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Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

4.

(a)

(b)

Candidates should attempt any FIVE questions. All questions carry equal marks.

---. '(

Let V and U be vector spaces over the field K and let V be of finite dimension. Let T: V. ----+ U be a linear Map. Prove that dim V = dim R(T) + dim N(T) Let S={(x,y,z)/x+y+z=O}, x, y, z being real. Prove that Sis a subspace of 1J3 . Find a basis of S.

(c) VerifY which of the following are linear transformations:

(a)

(i) T :13 ----+13 2 defined by T(x) = (2x,-x) (ii) T: 132 ----+ 133 defined by T (X, y) = ( xy, y, X) (iii) T: 132 ----+ 133 defined by T (X, y) = (X+ y, y, X) (iv) T: 13----+ 13 2 defined by T ( x) = (1, -1) LetT: M2,1----+ M2,3 be a linear transformation defined by (with usual notations)

r(~)=(~ ~ ~}rG)=(~ ~ ~) Find r(;}

(b) For what values ofY) do the following equations

(c)

(a)

(b)

(c)

(a)

(b)

x+y+z=1 x+2y+4z=7J x+4y+10z=ry2

Have a solution? Solve them completely in each case. Prove that a necessary and sufficient condition of a real quadratic form X' AX to be positive definite is that the leading principal minors of A are all positive.

State Cayley-Hamilton theorem and use it to calculate the inverse of the matrix---+ [ ~ ~] Transform the following to the diagonal forms and give the transformation employed:

x2+2y, 8x2-4xy+5/

Prove that the characteristic roots of a Hermitian matrix are all real and a characteristic root of a skew-Hermitian matrix is either zero or a pure imaginary number. If y=eax cos bx, prove that

yr2ayl+(a2+b2)y=O and hence expand e2xcos bx in powers of x. Deduce the expansion of eax and cos bx. If x = r sin 8 cos ~' y = r sin 8 sin ~' z = r cos8 then prove that

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5.

6.

7.

8.

(c)

(a)

(b)

(c)

(a)

(b)

(c)

(a)

dx2+d/+dz2 = dr2+r2 d82+r2 sin28 d~2 Find the dimensions of the rectangular parallelepiped inscribed m the ellipsoid x2 y2 2 2+-2 +.;.- = 1 that has greatest volume. a b c Prove that the volume enclosed by the cylinders

x2+y2=2ax, z2 = 2ax is 128a3/15

Find the centre of gravity of the volume formed by revolving the area bounded by the parabolas /=4ax and x2=4by about the x-axis. Evaluate the following integral in terms of Gamma function :

r\1 +xY (1-x)q dx, [P > -1,q > -1] and prove that

1(113)1(2/3)= ]Jlr. If ax2+2hxy+by2+2gx+2fy+c=O represents two intersecting straight lines, show that the square of the distance of the point of intersection of the straight lines from the origin is

c (a+ b)- !2- g2 ( 2 ) ----'------'---....,

2-- ab - h :;t 0

ab-h Discuss the nature of the conic

16x2-24xy+9/-104x-172y+ 144=0 in details. A straight line, always parallel to the plane of yz, passed through the curves x2+/=a2, z=O, and x2=ax, y=O; prove that the equation of the surface generated is

4 2 ( 2 )2( 2 2) x y = x -az a-x Tangent planes are drawn to the ellipsoid

x2 y2 2 2 -+-+-=1 a2 b2 c2

through the point (a,~,y). Prove that the perpendiculars to them from the origin generate the cone

(ax,~y,yz)2 = a2x2+b2/+c2z2 (b) Show that the locus of the foot of the perpendicular from the centre to the plane through the

extremities of three conjugate semi-diameters of the ellipsoid x2 y2 2 2 -+-+-=1 a2 b2 c2

IS a2x2 + b~2 + c2z2 = 3(x2+y2+z2)

(c) Define an osculating plane and derive its equation in vector form If the tangent and binormal at a point P of the curve make angles 8, ~respectively with the fixed direction, show that

(a)

(b)

(c)

( :::: )( ~:) = - (: J where k and 1 are respectively curvature and torsion of the curve at P. By eliminating the constants a, b obtain the differential equation of which xy=aex+be-x+x2 is a solution. Find the orthogonal trajectories of the family of the semi-cubical parabolas ay2=x3, where a is a variable parameter. Show that

(4x+3y+ 1)dx+(3x+2y+ 1)dy=O represents hyperbolas having the following lines as asymptotes

x+ =0 2x+ +1=0

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9.

10.

11.

12.

(d)

(a)

(b)

(c) (d) (a)

Solve the following differential equation: y(l +xy )dx+x(l-xy )dy=O

Find the curves for which the portion of y-axis cut off between the origin and the tangent varies as the cube of the abscissa of the point of contact. Solve the following differential equation:

(D2+4)y=sin2x, given that when x=O, then y=O and dy = 2. dx

Solve : (D3 -1 )y=xex+cos2x Solve : (x2D2+xD-4)y=x2

If ] ( x, y, z) = (y2 + z2 )Z + ( z2 + x2)] + ( x2 + y 2) k, then calculate J ].dx, where C consists of

c

(i) the line segment from (0,0,0) to (1,1,1) (ii) the three line segments AB, BC and CD, where A, B, C and D are respectively the

points (0,0,0), (1,0,0), (1,1,0) and (1,1,1) (iii) the curve x + ui + u2 j + u2 k, u from 0 to 1.

(b) If a and b are constant vectors, show that (i) div{xx(axx)} =-2x.a (ii) div { (a X x) X ( b X X)} = 2a. ( b X X)- 2b. (a X x)

(c) Obtain the formula

(a)

divA= ~---s{(_K_J112 A(i)} "g ax gii

where A(i) are physical components of A and use it to derive expressiOn of divA m cylindrical polar coordinates. Two equal rods, each of weight wl and length l, are hinged together and placed astride a smooth horizontal cylindrical peg of radius r. Then the lower ends are tied together by a string and the rods are left at the same inclination ~ to the horizontal. Find the tension in the string and if the string is slack, show that~ satisfies the equation

1 tan3 +tan=-

2r (b) Define central axis for a system of forces acting on a rigid body. A force Facts along the axis

of x and another force nF along a generator of the cylinder x2+y2=a2. Show that the central axis lies on the cylinder.

n2 (nx-z)2 +(1+n2r y2 =n4a2 (c) A semicircular area of radius a is immersed vertically with its diameter horizontal at a depth

b. If the circumference be below the centre, prove that the depth of centre of pressure is

(a)

(b)

3;r ( a2 + 4b2) + 32ab 4(3b;r+4a)

A particle is moving with central acceleration !J.(r5 -c\) being projected from an apse at a

distance c with a velocity )( 2; }'. Show that its path is the curve x4+y4=c4

A particle is projected with a velocity whose horizontal and vertical components are respectively u and v from a given point in a medium whose resistance per unit mass is k times

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the speed. Obtain the equation of the path and prove that if k is small, the horizontal range is approximately

2uv 8uv2k g 3g

(c) A particle slides down the arc of a smooth vertical circle of radius a being slightly displaced from rest at the highest point of the circle. Find the point where it will strike the horizontal plane through the lowest point of the circle.

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I -

Time Allowed: 3 hours Maximum Marks: 300

1. (a)

Candidates should attempt any jive Questions. ALL Questions carry equal marks.

SECTION- A If H is a cyclic normal subgroup of a group G, then show that every subgroup of H is normal in G.

20 (b) Show that no group of order 30 is simple.

20 (c) If p is the smallest prime factor of the order of a finite group G, prove that any subgroup of

index p is normal.

(a) 20

IfR is a unique factorization domain, then prove that any fER[x] is an irreducible element of R[x], if and only if either f is an irreducible element of R or f is an irreducible polynomial in R[x}.

20 (b) Prove that x2+ 1 and x2+x+4 are irreducible over F, the field of integers modulo 11. Prove also

that ~[ x] and 2F[ x] are isomorphic fields each having 121 elements.

(c) Find the degree of splitting field x5-3x3+x2-3 over Q, the field of rational numbers.

(a) If we metrize the space of functions continuous on [a,b] by taking. p(x,y) = J: !x(t)- y(t)/2 dt]

then show that the resulting metric space is NOT complete.

(b) Examine f(x,y,z) = 2xyz-4zx-2yz+x2 + i +z2 -2x-4y-4z

for extreme values.

(c) If U = 1+nx _ 1+(n+1)x

n nenx (n+1)e(n+l)x' 0O 0 1 + xa sin 2 x' (c) Evaluate

20

20

(a) 20

If u=e-x(x siny- ycosy), find v such that f(z)=u+iv is analytic. Also find f(z) explicitly as a function of z.

20 (b) Let f(z) be analytic inside and on the circle C defined by lzi=R and let z=eri8 be any point in

side C. Prove that

(R2 -r2 )f(Rei) f (rei(})= 11 217 r2 7r d Jo R 2 -2Rrcos(B+)+r 2

(c) Prove that all the roots of

(a)

z7 -5z3+ 12=0

lie between the circle lzl = 1 and lzl=2.

Find the region of convergence of the series whose n-th term is

( -1 r-1 z2n-l (2n -1)!

(b) Expand 1 f ( z) - --:-----:---:------:---:----

- (z+1)(z+3)

in a Laurent series valid for

(i) lzl>3 (ii) l

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7.

8.

9.

(a) Solve:

Is cosmx dx o x2 +1

(2x2 -/+z2 -2yz-zx-xy )p+(x2+2/+z2 -yz-2zx-xy )q=x2+/+2z2 -yz-zx-2xy

20

20 (b) Find the complete integral of

(y-x)(qy-px) = (p-q)2

20 (c) Use Charpit's method to solve

(a)

px + qy = z ~1 + pq

Find the surface passing through the parabolas z=O, y2=4ax

z=1, /-4ax and satisfying the differential equation

xr+2p=O

20

20 (b) Solve:

r+s-6t=ycosx 20

(c) Solve:

(a) 20

Classify each of the following dynamical systems according as they are

(i) scleronomic or rheonomic (ii) holonomic or nonholonomic (iii) conservative or non-conservative [I] a horizontal cylinder of radius a rolling inside a perfectly rough hollow horizontal

cylinder of radius b>a. [II] a particle constrained to move along a line under the influence of a force which is

inversely proportional to the square of its distance from a fixed point and a damping force proportional to the square of the instantaneous speed.

[III] a particle moving on a very long frictionless wire which rotates with constant angular speed about a horizontal axis.

10

(b) When the Lagrangian function has the form

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10.

