IAS (MAIN) MATHEMATICS PAPER - I - 2005

MATHEMATICS PAPER - I - 2005
Titne Allowed : Thre^ Hours Maximum Mark$ : 30O
Candidates should attempt Question 1 and S whtch are compii]sonj, rftM AHI/ THREE of the remaining questions seh£ting at least ONE (Question from each Section. Assume suitable data if considered necessan/ ami indicate the same dearly.
SECTION-A
1. y\ttt>nipt any five oi the following :
(a) rind the values of k for which the voctors (1,1,1,1), (1,3, -2, k), (2, 2k - 2. -k - 2H 3k - 1) and (3, k + 2, -3, 2k + 1) are linearly independent in R*. 12
(b) Let V be the vector space of polynomials in ^( of degree <, n over R. Prove that tha set (1, x, x"} is a basis for V. Extend this basis so til at it becomes a basis for the set of all polynomials in 12 (c) Show that the function given below is not continuous at the origin: 12 fOifxy^O ^^'^'^^ = Sif.y.O (d) let / : R be defined as Prove that/^ and/ exist at (0, 0), but/is not differentia ble at (0,0). ' ' 12 (e) If normals at the pomts of an ellipse vi^hosc eccentric angles are ct, P, and 5 meet in a point, then show that stn(p4'Y) + sin(y+a) + sin(a + [J) = 0 12 http://www.civilserviceindia.com 17d C.S-E MATHEMATICS (MAIN) - 2005 (f) A square ABCD having each diagonal AC and BD of length 2a, is folded along the diagonal AC so that the planes DAC and BAC are at right angle. Find the shortest distance between AB and DC 12 2. (a) Let T be a linear transformation on R', whose matrix relative to the standard basis of is "2 1 -l" 12 2 . 3 3 4 Find the matrix of T relative to the basis ^={aLl), (1,1,0), (0,1,1),} 15 (b) Find the inverse of the matriK given below using elementary row operations only : 15 ["2 0 -l' 5 1 0 0 13 (c) If S is a skew-l-lerniitian matrix, then show that A = (I + S) (I - S)'^ is a unitary matrix,can be oppressed in the above form provided -1 is not an eigenvalue of A. 15 (d) Reduce the quadratic form 6x\ + + '^^l - 4x^X2 - 4x3X3 + 4x3X1 to the sum of squares. Also find the corresponding linear transformation, index and signature, IS 3. (a) If u = X + y + uv = y + z and uuw = z, then find 3(u,v,w) ^5 http://www.civilserviceindia.com C.S,E. MATHEMATICS (MAIN) - 2005 (b) Evaluate 171 in terms of Beta function. IS (c) Evaluate 111 zdV , where y is the volume bounded below by the cone + y' = and above by the sphere + yi + = 1^ ]yifig on the positive side of the y-axis. 15 (d) Find the >;-coord in ate of the centre of gravity of the solid lying inside the cylinder x^ + y= = 2ax. between the plane £ 0 and the paraboliod + y' = az. 4. (a) A plane is drawn through the hne x + y = 1, ;i = 0 to
with the plane x + y + z = 5,
make an angle
3J
Show that two such planes can be drawn. Find their equations and the angle between them. 15
(b) Show that the locus of the centres of spheres of a co¬axial system is a straight line. 15
(c) Obtain the equation of a right circular cylinder on the circle through the points (a. 0,0), (0, b, 0) and (0,0, c) as the guiding curve. 15
(d) Reduce the following equation to canonical form and deterniine which surface is represented by it: 15
2x= - If + 2z^-10yz - 8zx - lOxy + 6x + I2y - 6i + 2 = 0
SECTION-B
5. Attempt any five of the following:
(a) Find the orthogonal trajectory of a system of co-axial circles + y* + 2gx + c - 0, where % is the parameter.
12
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CS.E. MATHEMATICS (MAIN) -2005
where P - ^. by reducing it to Clairauf s form by
using suitable substitutioiv 15
(c) Solve the differential equation (sinx -cDS\)y" - xsinxy' + ysinx = 0 given y = sinx is a solution of this equation.
(d) Solve the differential equation
xY' - Ixy' + 2y = xlogx, x>0 by variation of parameters. 7. (a) A particle is projected along the inner side of a smooth vetical circle of radius a so that its velocity at llie lowest point is u. Show that if 2ag < u' < Sag, the particle will leave the circle before arriving at the highest point and will describe a parabola whose latus rectum is
that 15
15
~21 u^a^
15
(b) Two praticles connected by a fine siring are constrained to move m a fine cycloidal tube in a vertical plane, The axis of the cycloid is vctical with vertex upwards. Prove that the tension in \\\^ string is constant throughout the motion,
(c) Two equal uniform rods AB and AC of the length a each, are freely joined at A, and arc placed symmetrically over two smoolh pegs on the same horizontal level at a distance c apart (3c < 2a). A weight equal to that of a rod, is suspended from the joint A, In the position of equitibriuii^, find the inchnation of cither rod with the horizontal by the principle of virtual work.
15
(d) A rectangular lamina of length 2a and breadth 2b is completelv immersed in a vertical plane, in a fluid, so that its centre is at a depth h and the side Za makes an
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C.S.E. MATHEMATICS (MAIN) - 2005
angle a with the horizontal. Find the position of the centre ot pressure.
15
8. (a) The parametric equation of a circular heh\ is
AAA
r = acosu i + asinu j + cuk where c is a constant and u is a parameter, Find the
unit tangent vector t at the point u and the arc' length measured from u = 0. Also find — where s is the arc
lei\gth. (b) Show that
15
curl
k X grad -1 + grad k.grad - 0
where r is the distance from the origin and k is the unit vector in the direction OZ 15
(c) Find the curvature and the torsion of the space curve
X = a (3a - u*)
y = 3au^
z = a (3u + u^) .
15
(d) Evaluate
k- J*-
J7s
by Gauss divergence theorem, where S is the surface of the cylinder + y^ = bounded by K = 0 and x = b. 15