IAS. MAINMATHEMATICS PAPER-I 2006

MATHEMATICS -
PAPER-I
2006
Time allowed: 3 hours
Maximum Marks: 309
INSTRUCTIONS
Each question is printed both in Hindi and in English.
Answers must be written in the medium specified in the Admis¬sion Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space provided for the purpose No marks will be given for the answers written in a medium other than that specified in the Admission Certificate.
Candidates should attempt Questions 1 and 5 which are com¬pulsory and any three of the remaining questions selecting at least one question from each Section.
Assume suitable data if considered necessary and indicate the same clearly.
The number of marks carried by each question is indicated at the end of the question.
SECTION 'A' Q. 1. Attempt anyyzve of the following:
(a) Let V be the vector space of all 2x2 matrices over the field F. Prove that V has dimension 4 by exhibiting a basis for V 12
(b) State Caylay-Hamilton theorem and using it find the in¬verse of
1 3
2 4
(c) Find a and b so that f (2) exists, where
1
12
f(x) =
X
if x>2
a + bx^, if x<2 12 THE TEAM VISION IAS (d) Eiqpress x"^ (1 - x^y dx in terms of Gamma function and 0 hence evaluate the integral /•I 0 x^V(l-x^dx 12 (e) A pair of tangents to the conic ax^ + by^ - i intercepts a constant distance 2 k on the y-axis. Prove that the locus of their point of intersection is the conic. ax2 (ax2 + by2 - 1) = bk^ (ax^ - l)^ 12 (f) Shew that the length of the shortest distance between the line z = X tan a, y = 0 and any tangent to the ellipse sin^ a + y^ = a^, z = 0 is constant. 12 Q. 2. (a) If T : is defined by T(x,y)-(2x-3y,x + y) compute the matrix of T relative to the basis ^={(1,2X(2,3)} 15 (b) Using elementary row operations, find the rank of the matrix 3 0 1 0 -2 0 2 2 -2 -3 1 2 -1 1 -2 1 15 (c) Investigate for what values of X and |i the equations x + y + z = 6 x + 2y + 3z=10 X + 2y + Xz = |i have (i) no solution; (ii) a unique solution; (iii) infinitely many solutions. 15 (d) Find the quadratic form q (x, y) corresponding to the sym¬metric matrix r A = 5 -3 V -3 8 Is this quadratic form positive definite ? Justify your answer. Q. 3. (a) Find the values of a and b such that a sin^ X X b log cos x 1 lim x->0
.-4
(b)If
z = xf show that
Vx/
+g
+ 2
+ y
dx^ xydxdy ' dy (c) Change the order of integration in
= 0
/•CO (.f-j p.~y
0 Jx y
dy dx
and hence evaluate it.
(d) Find the volume of the uniform elHpsoid
x^
15
15
15
15
15
Q. 4. (a) If PSP' and QSQ' are the two perpendicular focal
chords of a conic ^ = 1 + e cos 9, prove that
1 1
SP-SP' "^SQSQ' is constant.
(b) Find the equation of the sphere which touches the plane 3x + 2y - z + 2 = 0 at the point (1, -2, 1) and cuts orthogonally the sphere
+ y^ + - 4x + 6y + 4 = 0 15
(c) Show that the plane ax + by + cz = 0 cuts the cone xy + yz + zx = 0 in perpendicular lines, if
111
- + - + - = 0 15 a b c
(d) If the plane /x + my + nz = p passes through the extremities of three conjugate semidiameters of the ellipsoid
2 2 _2
x y z
a^ b^ c^ prove that
a^/^ + b^m^-f c2n2=:3p2 15 SECTION 'B' Q. 5. Attempt any five of the following:
(a) Find the family of curves whose tangents form an angle —
with the hyperbolas xy = c, c> 0. 12
(b) Solve the differential equation
dx-x2ydy = 0 12
2 ^3
xy^+e
V J
(c) A particle is free to move on a smooth vertical circular wire of radius a. It is projected horizontally from the lowest point with
velocity 2 ^/ga . Show that the reaction between the particle and the wire is zero after is time
- logics + A/6 ) 12 g
(d) The middle points of opposite sides of a jointed quadrilat¬eral are connected by hght rods of lengths /, /'. If T, T' be the tensions in these rods, prove that
T T'
7-y=o 12
(e) Find the depth of the centre of pressure of a triangular lamina with a vertex in the surface of the liquid and other two verti¬ces at tlepths b and c from the surface. 12
(f) Find the values of constants a, b and c so that the direc¬tional derivative of the function.
f = axy^ + byz + cz^ x^ at the point (1, 2, -1) has maximum magnitude 64 in the direction parallel to z-axis. 12 Q. 6. (a) Solve :
(l+y2)4-(x-e-^"'>)^==0 15
dx
(b) Solve the equation
X- p2 + yp (2x + y) + y^ = 0 using the substitution y = u and xy = v and find its singular solu¬tion, where
dy
(c) Solve the differential equation
3„ j2.. r
11^2x^ + 2^ = 10 dx^ dx^ x
V x^v
15
(d) Solve the differential equation
(D2-2D +2)y = e^tanx, D -by the method of
variation of parameters. 15 Q. 7. (a) A particle, whose mass is m, is acted upon by a
force m
^ a^^
towards the origin. If it starts from rest at a
distance a, show that it will arrive at origin in time
15
(b) If u and V are the velocity of projection and the terminal velocit^^ respectively of a particle rising vertically against a resis¬tance varying as the square of the velocity, prove that the time taken by the particle to reach the highest point is
— tan ^ g ^u^ 15
(c) Show that the length of an endless chain, which will hang over a circular pulley of radius c so as to be in contact with two-third of the circumference of the pulley is
+
471
log(2 + V3) 3 15
(d) A uniform rod of length 2a, can turn freely about one end, wliich is fixed at a height h (<2 a) above the surface of the liquid. If the densities of the red and liquid be p and o , show that the rod can rest either in a vertical position or inclined at an angle 9 to the vertical such that cos 9 = 2a'\ p-G Q. 8, (a)If A = 2i+k. B = i + j+k, C = 4i-3j-7k,de¬termine a vector ^ satisfying the vector equations. RxB=_CxB and R.A =0 15 (b) Prove that r^ r is an irrotational vector for any value of n, but is solenoidal only if n + 3 = 0. 15 (c) If the unit tangent vector { and binormal ^ make angles 9 and (j) respectively with a constant unit vector a, prove that sin9 d9 k ---=-- 25 sine}) d(|) X (d) Verify Stokes' theorem for the function F = X i - xy j integrated round the square in the plane z = 0 and bounded by the lines x = 9,y = 0, x = a andy = a, a> 0. 15