11.

L =qkqk-~(1-q2K) show that the generalized acceleration is zero.

20 (c) The ends of a uniform rod AB of length 2a cos 15 and weight W are constrained to slide on a

smooth circular wire of radius a fixed with its plane vertical. The end A is connected by an elastic string of natural length a and modulus of elasticity W/2 to the highest point of the wire. If 8 is the angle which the perpendicular bisector of the rod makes with the downward vertical, show that the potential energy V is given by

(a)

V=- ~a {cos(B-75)+2cosX(B+75)}+costant VerifY that 8=25 defines a position of equilibrium and investigate its stability.

30

A uniform rod oflength 2a which has one end attached to a fixed point by a light inextensible string of length 5a/12 is performing small oscillations in a vertical plane about its position of equilibrium Find the position at any time and show that the periods of its principal oscillations are

30

(b) a uniform circular disc of radius a and mass m rolls down a rough inclined plane without sliding. Show that the centre of the disc moves with constant acceleration 2/3 g sin a and the coefficient of friction !J. > 1/3 tan a, where a is the inclination of the plane.

30

(a) Show that the variable ellipsoid

is a possible form for the boundary surface of a liquid motion at any timet.

(b) Find the lines of flow in the 2-dimensional fluid motion given by ~+i'l' = -1/2n(x+iy)2e2int

20 (c) A source of strength m and a vortex of strength k are placed at the origin of the 2-dimensional

motion of unbounded liquid. Prove that the pressure at infinity exceeds that pressure at distance r from the origin by

1 (m2 +k2) -- p 2 r 2

20 12. (a) Compute to 4 decimal placed by using Newton-Raphson method, the real root of

x2 + 4 sin x = 0

20

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14.

15.

(b) Solve by Runge-Kutta method

dy =X+ y dx

with the initial conditions x0=0, y0=1 correct up to 4 decimal places, by evaluating up to second increment ofy. (Take h=0.1)

20

(c) Fit the natural cubic spline for the data

(a)

x:

y:

0

0

1

0

2

1

3

0

4

0

The standard deviations of two sets containing n1 and n2 are cr1 and cr2 respectively being measured from their respective means m1 and m2. If the two sets are grouped together as one set ofn1 + n2 members, show that the standard deviation cr of this set measured from its mean is given by

20

(b) Each coefficient in the equation Ax2+Bx+C=O is determined by throwing an ordinary die. Find the probability that the equation will have real roots.

20

(c) Prove that if A, B and C are random events in a sample space and if A, B, C are pair wise independent and A is independent of BuC, then A, B and C are mutually independent.

(a) Fit the curve y=aebx to the following data ( e being napierian base 2. 71828): x:

y:

0 2

5.012 10 4

31.62

20

(b) Let x12, x22 be independently distributed variates having chi-square distributions with n1, n2 degrees of freedom respectively. Derive the distribution of

F = x( I n1 xi I n2

20

(c) For a random sample of 10 horses fed on diet A, the increase in weight in kilograms in a certain period were

(a)

10, 6, 16, 17, 13, 12, 8, 14, 15, 9 For another random sample of 12 horses fed on diet B, the increase in the same period were

7, 13, 22, 15, 12, 14, 18, 8, 21, 23, 10, 17 kg Test whether the diets A and B differ significantly as regard the effect on increase in weight. You may use the fact that 5% value oft for 20 degrees of freedom is 2. 09.

20

Maximize

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16.

x1+x2::::;4.

x1-x2::::;2.

20

(b) the following table gives the cost for transporting material from supply points A,B,C,D to demand points E, F, G, H, J:

To E F G H J

A 8 10 12 17 15 B 15 13 18 11 9

From c 14 20 6 10 13 D 13 19 7 5 12

The present allocation is as follows : AtoE90; AtoF10; BtoF150; CtoF10 C toG 50; C to J 120; D to H 210; D to J 70 (i) Check if this allocation is optimum If not, find an optimum schedule. (ii) If in the above problem the transportation cost from A to G is reduce to 10, what will

be the new optimum schedule? 20

(c) Minimize z = Yl + Y2 + .... + Yn subject to YlY2 .. Yn = d and Yj20 for all j.

(a) Determine x1, x2, X3 so as to maximize z=-x12 -xl-x3 2+4xl+6x2

subject to the constraints x1+x2::::;2 2xl+3x2::::;12 Xl, X2Z2

20

20 (b) At what average rate must a clerk at a super-market work in order to insure a probability of

0.90 that the customer will not have to wait longer than 12 minutes? It is assumed that there is only one counter to which customers arrive in a Poisson fashion at an average rate of 15 per hour. The length of service by the clerk has an exponential distribution.

20 (c) There are 5 jobs, each of which must go through machines A, B and C in the order ABC.

Processing times are given in the following table: Job Processing times (hours)

A B C 1 8 5 4 2 10 6 9 3 6 2 8 4 7 3 6 5 11 4 5

Determine a sequence for five jobs that will minimise the elapsed time. 20

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Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

Candidates should attempt any FIVE questions. All questions carry equal marks.

---. '(

Show that the setS = {(1,0,0), (1,1,0), (1,1,1), (O,l,O)}spans the vector space R\R) but it is not a basis set. Define rank and nullity of a linear transformation T. IfV be a finite dimensional vector space and T a linear operator on v such that rank T2 = rank T, then prove that the null space ofT = the null space of T2 and the intersection of the range space and null space to T is the zero subspace ofV. If the matrix of a linear operator T on R2 relative to the standard basis {(1,0), (0,1)} is G ~),what is the matrix ofT relative to the basis B = {(1,1), (1,-1)}? Prove that the inverse of ( ~ ~) is

[ A-

1 0 J

c-1BA-1 c-1

where A, Care non-singular matrices and hence find the inverse of

1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1

If A be an orthogonal matrix with the property that-1 is not an eigenvalue, then show that a is expressible as (I-S) (S+srs1 for some suitable skew-symmetric matrix S. Show that any two eigenvectors corresponding to two distinct eigen values of (i) Hermitian matrix (ii) Unitary matrix are orthogonal. A matrix B or order n x n is of the form leA where A, is a scalar and A has unit elements everywhere except in the diagonal which has elements !J.. Find A, and !J. so that B may be orthogonal. Find the rank of the matrix

A=~~ ; 1 ~3 ~J ls 3 3 11 j by reducing it to canonical form

(c) Determine the following form as definite, semi-definite or indefinite: 2x12 + 2x22 + 3x/- 4x2x3- 4x3x1 + 2x1x2

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4.

5.

6.

7.

8.

9.

10.

(a)

(b)

(c)

(a)

Prove that f(x) = x2 sin ..!_, x:;tO and f(x) = 0 for x = 0 is continuous and differentiable at x = 0 X

but its derivative is not continuous there. If f(x), ~(x), 'l'(x) have derivatives when a:s;x::::;b, show that there is a value c of x lying between a and b such that

f(a) (a) lf/(a) f(b) (b) lf/(b) =O .. f f(c) (c) !f/(c)

Find the triangle of maximum area which can be inscribed in a circle.

Prove that fooo e -ax2

dx = ~ (a > 0) and deduce that fooo x 2ne -x2 dx = ;: [ 1.3.5 ... ( 2n -1)]. (b) Define Gamma function and prove that

(c)

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c) (a)

rnl(n+~)= 2./f_, f2n Show that the volume common to the sphere x2+y2+z2 = a2 and the cylinder x2+y2=ax is 2a3 9(3;z--4). If ax2+2hxy+b/+2gx+2fy+c=O represents a pair of lines, prove that the area of the triangle formed by their bisectors and axis of x is

(a-b) 2 +4h2 ca-g2 2h ab-h2

Find the equation of the director circle of the conic l/r = 1 + e cos 8 and also obtain the asymptotes of the above conic. A line makes angles a, ~' y, 8 with the diagonals of a cube. Prove that

cos2a + cos2~ + cos2y + cos28 = 4/3

Prove that the centres of the spheres which touch the lines y = mx, z = c; y = -mx, z = -c lie upon the conicoid mxy + cz(l + m2) = 0. Find the locus of the point of intersection of perpendicular generator of a hyperboloid of one sheet. A curve is drawn on a parabolic cylinder so as to cut all the generators at the same angle. Find its curvature and torsion. Determine the curvature for which the radius of curvature is proportional to the slope of the tangent. Show that the system of confocal conics

x2 i - 2 - + - 2-- = 1 is self-orthogonal. a +A b +A Solve {y(l+1/x)+cosy} dx+ {x+logx-xsiny}dy=O

Solve y d2y- 2(dyJ2 = y2 dx2 dx

Solve d2

; + cv6 y =a cos cvt and discuss the nature of solution as cv dt2

dt Solve (n4 + D 2 + 1 )ye-x12 cos( x.fi I 2). Prove that the angular velocity of rotation at any point is equal to one half of the curl of the velocity vector V.

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11.

12.

(b) Evaluate Jfs 11x FiidS where S is the upper half surface of the unit sphere x2+l+i=1 and F = zF + x} + yk .

(c)

(a)

a A Show that _P_ is not a tensor even though Ar is a covariant tensor or rank one.

axq The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is plog[l+7'] where !J. is the coefficient of friction.

(b) A solid hemisphere is supported by a string fixed to a point on its rim and to point on a smooth vertical wall with which the curved surface of the sphere is in contact. If 8 and ~ are the inclinations of the string and the plane base of the hemisphere to the vertical, prove that

3 tan=-+tane.

8 (c) A semi circular lamina is completely immersed in water with its plant vertical, so that the

extremity A of its bounding diameter is in the surface and the diameter makes with the surface and angle a. Prove that if E be the centre of pressure and ~ the angle between AE and the diameter,

(a)

"' 3;r + 16tana tan'f'=-----16+15;rtana

A point executes simple harmonic motion such that in two of its positions, the velocities are u and v and the corresponding accelerations are a and ~ Show that the distance between the

. . . v2 -u2 positiOns IS .

a-fJ (b) A particle moves under a force

m!J.{3au4-2(a2-b2)u5}, a>b

and is projected form an apse at a distance a+ b with velocity ~ . show that its orbit is r = a+b

a+ b cos8. (c) A particle is projected upwards with a velocity u in a medium whose resistance varies as the

square of the velocity. Prove that it will return to the point of projection with velocity u v f . v ( -1 u anh-1 u ) h . h . al 1 . v = 1 a ter a time- tan -t - w ere VIS t e termm ve ocity.

"\IU2+V2 g V V

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I -

Time Allowed: 3 hours Maximum Marks: 300

1.

