represents the volume of the
(a) rectangle (b) sphere
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(c) parallelepiped (d) ellipse 1
(iii) The curvature k is determined
(a) only in magnitude (b) only in sign
(¢) both in magnitude and sign (d) neither in magnitude nor sign 1
(iv) A plane determined by the tangent and binormal at P(?) to the curve T = 1(S) is
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(a) osculating plane (b) rectifying plane
(¢) normal plane (d) none of these 1
(v) Which of the following quantity is defined ?
(a) div (div ?) (b) curl (div F)
(c) grad (curl f) (d) grad (div ?)
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(vi) A vector fis solenoidal if
(a) curl =0 (b) div f=0
(c) grad =0 (d) grad (div t-”) =0 1
(vii) The equations of the sphere and the plane taken together represent a .
(a) small circle (b) imaginary circle
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(c) great circle (d) none of these 1
(viii)The equations of the sphere and the plane taken together represent a .
(a) sphere (b) plane
(c) straight line (d) circle 1
(ix) Every section of a right circular cone by a plane perpendicular to its axis is _
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(a) a sphere (b) a cone
(c) a circle (d) a cylinder I
(x) The gereral equation of the cone passing through the coordinate axes is __
(a) fvz+gzx +hxy =0 (b) yz+zx+xy=0
(¢) ax* +by* + ¢cz2 =10 (d) x*+y*+22= 1
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UNIT—I1
2. (a) Show that 4x (B x E‘),Q;x (G xd)éxlax B) arc coplanar. 5
(b) If 4,b,¢ be three unit vectors such that 3 x (_h x é) =5 b, find the angles which @ makes with p and ¢, b and ¢ being non-parallel. 35
3. (p) If f is a vector function of t and u is a scalar function of t, then prove that :
d (=) df du:
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—Wuf)=u—+— 5
dt ( ) de dt
(q) FEvaluate If a dt. where
I
F=5C¢T+tj-t"k 5
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UNIT—il
4. (a) Prove that helices arc the onlv twisted curves whose Darboux's vector has a constant direction. 5
(b) For the curve x = 31, y = 3t/ 7 = 2t at the point t = I, find the equations for osculating plane, normal planc and rectitying planc. 5
5. (p) Forthe curve x = a(3t -t ),y = 3at’, z = a(3t + t'), show that the curvature and torsion are equal. 5
(q) If '=dxf,i'=dxn.b'=d=b, then find the vector d. 5
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6. (a) If #=xi+yj+zk, then show that div(" 7)=(n+3)". 4
(b) Find the directional derivative of ¢ = xy? + yz? at the point (2, — 1, 1) in the direction of the vector i +2j+2k. 3
(c) If & = 3x’y — y*2%, find grad ¢ at the point (1, — 2, — 1). 3
7. (p) If F<{2x + y?Ji + By —4x)], evaluate .fF'dF along the parabolic arc y = x?joining (0, 0) and (1, 1). 5
(q) Apply Green's theorem to prove that the area enclosed by a simple plane curve C is 2 ,[ (xdy -y d"). Hence find the area of an ellipse whose semi-major and semi-minor axes are of lengths a and b. 3+2
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UNIT—IV
8. (a) Find the equation of a sphere for which the circle x2 + y?> + 22 +7y — 2z + 2 = (,
2x + 3y + 4z = 8 is a great circle. 5
(b) Find the equation of the sphere circumscribing the tetrahedron whose faces arc :
X 'y z
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= = = — Jite P ]
x=0,y=0,z O‘a b+c ; 5
9. (p) State and prove the condition for the orthogonality of two spheres. 1+4
(q) Find the coordinates of the centre and radius of the circle x + 2y + 2z = 15 ;
X+y+22-2y-4z=11. S
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UNIT—V
10. (a) Find the equation of the cone whose vertex is at the point (a, B, y) and whose generators touch the sphere x* + y* + 22 = a%, 5
(b) Find the equation of right circular cone whose vertical angle is 90° and its axis is along the line x=-2y=12z. 5
1. (p) Find the equation to the cylinder whose generators are parallel to the line -’li = -_15 = and the guiding curve is the ellipse x> +2y* =1, z = 3. 5
(q) Find the equation of the right circular cylinder of radius z whose axis passes through (1, 2, 3) and has direction cosines proportional to 2, — 3, 6. S
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