Candidates should attempt any jive Questions. ALL Questions carry equal marks.

SECTION- A (a) If G is a cyclic group of order n and p divides n, then prove that there is a homomorphism of

G onto a cyclic group of order p. what is the Kernel ofhomorphism? (b) Show that a group or order 56 cannot be simple.

20 (c) Suppose the H, K are normal subgroups of a finite group G with H a normal subgroup of K.

(a)

(b)

If P = K!H, S = GIH, then prove that the quotient groups S/P and G/K are isomorphic.

If Z is the set of integers then show that z[ .J=3]={a+Hb:a,bEZ} is not a unique factorization domain.

Construct the addition and multiplication table for z3 [x] /

20

20

(c) where Z3 is the set of integers modulo 3 and is the ideal generated by (x2+ 1) in Z3[x]. Let Q be the set of rational number and Q(2112, 2113) the smallest extension field of Q

. . 21/2 21/3 p d h b . c Q(21/2 21/3) Q contammg , . m t e as1s 10r , over . (a) Does the set C of all complex numbers form a metric space under

( ) _ I Z1 + Z2 I ? d 2 b 2 2 - 112 {(1+ I z11 2)(1+ I z2l2)}

Justify your claim

(b) Show that x = 0 is a point of non-uniform convergence of the series whose nth term is n2 xe-n2x2 -(n-1 )2 X e-(n-1)2X2.

(c) Find all the maxima and minima of f(x,y) = x3+y3 -63(x+y )+ 12xy.

(a)

(b)

Examine for Riemann integrability over [0,2] of the function defined in [0,2] by f ( x) = { x + x2 , for rational values of x

x2 + x3 , for irrational values of x

Prove that Joo sm x dx converges and conditionally converges. 0 X

20

20

20

20

20

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(c) Evaluate Iff dxdydz x+ y+x+1

over the volume bounded by the coordinate planes and the plane x + y + z = 1. 20

5. (a) In the finite z-plane, show that the function 1 f(z) =sec-z

has infinitely many isolated singularities in a finite intervals which includes 0. 20

(b) Find the orthogonal trajectories of the family of curves in the xy-plane defined by e-x(x sin y-y cos y)=a

where a is real constant. 20

(c) Prove that (by applying Cauchy Integral formula or otherwise) rff 2n 1.3.5 ... (2n-1)

cos ()d() = 2;r o 2.4.6 ... 2n

where n = 1, 2, 3 .......... 20

6. (a) If cis the curve y = x3 -3x2+4x-1 joining the points (1, 1) and (2,3) find the value of L(12z2 -4iz}tz

20

(b) oo n

Prove that I ( z ) converges absolutely for lzl::::;l. n=l n n + 1

20 (c) Evaluate

foo dx o x6 +1

by choosing an appropriate contour. 2

7. (a) Find the surface whose tangent planes cut off an intercept of constant length R from the axis ofz.

20 (b) Solve (x3+3x/)p + (y3+3x2y)q=2(x2+/)z.

20 (c) Find the integral surface of the partial differential equation (x-y)p+(y-x-z)q=z.

thought the circle z =1, x2+/=l. 20

8. (a) Using Charpit's method find the complete integral of2xz-px2-2qxy+pq=O. 20

(b) Solve r-s+2q-z=x~2 . 20

(c) Find the general solution of x

2r-/t+xp-yq =log x.

20

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9.

10.

11.

(a) (I) (II)

SECTION- B

Consider the two dynamical systems: A sphere rolling down from the top of a fixed sphere. A cylinder rolling without slipping down a rough inclined plane. (i) State whether (I) is rheonomic, holonomic and conservative and justify your claim (ii) Give reasons for (II) to be classified as scleronomic, holonomic. Is it conservative?

10 (b) Find

(i) the Lagrangian (ii) the equations of motion for the following system: A particle is constrained to move in a plain under the influence of an attraction towards to origin proportional to the distance from it and also of a force perpendicular to the radius vector inversely proportional to the distance of the particle from the origin in anticlockwise direction.

30 (c) A heavy uniform rod rotating in a vertical plane falls and strikes a smooth inelastic horizontal

plane. Find the impulse,

(a) 20

The door of a railway carriage has its hinges, supposed smooth, towards the engine, which starts with an acceleration f. Prove that the door closes in time.

[a2 + K2 Jl/2 rr/2 d()

2af fo -&8 with an angular velocity ~

22af 2 where 2a is the breadth of the door and K its radius of

a +K gyration about a vertical axis through G, the centre of mass.

30 (b) A solid homogeneous sphere is resting on the top of another fixed sphere and rolls down it.

(a)

Write down the equations of motion and find the friction. When does the upper sphere leave the lower sphere if (i) both the spheres are smooth (ii) the upper sphere is sufficiently rough so as not to slip.

30 Show that

-2xyz ( x2- i)z y U= 2'V= 2'W= 2 2

( x2 + y2) ( x2 + y2) x + y are the velocity components of a possible liquid motion. Is this motion irrotational?

(b) Steam is rushing from a boiler through a conical pipe, the diameters of the ends of which are D and d; if v and v be the corresponding velocities of the steam and if the motion be supposed to be that of divergence from the vertex of the cone, prove that

~= D 2 e(v2 -v2 )!2K v d 2

where K is the pressure divided by the density and supposed constant. 20

(c) Prove that for liquid circulating irrotatinally in part of the plane between two non-intersecting circles, the curves of constant velocity are Cassini's ovals.

20

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12.

13.

14.

(a)

(b)

Find correct to 3 decimal places the two positive roots of 2ex -3x2=2.5644. 20

Evaluate approximately J3 x4dx Simpson's rule by taking seven equidistant ordinates. -3

Compare it with the value obtained by using the trapezoidal rule and with exact value. 20

(c) Solve dy/dx = xy

(a)

for x=l.4 by Runge-kutta method, initially x=1, y=2 (Take h=0.2). 20

The intelligence quotient of 480 student is as follows :

Class marks X Frequency f 70 4 74 9 78 16 82 28 86 45 90 66 94 85 98 72 102 54 106 38 110 27 114 18 118 11 122 5 126 2

Find the moment coefficient of skewness for the intelligence quotient. It is skewed to the left or right?

20

(b) If the correlation coefficient of zero order in a set of P variates were equal to p, show that every partial correlation of the s-th order is

p 1+sp

20

(c) Fit a second degree parabola to the following data taking x as the independent variable:

(a)

X: 1 2

y: 2 6

3

7

4

8

5

10

6

11

7

11

8

10

9

0

20

An event A is known to be independent of the event B, BuC and BnC. Show that it is also independent of C.

20

(b) If the probability that an individual suffers a bad reaction from injection of a given serum is 0.001, determine the probability that out of2000 individuals

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16.

(i) exactly 3 will suffer a bad reaction (ii) more than 2 will suffer a bad reaction.

20

(c) A machine has produced, in the past, washers having a thickness of 0.050 inches. To determine whether the machine is in proper working order, a sample of 10 washers is chosen for which the mean thickness is 0.053 inches and the standard deviation is 0.003 inches. Test the hypothesis that the machine is in proper working order using a level of significance of

(a)

(i) 0.05 (ii) 0.01.

Use Simplex method to solve:

Minimize

Subject to xo = x1 - 3x2 + 2x3

3xl-x2+2x3::::;7

-2xl+4x2::::;12

-4xl+3x2+8x3::::;10

20

(b) A Departmental Head has four subordinates and four tasks are to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the times each man would take to perform each task is given in the effectiveness matrix below. How should the tasks be allocated one to one man, so as to minimize the total man hours?

20 Man

I II III IV A 8 26 17 11

Task B 13 28 14 26 c 38 19 18 15 D 19 26 24 10

(c) Minimize z = (yl)2 + (y2i + (y3i Subject to Y1 + Y2 + Y3 215,

Yl, Y2, Y3 2 0, by dynamic programming.

20

(a) Use Kuhn-Tucker conditions to Minimize f(x) = (x1)2 + (x2)2 + (x3)2

subject to 2x1 + x2- 5::::; 0 X1 + X3- 2::::; 0

1-xl::::;o

2-x2::::;o

-x3::::;o

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20 (b) On an average 96 patients per 24-hour day require the service of an emergency clinic. Also

on average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic Rs. 100 per patient treated to obtain an average servicing time of 10 minutes and that each minute of decrease in this average time would cost Rs. 10 per patient treated. How much would have to budgeted by the clinic to decrease the average size of the queue from 1~ patients to ~ patient?

20 (c) A transport manager finds from his past records that the costs per year of running a two-wheeler

whose purchase cost is Rs. 6000/- are given below:

Year Running Costs Resale (Salvage) price

1 1000 3000

2 1200 1500

3 1400 750

4 1800 375

5 2300 200

6 2800 200

7 3400 200

8 4000 200

At what age the two wheeler should be replaced? (Note that the money value is constant in time)

20

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Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

(a)

Candidates should attempt any FIVE questions. All questions carry equal marks.

Show that f1(t) = 1, f2(t) = t-2, f3(t) = (t-2)2 form a basis ofP3, the space of polynomials with degree ::::; 2. Express 3t2- 5t+4 as a linear combination of f1, f2, h

20 (b) If T : V 4(R) ----+ V 3(R) is a linear transformation defined by

T(a,b,c,d) = (a-b+c+d, a+2c-d, a+b+3c-3d) For a, b, dER, then verify that Rank T +Nullity T = dim V 4 (R) .

20 (c) If T is an operator on R3 whose basis is

B = {(1,0,0), (0,1,0), (0,0,1)} such that

(a)

[r:B]=~~ ~ ~1l l-1 -1 0 J find the matrix ofT with respect to a basis B1 = {(0,1,-1), (1,-1, 1), (-1, 1, 0)}.

If A = [ aij] is an n x n matrix such that aii = n, aij = r if i :;t:j, show that

[A-(n-r) I] [A-(n-r+nr) I]=O. Hence find the inverse of the n x n matrix B = [bij] Where bii = 1, bij = p when i:;t:j and

1 p:;t:1, p:;t:--. 1-n

20

20 (b) Prove that the eigen vectors corresponding to distinct eigen values of a square matrix are

linearly independent. 20

(c) Determine the eigenvalues and eigenvectors of the matrix

(a)

A~l~ ~ ~j 20

Show that a matrix congruent to a skew-symmetric matrix is skew-symmetric. Use the result to prove that the determinant of skew-symmetric matrix of even order is the square of a rational function of its elements.

20

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3.

(b) Find the rank of the matrix 0 c -b a' -c 0 a b' b -a 0 c'

-a' -b' -c' 0 where aa'+bb'+cc'=O and a,b,c are all positive integers.

20 (c) Reduce the following symmetric matrix to a diagonal form and interpret the result in terms of

quadratic forms:

A=~~ ~ ~1l l-1 3 1J (a) f(x) is defined as follows :

~(b 2 -a2 ) for O

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10.

11.

12.

(c) Solve (D2-4D+4)y = 8x2 e2x sin 2x d

where D=-. dx

20 (a) Show that rnr is an irrotational vector for any value ofn, but is solenoidal only ifn = -3.

20 (b) If F=yt +(x-2xz)]-xyk,evaluate

ffs( 11 x P).ndS

(c)

(a)

Where S is the surface of the sphere x

2+/+z2=a2 above the xy plane.

Prove that {1 } = _j_(log fi). zk axk

20

20 Show that the length of an endless chain which will hang over a circular pulley of radius a so as to be in contact with two-thirds of the circumference of the pulley is

a{ log( :+FJ) + 4;} 20

(b) A smooth rod passes through a smooth ring at the focus of an ellipse whose major axis is horizontal and rests with its lower end on the quadrant of the curve which is farthest removed from the focus. Find its position of equilibrium and show that its length must at least be

( 3: + : .J1 + 8e2 J where 2a is the major axis and e, the eccentricity. 20

(c) The height of a balloon is calculated from the barometric pressure reading (p) on the assumption that the pressure of the ai varies as the density. Show that if the pressure actually varies as the nth power of the density, there will be an error

(a)

h"l n~l {~-(~f}-log ~ J in the calculated height where ho is the height of the homogeneous atmosphere and Po is the pressure at the surface of the earth.

20 If in a simple Harmonic Motion, the velocities at distances a, b, c from a fixed point on the straight line which is not the centre of force, be u, v, w respectively, show that the periodic

u2 2

timeTisgivenby 4~ (b-c)(c-a)(a-b)= a T 1

v2 w2

b c 1 1

20 (b) A gun is firing from the sea-level out to sea. It is mounted in a battery h meters high up and

fired at the same elevation a. Show that the range is increased by - 1 + 2 ~ 2 -1 of 1 [( 2 h )112 l

2 u sm a

itself, u being the velocity of projectile. 20

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(c) A particle of mass m is projected vertically under gravity, the resistance of the air being mk 2

times the velocity. Show that the greatest height attained by the particle is ~[ A-log(l +A) J g

where V is the terminal velocity of the particle and 'A V is the initial vertical velocity. 20

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i:,\ : ;' 'I I -

Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

(a)

Candidates should attempt any jive Questions. ALL Questions carry equal marks.

SECTION- A If G is a group such that (abt = anbn for three consecutive integers n for all a,b in G, then prove that G is abelian.

20 (b) Can a group of order 42 be simple? JustifY your claim

20 (c) Show that the additive group of integers modulo 4 is isomorphic to the multiplicative group

of the non-zero elements of integers modulo 5. State the two isomorphisms. 20

(a) Find all the units of the integral domain of Gaussian integers. 20

(b) Prove or disprove the statement : The polynomial ring I [x] over the ring of integers is a principal ideal ring.

20 (c) If R is an integral domain (not necessarily a unique factorization domain) and F is its field of

(a)

quotients, then show that any element f(x) in F(x) is of the form f ( x) = fo ( x) where f0(x) E a

R[x], aER.

If dis a metric on a nonempty set X, then is the function e defined by e(a,b) =min (1, d(a,b) for a,b E X, also a metric on X? Substantiate your claim

20

20 (b) Let C[O,l] denote the collection of all real continuous functions defined on I = [0, 1]. Suppose

d and e are metrics on C [0,1] defined by d(f,g) =sup {lfx)-g(x)l: xEI} e(f,g)= f~lf(x)-g(x)dxl. Is the topology 'Td induced by d coarser then "Te, the topology induced by e? JustifY your answer.

(c) Examine the (i) absolute convergence (ii) uniform convergence Of the series (1-x)+x(l-x)+x\1-x)+ ... in [ -c, 1] where O

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8.

9.

10.

xy = x+y, z =1. 20

(c) Obtain a computer solution ofpq = xrn yn z2l. 20

(a) Use Charpit's method to solve 16 p2z2 + 9q2z2 + 4z2 - 4 = 0.

Interpret geometrically the complete solution and mention the singular solution. 20

(b) Solve (D2 + 3 DD' + 2 D' 2)z = x+y, by expanding the particular integral in ascending powers ofD, as well as in ascending powers ofD'.

20 (c) Find a surface satisfYing (D2 + DD')z = 0

(a)

and touching the elliptic paraboloid z = 4x2 + / along its section by the plane y = 2x + 1.

SECTION- B

20

What is D' Alembert's principle ? An inextensible string of negligible mass hanging over a smooth page at A connects the mass m1 on a frictionless inclined plane of angle 8 to another mass m

2. UseD' Alembert's principle to prove that the masses will be in equilibrium if m2 =

m1 sin 8. 20

(b) Two mass points of mass m1 and m2 are connected by a string passing through a hole in smooth table so that m1 rests on the table surface and m2 hangs suspended. Assuming m2 moves only in a vertical line, what are the generalized coordinates of the system ? Write down Lagrange's equations of motion and obtain a first integral of the equations of motion.

20 (c) Twelve equal uniform rods are smoothly joined at their ends so as to form a cubical

framework, which is suspended from a point by a string tied to one comer and kept in shape by a light string occupying the position of a vertical diagonal. Suppose that the string supporting the framework is cut, so that it falls and strikes a smooth inelastic horizontal plane. Find the impulsive reaction of the plane.

(a) 20

A small light ring is threaded on a fixed thin horizontal wire. One end of a uniform rod, of mass m and length 2a is freely attached to the ring. The coefficient of friction between the ring and the wire is !J.. The system is released from rest when the rod is horizontal and in the vertical plane containing the wire. If the ring slips on the wire when the rod has turned through an angle a, then prove that

!J-(10 tan2 a+ 1) = 9 tan a. 30

(b) A uniform rod AB held at an inclination a to the vertical with one end A in contact with a rough horizontal table. If released, then prove that the rod will commence to slide at once if the coefficient of the friction !J. is less than

3sinacosa 1+3cos2 a

30

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11.

12.

(a) The particle velocity for a fluid motion referred to rectangular axes is given by

A cos-cos-, 0, Asm-sm-( ~X ~z . ~X . ~ZJ 2a 2a 2a 2a

where A, a are constants. Show that this is a possible motion of an incompressible fluid under no body forces in an infinite fixed rigid tube

-a ::::; x ::::; a, 0 ::::; z ::::; 2a.

Also find the pressure associated with this velocity field.

20 (b) Determine the streamlines and the path lines of the particles when the velocity field is given

by

(c)

(a)

( X y Z J

------

1+t'1+t'1+t .

20

Between the fixed boundaries () = !!..._ and () = _!!..._, there is a two dimensional liquid motion 4 4

due to a source of strength m at the point r = a, 8 = 0 and an equal sink at the point r = b, q = 0. Use the method of images to show that the stream function is

Show also that the velocity at (r, 8) is

4m(a4 -b4 )r3

Find the positive root of the equation

x2 x3 ex =1+x+-+-e03x

2 6

correct to five decimal places.

20

20 (b) Fit the following four points by the cubic splines.

0 1 2 3

xi 1 2 3 4

Yi 1 5 1 1 8

Use the end conditions y"o = y"3 = 0.

Hence compute (i) y (1.5) (ii) y'(2)

20

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13.

(c) Find the derivative off(x) at x = 0.4 from the following table:

(a)

X 0.1 0.2 0.3 0.4

y=f(x) 1.10517 1.22140 1.34986 1.49182

Heights of 100 students at a college are as follows :

Heig_ht (jn units~ Number o[students

60-62 5

63-65 18

66-68 42

69-71 27

72-74 8

[1 unit= 2.5 em] Find the moment coefficient of skewness. Also obtain its value when Sheppard's corrections for grouping are used.

20 (b) The frequency distributions of the final grades of 100 students in Mathematics and Physics

are shown in the following table:

Mathematics grades

Total 40-49 50-59 60-69 70-79 80-89 90-99

fy

90-99 2 4 4 10

80-89 1 4 6 5 16

70-79 5 10 8 1 24

60-69 1 4 9 5 2 21

50-59 3 6 6 2 21

40-49 3 5 4 12

Total 7 15 25 23 20 10 100

fx

For this data, compute the standard error of estimate Sx, v and their covariance Sxv. 20

(c) A college entrance examination consisted of 3 tests in Mathematics, English and General Knowledge. To test the ability of the examination to predict performance in a Statistics course, data concerning a sample of 200 students were gathered and analyzed. Letting

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14.

15.

(a)

X1 =grade in Statistics course

X2 = score on Mathematics test

X3 = score on English test

X4 = score on General Knowledge test

the following calculations were obtained :

the mean of

the standard deviation X1 = S1 = 10, S2 = 5, S3 = 3, S4 = 6;

the linear correlation coefficient between X1 and X2 = r12 = 0.90, r13 = 0.75, r14 = 0.80, r23 = 0.70, r24 = 0,70, r34 = 0.85.

Find the least square regression equation of X1 and X2, X3 and X4. What is the predicted grade in Statistics of a student who scores 30 in mathematics, 18 in English and 32 in general knowledge?

The contents ofurns 1,2,3 are as follows :

1 white, 2 black, 3 red balls

2 white, 1 black, 1 red balls

4 white, 5 black, 3 red balls

20

One urn is chosen at random and two balls drawn. They happen to be white and red. What is the probability that they came from urn 2 or 3?

20

(b) Ten percent of the tools produced in a certain manufacturing process tum out to the defective. Find the probability that in a sample of 10 tools chosen at random, exactly two will be defective by using

(i) the binomial distribution (ii) the Poisson approximation to the binomial distribution.

20

(c) Show that in a 2 x 2 contingency table wherein the frequencies are

(a)

a b c d

Chi-square (x2) calculated from independent frequencies is

2 (a+b+c+d)(ad-bc)2

X =~--~--~~~~~~

Maximise

Subject to

(a +b)( c+d)(b +d)( a +c) z = 3xl + 2x2 + 5x3 (by Simplex method) X1 + 2X2 + X3 ::::; 430

3xl + 2x3::::; 460

x1 + 4x2::::; 420

20

20

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(b) Consider the following data :

Destinations

1 2 3 Capacities

1 2 2 3 10

Sources 2 4 1 2 15

3 1 3 X 40

Demands 20 15 30

The cost of shipment from third source to the third destination is not known. How many units should be transported from the sources to the destinations so that the total cost of transporting all the units to their destinations is a minimum?

20

(c) Divide a quantity b into n parts so as to maximize their product. Let fn(b) denote the value.

(a)

Show that f1(b) = b, fn(b) =max {zfn-l(b-z)}. O::::;z::::;b Hence find fn(b) and the division that maximises it.

Apply Wolfe's method for solving the quadratic programming problem

Maximise

Zx = 4xl + 6xr2(xl)2 -2xlxr2(x2)2

subject to x1+2x2::::;2

20

20 (b) A car hire company has one car at each of five depots a, b, c, d, e. A customer requires a car

in each town namely A, B, C, D, E. Distance (in kilometers between depots (origins) and towns (destinations) are given in the following distance matrix:

a b c d e

A 160 130 175 190 200

B 135 120 130 160 175

c 140 110 155 170 185

D 50 50 80 80 110

E 54 34 70 80 105

How should cars be assigned to customers so as to minimize the distance travelled?

20

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(c) A person is considering to purchase a machine for his own factory. Relevant data about alternative machines are as follows:

Machine A MachineD Machine C

Present investment(Rs.) 10,000 12,000 15,000

Total annual cost(Rs.) 2,000 1,500 1,200

Life (years) 10 10 10

Salvage Value (Rs.) 500 1,000 1,200

As an adviser to the buyer, you have been asked to select the best machine, considering 12 per cent annual rate of return. You are given that

(i) Single payment present worth factor@ 12% for 10 years= 0.322. (ii) Annual series present worth factor@ 12% for 10 years= 5.650.

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Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

4.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

Candidates should attempt any FIVE questions. All questions carry equal marks.

Let T be the linear operator in R3 defined by T(x1, x2, x3) = (3xl + X3, -2x1 + x2, -x1 + 2x2 + 4x3). What is the matrix ofT in the standard ordered basis for R3? What is a basis of range space of T and a basis of null space ofT ? Let A be a square matrix of order n. Prove that AX= b has a solution if and only if bERn is orthogonal to all solutions Y of the system ATY= 0. Define a similar matrix. Prove that the characteristic equation of two similar matrices is the same. Let 1, 2, 3 be the eigen-values of a matrix. Write down such a matrix. Is such a matrix unique?

Show that A=~ ~1 ~6 ~6l is diagonalizable and hence determine A5. l3 -6 -4J

Let A and B be matrices of order n. Prove that if (I -AB) is invertible, then (1-BA) is also invertible and (I-BAr1 =I+ B(I-ABr1 A Show that AB and BA have precisely the same characteristic values. If a and b are complex numbers such that lbl = 1 and H is a Hermitian matrix, show that the eigen-values of ai + bH lie on a straight line in the complex plane. Let A be a symmetric matrix. Show that A is positive definite if and only if its eigen-values are all positive. Let A and B be square matrices of order n. Show that AB-BA can never be equal to unit matrix.

(c) Let A = [ aij]; i, j = 1, 2, ... , n and

(a)

(b)

I aii I> I I ail I for every i = 1, 2, .... n. Show that A is a non-singular matrix. Hence or j'Fi

otherwise prove that the eigen-values of A lie in the discs I A -aii 1::::; IlaiJI,i = 1,2, .... ,n

j'Fi in the complex plane. If g is the inverse off and

f'(x) =-1-3' 1+x

prove that g'(x) = l+[g(x)]3. Taking the nth derivative of (xn)2 in two different ways, show that

n2 n2(n-1)2 n2(n-1)2(n-2)2 (2n)!

1 + 2 + 2 2 + 2 2 2 + ... to ( n + 1) terms = --2 1 1 .2 1 .2 .3 (n!)

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5.

6.

7.

8.

9.

(c)

(a)

Let f(x,y), which possesses continuous partial derivatives of second order, be a homogeneous function of x andy of degree n. Prove that

x2fxx + 2xyfxy + /fyy = n(n-1 )f

Find the area bounded by the curve

[+iJ2 .__y2 4 9 4 9 (b) Let f(x), x 21 be such that the area bounded by the curve y = f(x) and the lines x = 1, x=b is

equal to ~1 + b2 - .J2 for all b21. Does f attain its minimum? If so what is its value? (c) Show that

(a)

1 ( J 1 ( ~ J 1 ( ~ J ... r ( n ~ 1 J = (~) n ~ 1 Two conjugate semi-diameters of the ellipse x2 y2 2+-2 = 1 cut the circle x

2+y2=r2 at P and Q. a b Show that the locus of the middle point of PQ is

a2 {(x2 + y2r -r2x2 }+b2 {( x2 +if -r2i} = 0

(b) If the normal at one of the extremities of latus rectum of the conic .!_ = 1 + e cos(), meets the r

curve again at Q, show that l ( 1 + 3e2 + e 4 )

SQ = ----7---------,--'-(1+e2-e4) '

where S is the focus of the conic. (c) Through a point P(x', y', z') a plane is drawn at right angles to OP to meet the coordinate

5 axes in A, B, C. Prove that the area of the triangle ABC is r , where r is the measure of

(a)

(b)

(c)

(a)

2x'y'z' OP. Two spheres of radii r1 and r2 cut orthogonally. Prove that the area of the common circle is

;rr2r2 1 2

r2 +r2 1 2 Show that a plane through one member of the /c-system and one member of ~J.-system is tangent plane to the hyperboloid at the point of intersection of the two generators. Prove that the parallels through the origin to the binormals of the helix

x=acos8, y=asin8, z = k8 1. h . h 2( 2 2) k2 2 1e upon t e ng t cone a x +y = z . Determine a family of curves for which the ratio of they-intercept of the tangent to the radius vector is a constant.

(b) Solve

(c)

(a)

Test whether the equation (x+y)2 dx- (/-2xy-x2) dy=O is exact and hence solve it. Solve

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(b) Determine all real valued solutions ofthe equation

(c)

. . dy y"'-1y"+y'-1y = 0, y' = dx

Find the solution of the equation y" + 4y = 8 cos 2x, give that y = 0 andy'= 2 when x = 0.

10. (a) Consider a physical entity that is specified by twenty-seven numbers Aijk in a given coordinate system In the transition to another coordinate system of this kind, let Ajk Bjk

(b)

(c)

transform as a vector for any choice of the anti-symmetric tensor Bjk Prove that the quantities Ajk - Akj are the components of a tensor of third order. Is Ajk the components of a tensor? Give reasons for your answer. Let the region V be bounded by the smooth surface S and let n denote outward drawn unit normal vector at a point on S. If~ is harmonic in V, show that

I a ds = o. san In the vector field u(x), let there exist a surface on which u=O. Show that, at an arbitrary point of this surface, curl u is tangential to the surface or vanishes.

11. (a) Prove that for the common catenary the radius of curvature at any point of the curve is equal to the length of the normal intercepted between the curve and the directrix.

12.

(b)

(c)

(a)

Two uniform rods AB and AC, smoothly jointed at A, are in equilibrium in a vertical plane. The ends B and C rest on a smooth horizontal plane and the middle points of AB and AC are

connected by a string. Show that the tension of the string is ( W ) , where W is the tanB+tanC

total weight of the rods and B and C are the inclinations to the horizontal of the rods AB and AC. A semi-ellipse bounded by its minor axis is just immersed in a liquid the density of which varies as the depth. If the minor axis be in the surface, find the eccentricity in order that the focus may be the centre of pressure. Two bodies, of masses M and M', are attached to the lower end of an elastic string whose upper end is fixed and hang at rest; M' falls off. Show that the distance of M from the upper end of the string at time t is

a+b+cco{ Jf~} where a is the unstretched length of the string, and b and c the distances by which it would be stretched when supporting M and M' respectively.

(b) A particle of mass m moves under a central attractive force mp[:, + 8;,2 J and is projected from an apse at a distance c with velocity 3~ . Prove that the orbit is

c

r =ccos(2:}

and that it will arrive at the origin after a time ;r c2

--s~ (c) If t be the time in which a projectile reaches a point P in its path and t' the time from P till it

reaches the horizontal plane through the point of projection, show that the height of P above the horizontal plane is _.!_ gtt' and the maximum height is .!. g (t + t ') 2 .

2 8

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-1 ~- ~

Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

(a)

Candidates should attempt any jive Questions. ALL Questions carry equal marks.

SECTION- A

Let G be a finite set closed under an associative binary operation such that ab = ac implies b = c

and ba = ca implies b = c for all a, b, c E G. Prove that G is a group.

20 (b) Let G be a group of order pn, where p is a prime number and n> 0. Let H be a proper

subgroup of G and N(H) = { x E G : x-1 h x E H \/ h EH}. Prove that N(H) -::F H. 20

(c) Show that a group of order 112 is not simple.

(a) 20

Let R be a ring with identity. Suppose there is an element a of R which has more than one right inverse. Prove that a has infinitely many right inverses.

20 (b) Let F be a field and let p(x) be an irreducible polynomial over F. Let< p(x) > be the ideal

generated by p(x). Prove that< p(x) >is a maximal ideal.

(c)

(a)

Let F be a field of characteristic p-::F-0. Let F(x) be the polynomial ring. Suppose f(x) = ao + a1x + ... + anxn

is an element ofF[x]. Define f'(x) = a1 + 2a2x + ... + n anxn-1

Iff'(x) = 0, prove that there exists g(x) E F[x] such that f(x) = g(xP).

20

Let K and F be nonempty disjoint closed subsets of I R2. If K is bounded, show that there exists 8>0 such that d(x, y) 28 for xEK and yEF' where d(x,y) is the usual distance between x andy.

20 (b) Let f be a continuous real function on I R such that f maps open interval onto open intervals.

Prove that f is monotonic. 20

(c) Let enzO for all positive integers n such that is convergent. Suppose {sn} is a sequence of distinct points in (a, b). For x E [a,b], define

a(x) = ~Cn. {n: X> Sn}

Prove that a is an increasing function. Iff a continuous real function on [a, b], show that b J fda= 2:Cnf(sn) a n

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5.

6.

7.

(a) Suppose f maps an open ball U c I Rn into I Rrn and f is differentiable on U. Suppose there exists a real number M > 0 such that II f'(x) II::::; M for all x E U. Prove that

lf(b)- f(a) I::::; M I b-al for all a, b E U.

20 (b) Find and classify the extreme values of the function

f(x,y) = x2 + y2 + x + y + xy 20

(c) Suppose a is a real number not equal to nn for any integer n. Prove that

(a)

(b)

(c)

(a)

JJ e -(x2+2xycosa+i)dxdy = ~ 0 0 2sma

20 Let u(x, y) = 3x2y + 2x2 - y3 - 2/. Prove that u is a harmonic function. Find a harmonic function v such that u + iv is an analytic function of z.

20

Find the Taylor series expansion of the function f ( z) = + around z = 0. Find also the z +9

radius of convergence of the obtained series. Let C be the circle lzl=2 described counter clockwise. Evaluate the integral

cosh;rz

f dz z(z2 +1) Let a 2 0. Evaluate the integral

00

I cos ax dx x2 +1 0 with the aid of residues.

20

20 (b) Let f be analytic in the entire complex plane. Suppose that there exists a constant A> 0 such

that I f(z) I ::::; A I z I for all z. Prove that there exists a complex number a such that f(z) = az for all z.

20 00

(c) Suppose a power series I anzn converges at a point zo:;tO. Let z1 be such that lz1l< lzol and

(a)

(b)

n=O z1:;tO. Show that the series converges uniformly in the disc {z: lzl::::; lz1l}.

20 In the context of a partial differential equation of the first order in three independent

variables, define and illustrate the terms: (i) the complete integral (ii) the singular integral Find the general integral of

aw aw aw (y+z+w)-+(z+x+w)-+(x+ y+w)-= x+ y+z ax ay az

20 (c) Obtain the differential equation of the surfaces which are the envelopes of a one-parameter

family of planes. 20

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9.

10.

(a) Explain in detail the Charpit' s method of solving the nonlinear partial differential equation

( az az) f x,y,z, ax' ay =0 20

(b) Solve azazaz 3 ---=z x1x2x3 axl ax2 ax3

20 (c) Solve

(a)

(Dx3 - 7Dx Dy2 - 6Dy3) z =sin (x + 2y) + e3x+y

SECTION- B How do you characterize (i) the simplest dynamical system? (ii) the most general dynamical system? Show that the equations of motion

d aL ar ----=Qk,(k=1,2, ... ,n) dt aqk aqk

20

correspond to a non-conservative but sceleronomic and holonomic system with n degrees of freedom, where q, qk, Qk are respectively the generalized coordinates, the generalized velocities and the generalized forces.

20 (b) A solid uniform sphere has a light rod rigidly attached to it which passes through the centre.

The rod is so joined to a fixed vertical axis that the angle 8 between the rod and the axis may alter but the rod must tum with the axis. If the vertical axis be forced to revolve constantly with uniform angular velocity, show that 82 is of the form

n2 (cos 8- cos ~)(cos a- cos 8)

where n, a, ~' are certain constants. 20

(c) A uniform rod of length 20 ems which has one end attached to a fixed point by a light

(a)

inextensible string of length 4_!_ ems, is performing small oscillations in a vertical plane 6

about its position of equilibrium. Find its position at any time and the periods of principal oscillations.

20 A carriage is placed on an inclined plane making an angle a with the horizon and rolls down without slipping between the wheels and the plane. The floor of the carriage is parallel to the plane and a perfectly rough ball is placed freely on it. Show that the acceleration of the carriage down the plane is

14M +4m'+14m . ------gsma 14M +4m'+21m

where M is the mass of the carriage excluding the wheels, m the sum of the masses of the wheels which are uniform circular discs and M' that of the ball which is a homogeneous solid sphere (the friction between the wheels and the axes is neglected). Show that for the motion to be possible, the coefficient of friction between the wheels and the plane must exceed the constant

7(M +m)+2M' --'---------'----tan a 14M +21m+4M'

30

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12.

13.

(b) A sphere of radius a is projected up an inclined plane with velocity V and angular velocity n in the sense which would cause it to roll up; if V> a n and the coefficient of friction greater

(a)

than ~tan a , show that the sphere will cease to ascend at the end of a time 7

5V +2an 5gsina

where a is the inclination of the plane.

Determine the restrictions on f1, f2, f3 if x2 y2 2 2 h (t)2+ !2 (t)-2 + h (t)2 = 1 a b c

is a possible boundary surface of a liquid.

30

20 (b) If a, b, c, d, e, fare arbitrary constants, what type of fluid motion does the velocity

(a + by - cz, d - bx + ez, f + ex - ey) represent?

20 (c) If the fluid fill the region of spaces on the positive side ofx-axis, which is a rigid boundary

and ifthere be a source +mat the point (0, a) and an equal sink at (0, b) and if the pressure on the negative side of the boundary be the same as the pressure of the fluid at infinity, show that the resultant pressure on the boundary is

(a)

(b)

(a -b )2 ;rpm 2 --'----.,------~ ab(a+b)

where pis the density of the fluid. 20

Find the positive root of logex =cos x nearest to five places of decimal by Newton-Raphson method.

20 3.4 3

Find the value of J f ( x) dx from the following data using Simpson's S th rule for the 1.6

interval (1.6, 2.2) and ..!:..rd, rule for (2.2, 3.4) : 3

X 1.6 1.8 2.0 f(x) 4.953 6.050 7.389 X 2.6 2.8 3.0 f(x) 13.464 16.445 20.086

2.2 9.025 3.2 24.533

2.4 11.023 3.4 29.964

20 (c) For the differential equation

(a)

: = y - x2 , y ( 0) = 1 starting values are given as

y(0.2) = 1.2186, y(0.4) = 1.4682 and y(0.6) = 1.7379. Using Milne's predictor corrector method advance the solution to x = 0.8 and compare it with the analytical solution. (Carry four decimals).

20 A and B play a game of dice. A wins if he throws 11 with 3 dice before B throws 7 with 2 dice. B wins is he throws 7 before A throws 11. A starts the game and they throw alternately. What are the odds against A winning the game ultimately?

20

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15.

(b) One bag contains three identical cards marked 1, 2, 3 and another contains two cards marked 1, 2. Two cards are randomly chosen one from each bag and the numbers observed. Denote the minimum of the two by X and the sum of the two by Y. Find the joint probability distribution of the random variable pair (X, Y). Find also the two marginal distributions and the conditional distribution of X when Y::::; 4.

20 (c) IfX1 = Y1+Y2, X2 = Y2+Y3 and X3=Y3+Y1, where Y1, Y2, Y3 are uncorrelated variates with

mean zero and standard deviation unity, find the multiple correlation of X1 on X2 and X3 and partial correlation ofX1 and X2.

(a) 20

Tests of fidelity and selectivity of 190 radio receivers produced the results shown in the following table:

Fidelity Low Average High Total

Low 6 12 32 50

Selectivity Average 33 61 18 112 High 13 15 0 28

Total 52 88 50 190 Apply the chi-square test at significance level of 0.01 and show that there is indeed a relationship between fidelity and selectivity. Th al f h. 1 o/t 1 1 . d ffi d b 1 w: ev ues o c 1-square at o eve or vanous egrees o ree om are g1ven eo

No. of degrees of freedom Chi-square value 3 11.345 4 13.277 5 13.086 6 16.812 7 18.475

20 (b) For the Gamma distribution given by the density function

{

1 xa-le-x!f3 forx>O,a>0,/]>0

f(x)= fJa~ 0 otherwise

20 find the moment coefficient of skewness.

20 (c) State and prove Chebyshev's lemma for a discrete distribution and deduce from it the weak

law of large numbers. 20

(a) Solve the following linear programming problem: Maximize:

subject to = 15 2x1 + x2+ 5x3 = 20

X1 + 2X2 + X3 + X4 = 10

20

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16

(b) Solve the transportation problem below for minimizing the cost

Costs Store

Availability 1 2 3 4 5 6

1 9 12 9 6 9 10 5

Warehouse 2 7 3 7 7 5 5 6 3 6 5 9 11 3 11 2 4 6 8 11 2 2 10 9

Requirement 4 4 6 2 4 2 22 20

(c) There are five jobs each of which must go through two machines A and Bin the order A, B.

(a)

Processing times are given below: Job 1 2 3 4 5 Time for A (in hours) : 7 3 11 5 12 Time for A (in hours): 4 8 9 10 6 Determine a sequence for the jobs that will minimise the elapsed time. Compute the total idle times for the machines in this period.

Using dynamic programming technique solve the problem below: M . . . 2 2 2 mnmze z=x1 +x2 +x3 subject to x1+x2+x3 2 30

x120,x220,x320.

20

20 (b) If P n represents the probability of finding n in the long run in a queuing system with Poisson

arrivals having parameter A and exponential service times with parameter !J., show that APn-1- (A+!J.)Pn + !J. Pn+l = 0 for n > 0

and -A Po+ !J. P1 = 0

Solve these difference equations and obtain Pn in terms of p = ~. Jl

20 (c) F h" h 11 d "1 bl or a mac me t e o owmg ata are avm a e.

Salary of Loss due to Year Cost of Spares Maintenance break-downs Resale value (Rs.) Staff (Rs.) (Rs.) (Rs.)

0 - - - 20000 1 100 1600 500 14000 2 500 1600 700 12000 3 700 1600 500 10000 4 900 2000 1000 6000 5 1300 2400 1500 3000 6 1600 2400 1600 800

Determine the optimum policy for replacement of the above machine. 20

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Time Allowed: 3 hours Maximum Marks: 300

Candidates should attempt any FIVE questions. All questions carry equal marks.

---. '(

1. (a) R4, let W1 be the space generated by (1, 1, 0,-1), (2, 6,0) and (-2, -3, -3, 1) and let W2 be the space generated by(-1,-2,-2,2), (4;6,4,-6) and (1,3,4,-3). Find a basis for the space W 1 + W2.

(b) Let V be a finite dimensional vector space and v E V, v -::F 0. Show that there exists a linear functional f on V such that f(v) -::F 0.

(c) Let V = R3 and v1, v2, v3 be a basis ofR3. LetT: V----+ V be a linear transformation such that T(v1)=v1+v2+v3, 1 ::::; i ::::; 3. By writing the matrix ofT with respect to another basis, show that the matrix

~ ~ ~ ~ l is similar to ~ ~ ~ ~ l. l1 1 1J lo o oJ

2. (a) Let V= R3 and T: V----+ V be the linear map defined by T(x, y, z) = (x + z, -2x + y, -x +2y+z).

3.

4.

What is the matrix ofT with respect to the basis (1, 0, 1), (-1, 1, 1) and (0, 1, 1)? Using this matrix, write down the matrix ofT with respect to the basis (0, 1, 2), (-1, 1, 1) and (0, 1, 1).

(b) Let V and W be finite dimensional vector spaces such that dim V 2 dim W. Show that there is always a linear map ofV onto. W.

(c) Solve x+y-2z=1 2x-7z=3 x+ y-z=5

by using Cramer's rule. (a) Find the inverse of the matrix

0 1 0 0 0 0 1 0 0 0 0 1

by computing

1 0 0 0 its characteristic polynomial.

(b) Let A and B be n x n matrices such that AB = BA. Show that A and B have a common characteristic vector.

(c) Reduce to canonical form the orthogonal matrix

(a) lx -x xj x x -x X % % Find the asymptotes of the curve 4(x4+ y4)- 17x2/- 4x(4/- x2) + 2(x2- 2) = 0 and show that they pass through the points of intersection of the curve with the ellipse x2 + 4y2 = 4.

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8.

(b) Show that any continuous function defined for all real x and satisfying the equation f (x) = f (2x + 1) for all x must be a constant function.

(c) Show that the maximum and minimum of the radii vectors of the sections of the surface

(a)

2 x2 y2 2 (x2 + y2 +z2) =-+-+.:.__ a2 b2 c2

by the plane AX+ !J.Y + vz = 0

are given by the equation a2 A 2 b2 Jl2 c2v2 -----::~ + + = 0 1-a2r2 1-b2r2 1-c2r 2

If u = f (~, y ,.:.) , prove that y Z X

au au au x-+y-+z-=0

ax ay az (b) Evaluate

0000 -y f f-e dxdy 00 y

(c) The area cut off from the parabola y2 = 4ax by the chord joining the vertex to an end of the latus rectum is rotated through four right angles about the chord. Find the volume of the solid so formed.

(a) (b)

(c)

(a)

(b)

(c)

(a)

(b)

Find the equation of the common tangent to the parabolas /=4ax and x2 = 4by. If the normal at any point 't1' of a. rectangular hyperbola xy = c2 meets the curve again at the

. ' ' h 3 1 pomt h , prove t at t1 h =- . A variable plane is at a constant distance p from the origin and meets the axes in A, B and C. Through A, B, C the planes are drawn parallel to the coordinate planes. Show that the locus of their point of intersection is given by x-2 + y-2 + z-2 = p-2. Find the equation of the sphere which passes through the points (1, 0, 0), (0, 1, 0), (0, 0, 1) and has the smallest possible radius.

x2 y2 2 The generators through a point P on the hyperboloid 2 + - 2 - ;.- = 1 meet the principal

a b c elliptic section in two points such that the eccentric angle of one is double that of the other. Show that P lies on the curve

a ( 1- 3t2 ) bt ( 3- t 2 ) x = 2 , y = 2 , z = ct . 1+t 1+t

A curve is drawn on a right circular cone, semi-vertical angle a, so as to cut all the generators at the same angle ~ Show that its projection on a plane at right angles to the axis is an equiangular spiral. Find expressions for its curvature and torsion. Find the curves for which the sum of the reciprocals of the radius vector and polar sub-tangent is constant. Solve:

x2 (y- px) = yp2,p =: (c) ysin2xdx- (1 + y 2 + cos2 x )dy = 0

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10.

11.

12.

(a) d2y dy

--2 +2-+10y+37sin3x=O dx dx

Find the value ofy when x = n/2, if it is given that y = 3 and dy = 0, when x =0. dx

(b) Solve:

(c) Solve:

(a) (b)

d 3y d 2y dy x3 --+3x2 --+x-+ y = x+logx

dx3 dx 2 dx State and prove 'Quotient law' of tensors. If xr + y} + zk and r =I r I ' show that. (i) r X gradj (r) = 0 (ii) div(rnr)=(n+3)rn

(c) VerifY Gauss' divergence theorem for

(a)

F=xyF +z2 }+2yzk on the tetrahedron

x=y=z=O,x+y+z=1 A body of weight W is placed on a rough inclined plane whose inclination to the horizon is a greater than the angle of friction 'A. The body is supported by a force acting in a vertical plane through the line of greatest slope and makes an angle 8 with the inclined plane. Find the limits between which the force must lie.

(b) A body consisting of a cone and a hemisphere on the same base rests on a rough horizontal table, the hemisphere being in contact with the table. Show that the greatest height of the cone, so that the equilibrium may be stable, is .J3 times the radius of the sphere.

(c) A hollow cone without weight, closed and filled with a liquid, is suspended from a point in the rim of its base. If~ be the angle which the direction of the resultant pressure makes with the vertical, then show that

(a)

"' 28 cot a + cot3 a cot'f'=------

48 a being the semi-vertical angle of the cone. One end of a light elastic string of natural length a and modulus 2 mg is attached to a fixed point 0 and the other to a particle of mass m. The particle is allowed to fall from the position of rest at 0. Find the greatest extension of the string and show that the particle will reach 0 again after a time

( ;r + 2-tan-1 2 )I: (b) A stone is thrown at an angle a with the horizon from a point in an inclined plane whose

inclination to the horizon is ~' the trajectory lying in the vertical plane containing the line of greatest slope. Show that if 8 be the elevation of that point bf the path which is most distant from the inclined plane, then

2 tan 8 = tan a + tan ~ (c) A particle moves under gravity on a vertical circle, sliding down the convex side of smooth

circular arc. If its initial velocity is that due to a fall to the starting point from a height h

above the centre; show that it will fly off the circle when at a height 2h above the centre. 3

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Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

4.

(a)

(b)

(c)

(a) (b)

(c) (a)

(b) (c)

(a)

(b)

Candidates should attempt any jive Questions. ALL Questions carry equal marks.

SECTION- A Let R be the set of real numbers and G={(a,b) I a,b E R, a:;to}. * G x G ----+ G is defined by (a, b)* (c, d)= (ac, be+ d). Show that (G, *)is a group. Is it abelian? Is (H, *)a subgroup of (G, *),when H = {(1, b) I bER}? Let fbe a homomorphism of a group G onto a group G' with kernel H. For each subgroup K' of G' define K by K={xEG I f(x)EK'}. Prove that (i) K' is isomorphic to K I H. (ii) Gl K is isomorphic to G' I K'. Prove that a normal subgroup H of a group G is maximal, if and only if the quotient group G/H is simple. In a ring R, prove that cancellation laws hold, if and only ifR has no zero divisors. If S is an ideal of a ring R and T any subring or R, then prove that S is an ideal of S+T={s+t I sES, tET}. Prove that the polynomial x2 + x + 4 is irreducible over the field of integers modulo 11. Let F be the set of all real valued bounded continuous functions defined on the closed interval [0, 1]. Let d be a mapping ofF x F into R, the set of real numbers, defined by

1

d(f,g)= jlf(x)-g(x)ldx 0

for all f, gin F. Verify that dis a metric for F. Prove that a compact set in a metric space is a closed set. Let C[a, b] denote the set of all functions f on [a, b] which have continuous derivatives at all points ofl=[a, b]. For f, g, E C[a, b] define

d(f,g) = f(a)-g(b)l + sup{lf'(x)-g'(x)l, xEI}. Show that the space (C[a, b], d) is complete. A function fis defined in the interval (a, b) as follows:

f(x) = q-2, when x=pq-1, f(x) = q-3, when x=(pq-1) 112

where p, q are relatively prime integers; f(x) = 0, for all other values ofx. Is fRiemann integrable? Justify your answer. Test for uniform convergence, the series

00 2n )2n-1) ~ 1+x2n

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6.

7.

8.

(c) Evaluate

(a)

~~ I I sin xsin -l (sin xsin y )dxdy 0 0

Sketch the ellipse C described in the complex plane by Z=AcosAt + iBsinAt, A>B, where tis a real variable and A, B, A are positive constants. If C is the trajectory of a particle with z(t) as the position vector of the particle at timet, identify with justification (i) the two positions where the acceleration is maximum, and (ii) the two positions where the velocity is minimum

(b) Evaluate 1. 1-cosz lm ----,------c-z~O sin ( z2)

f(z) __ sin fz z. (c) Show that z =0 is not a branch point for the function v z Is it a removable

(a)

(b)

singularity? Prove that every polynomial equation ao+a1z+a2i+ ... +anzn=O, an:;tO, n21 has exactly n roots. By using residue theorem, evaluate

ooi loge ( x2 + 1) 2 dx

O X +1 (c) About the singularity z = -2, find the laurent expansion of

(a)

(b) (c)

(a)

(b)

(z-3)sin-1-z+2

Specify the region of convergence and the nature of singularity at z=-2. (i) Find the differential equation of all spheres of radius A having their centre in xy-

plane. (ii) Form differential equation by eliminating f and g from z=f(x2-y)+g(x2+y). Solve: z\p2+q2+ 1) = C2 Find the integral surface of the equation

(x- y)/p + (y- x)x2q = (x2 + /)z passing through the curve xz = a3, y = 0 Apply Charpit's method to find the complete integral of z = px + ay + p2 +q2. Solve:

82z 82z - 2 +-2 = cosmxcosny. ax ay

(c) Find a surface passing through the lines z = x = 0 and z - 1 = x - y = 0 satisfying a2 z - 4 a2 z + 4 a2 z = o . ax2 axay ay2

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10.

11.

12.

(a)

SECTION- B

A uniform rod OA of length 2a free to tum about its end 0 revolves with uniform angular velocity co about the vertical OZ through 0 and is inclined at a constant angle a to OZ. Show

that the value of a is either zero or cos-1 ( 3g 2 J. 4acv

(b) Six equal uniform rods form a regular hexagon loosely jointed at the angular points, and rests on a smooth table, a blow is given perpendicular to one of them at its middle point. Show that the opposite rod begins to move with one- tenth of the velocity of the rod that is struck.

(c) A cylinder of mass m, radius Rand moment of inertia 1 about its geometrical axis rolls down a hill without slipping under the action of gravity. If the velocity of the centre of mass of the cylinder is initially v0, fine the velocity after the cylinder has dropped through a vertical distance h.

(a) A perfectly rough circular hoop of diameter 24 em rolls on a horizontal floor with velocity V em/sec towards an inelastic step of height 4 em, the plane of the hoop being vertical and perpendicular to the edge of the step. Prove that hoop can mount the step without losing contact at any stage if 2.4.[2i > V > 2.4.ji.

(b) A homogeneous sphere rolls down an imperfectly rough fixed sphere, starting from rest at the highest point. If the sphere separates when the line joining their centres makes an angle 30 with the vertical, show that the coefficient of friction !J. satisfies the following equation:

(a)

efl7rl3 = 3.J3 + 6J.1 4(1-2j.12 ).

_ -yi +xj Show that the motion specified by q = is a possible form for an incompressible

x2+i fluid. Determine the stream lines. Show that the motion is irrotational and find the velocity potential.

(b) A sphere is at rest in an infinite mass of homogeneous liquid of density p. the pressure at infinity being m. Show that, if the radius R of the sphere varies in any manner, the pressure at the surface of the sphere at any time is

(c) Find the stream function of two-dimensional motion due to two equal sources and an equal sink situated midway between them

(a)

In a region bounded by a fixed quadrantal arc and its radii, deduce the motion due to a source and an equal sink situated at the end of one of the bounding radii. Show that the stream line leaving either end at an angle n/6 with the radius is

r2

sin ( ~ + () J = a 2 sin ( ~ - () J, where a is the radius of the quadrant.

Describe Newton-Raphson method for finding the solutions of the equation f(x) = 0 and show that the method has a quadratic convergence.

(b) The following are the measurements t made on a curve recorded by the oscillograph representing a change of current i due to a change in the conditions of an electric current:

t: 1.2 2.0 2.5. 3.0

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13.

14.

1 : 1.36 0.58 0.34 0.20

Applying an appropriate formula interpolate for the value of i when t = 1.6.

(c) Solve the system of differential equations

(a)

dy dz -=xz+1, -=-xy dx dx

for x = 0.3 given that y= 0 and z =1 when x= 0, using Runge-Kutta method of order four.

For a statistical distribution the mean, variance, y1 and ~2 are respectively 10, 16, + 1 and 4. Find the first four moments about the origin.

(b) X1 and X2 are random variables with means !J-1, !J-2 equal variances cr2 and positive correlation r. If for some given 8

u1 = x1 cos8 + x2 sin8

U2 = -X1 sin8 + X2 cos8

find the coefficient of correlation p between U 1 and U2 and prove that 0 ::::; p 2 r. Deduce that

X1hX2 and X2h X 1 are not correlated random variables even ifX1 and X2 are correlated. (c) For the data given below fit a straight line and a parabola of the second order using the

principle ofleast squares and determine which of these curves is a better fit.

(a)

x:

y:

0

1

1

5 2

10

3

22

4

38

Ifx be one of the first hundred natural numbers chosen at random, find the probability that

100 x+->50.

X

(b) Among the three hundred employees of a company 240 are union members while the others are not. If eight of the employees are to be chosen to serve on a committee which administers the pension fund, find the probability that five of them will be union members while the others are not, using

(i) hyper-geometric distribution and (ii) binomial approximation.

(c) Genetic theory states that children having one parent of blood type M and the other of blood type N will always be of the three types M, MN or N and that the proportions of these three types will be on the average as 1 : 2 : 1. A report states that out of 300 children having one M parent and one N parent, 30% were found to be of type M, 45% of type MN and the remaining of type N. Test the validity of the genetic theory using chi-square test at 5% level of significance. The following extract from chi-square tables may be used.

Number of Probability of a deviation greater than chi-square Degree of freedom .01 .02 .05 .10 .90 .95 .98 .99

1 6.635 5.412 3.841 2.706 .0158 .00393 .000628 .000157

2 9.210 7.824 5.991 4.605 .211 .103 .0404 .0201

3 11.341 9.837 7.815 6.251 .584 .352 .185 .115

4 13.277 11.668 9.488 7.779 1.064 .711 .429 .297

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15.

16.

(a) Solve that linear programming problem: Maximize

subject to ::::;4 X2 ::::; 6

3xl + 2x2 ::::; 18

If the cost coefficient of x1 is kept fixed, find the range for the cost coefficient of x2 without affecting the optimal solution.

A tax consulting firm has four service stations (counters) in its office to receive people who have problems and complaints about their income, wealth etc. The number of arrivals averages 80 persons in an eight hour service day. Each tax adviser spends an irregular amount of time serving the arrivals which have been found to have an exponential distribution. The average service time is 20 minutes. Calculate the average number of people waiting to be serviced, average time a person spends in the system and the average waiting time for a person. What is the expected number of idle tax advisers at any specified time:

(c) Solve the assignment problem represented by the following for minimisation of costs. Find also alternate solutions if any.

(a)

I II III IV v VI

A 11 24 60 13 21 29

B 45 80 74 52 65 50 c 43 30 93 39 47 35 D 76 44 29 51 41 34 E 38 13 59 24 27 27 F 5 58 55 33 19 30

A company has four plants P1, P2, P3, P4 from which it supplies to three markets M1, M2, M3. Determine the optimal transportation plan using Modi method from the following data giving the plant to market shifting costs, quantities available at each plant and quantities required at each market:

Plant Required at

Pl P2 P3 P4 Market M1

21 16 25 13 11 Market M2

17 18 14 23 13 M3

32 27 18 41 19

Available at plant 6 10 12 15 43

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(b) Determine the maximum value of z=P1P2 ... Pn

subject to the constraints n

2:CiPi::::; x, 0::::; Pi::::; 1(i = 1,2, .... ,n) i=l

(assume that Ci > x for all i) (c) Determine the optimal sequence of jobs that minimizes the total elapsed time required to

complete the following jobs and find the total elapsed time. The jobs are to be processed on three machines M1, M2, M3 in the same order M1, M2, M3 and processing times are as below:

Job

A B c D E F G

Machine M1 3 8 7 4 9 8 7

Machine M2 4 3 2 5 1 4 3

Machine M3 6 7 5 11 5 6 12

Find also the idle times for the -three machines.

*******

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Time Allowed: 3 hours Maximum Marks: 300

1.

2.

3.

4.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

Assume suitable data if considered and indicate the same clearly. Candidates should attempt any jive questions.

All questions carry equal marks.

Let V be the vector space of polynomials over R. Find a basis and dimension of the subspace W of V spanned by the polynomials

v1= t3- 2t2 + 4t + 1, V2 = 2t3- 3t2 + 9t- 1, V3 = t3 + 6t- 5, V4 = 2t3- 5t2 + 7t +5. VerifY that the transformation defined by

T(x1, X2) = (x1 + X2, X1 - X2, X2) is a linear transformation from R2 into R3. Find its range, null space and nullity. Let V be the vector space of 2 x 2 matrices over R. Determine whether the matrices A, B, C E V are dependent where

A= [1 2] B = [3 -1] C = [ 1 -5] 31' 22' -40

Let a square matrix A of order n be such that each of its diagonal elements is !J. and each of its off diagonal elements is 1. If B =leA is orthogonal, determine the values of 'A and !J..

Show that A=~ ~1 ~1 ~ l is diagoalisable over R and find a matrix P such that P-1 AP is l2 2 3 J

diagonal. Hence determine A 25 . Let A = [ aij] be a square matrix of order n such that [ aij] ::::; M \/ i, j = 1, 2, ... n. Let 'A be an eigen-value of A Show that I 'AI::::; nM.

Define a positive definite matrix. Show that a positive definite matrix is always non-singular. Prove that its converse does not hold.

(b) Find the characteristic roots and their corresponding vectors for the matrix

(c)

(a)

(b)

l ~ ~: ~~j Find an invertible matrix P which reduces Q(x,y,z) = 2xy + 2yz + 2zx to its canonical form Suppose

f(x) = 17x12 - 124x9 + 16x3 - 129x2 + x- 1. Determine ~(f-1 ) at x = -1 if it exists.

dx Prove that the volume of the greatest parallelopiped that can be inscribed in the ellipsoid

x2 y 2 z 2 . 8abc

-+-+-=1 1S --a2 b2 c2 3J3

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5.

6.

7.

8.

9.

(c)

(a)

(b)

(c)

(a)

(b)

Show that the asymptotes of the curve (x2-/) (/-4x2) + 6x3 - 5x2y- 3x/ + zy3 - x2 + 3xy- 1 = 0

cut the curve again in eight points which lie on a circle of radius 1. An area bounded by a quadrant of a circle of radius a and the tangents at its extremities revolves about one of the tangents. Find the volume so generated.

Show how the change of order in the integral fooo fooo e -xy sin x dx dy leads to the evaluation of

foosmx . --dx. Hence evaluate It. 0 X Show that in .[;; ~ n + _!_ = J;2;r 1 where n > 0 and .[;; denotes gamma function. 2 2 n-Let P be a point on an ellipse with its centre at the point C. Let CD and CP be two conjugate diameters If the normal at P cuts CD in F, show that CD.PF is a constant and the locus ofF is

a2 b2 [a2 -b2]2 x2 y2 2+-2 = 2 2 where 2 +-2 = 1 equation of the given ellipse. X y X +y a b A circle passing through the focus of conic section whose latus rectum is 21 meets the conic in four points whose distances from the focus are YI, Y2, y3 and y4Prove that

1 1 1 1 2 -+-+-+-=-. Y1 Y2 Y3 Y4 l

(c) Determine the curvature of the circular helix r(t)=(acost)t +(asint)}+(bt)k and an

(a) (b)

(c)

(a)

(b)

(c)

(a)

equation of the normal plane at the point ( 0, a, ;r: J. Find the reflection of the plane x + y + z - 1 = 0 in plane 3x + 4z + 1 = 0 Show that the point of intersection of three mutually perpendicular tangent planes to the

2 2 2 11 . .d X y Z 1. h h 2 2 2 2 b2 2 e 1ps01 2 + - 2 + 2 = 1 1es on t e sp ere x + y + z = a + + c .

a b c Find the equation of the spheres which pass through the circle x2 + / + z2- 4x- y + 3z + 12 = 0, 2x + 3y- 7z = 10 and touch the plane x-2y + 2z = 1. Solve the initial value problem

dy X -= 2 3'y(0)=0 dx X y+ y

Solve (x2-/+3x-y)dx + (x2-/ +x-3y)dy = 0. Assume that a spherical rain drop evaporates at a rate proportional to its surface area. If its radius originally is 3 mm, and one hour later has been reduced to 2 mm. find an expression for the radius of the rain drop at any time. Solve

d 4y d 3y d 2y dy --+ 6--+ 11--6- = 20e - 2x sin x dx4 dx3 dx2 dx

(b) Make use of the transformation y(x) = u (x) sec x to obtain the solution of y"- 2y' tan x + 5y = 0, y'(O) = 0,. y'(O) = .J6

(c) Solve

(1+2x)2 d2;' -6(1+2x)dy +16y=8(1+2x)2

dx dx y(O) = 0, y'(O) = 2.

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