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Time: 3 hrs.

4

cy 2"-r cos(nnl2-zx) oy z"*r cos(nnl2-2x)

(06 Marks)

(06 Marks)

(04 Mark)(04 Marks)

. (06 Mark)

three dimensional Cartesian coordinates to(06 Mark)

ii) If F=3x2i-xyj+(a-3)xzk is Solenoidal then aisequal to A)0iii) lf F=(x+y+1)i+.j-(x+y)k then F.curlF is-. e) O B) x+yiu) The scale factors for cylindrical coordinate system (p, {, z) are given by.'l

A) (p, l, l) B) (1, p, 1) c) (1, l, p)

Choose your answers for the following :

i) If i=xi+yj+zk ttren div R

Prove that curlA = g rad(divA) - V2A .

Find the constants a, b, c such that the vector

A)0 B)3 C) -3

D) none ofthese(06 Marks)b.

c.

First/Second Semester B.E. Degree Examination. June/July 2013Engineering Mathematics - I

Max. Marksil00Notet I. Answer any FII/E full queslions, choosing at least tteolrom eacl, parl

2. Aneeer all objective type queslions only on OMR sheel page S ofthe answer booklet.3. Arr$tef lo objectire Upe quesliorrs oh sheels other lhan OMR toill ,iol be talued

PART _ Aa. Choose vour answers fbr the followins :

r) lt ) : J- then yn rs e1 (3log s)'e5. ny (s log:)" e5"

ii) If y = s652 x then yn is

e1 2"*r cos(nnl2+2x) s1 2'-rcos(nnl2+2x)

iii) The Lagrange's mean value theorem 1b. the firnction f(x) = g* il the inteNal [0, l] is

A) C:0.5413 B)C= 2.3 C) 0.3 D) None ofthesc

iu) Expansionof log(l+e")inporversofxis_. A)log2-x/2+x2/8+xa llg2+-

B)log2+x/2+x2l8 - xn ltg2+- c)log2+x/2+x2l8 +xn 1192+- pJ roez-I-}1-,I1* -'28t921',. tryr;,+y-],i,=2xprorethat(x2-l)yD+2+(2n+l)xy,,+r +(n2-m2 1y,, =0. (06 Marks)

c. Verily the Rolle'sthcorem lbr the tunctions: l (r) = ?'(sin r - cos x) in (rrl4,57rl4). (06 Marks)

d. By using Maclaarin's theorem expand log sec x up to the term containing x6, (04 Mark)a. Choose your ansrvers lor the lbllouing : (04 Mark)

i) The indeterminate tbrm of 1i6 u'd;, at 1og\y^) at log(/b) c) I D) -lx+0 X '

ii) The angle between the radius vector and the tangent-for the curves r = a(l - cos0) is

L) 0t2 B) -0 l2 C) 1rl2+0 D\ n/2-0/2.iii) The polar lbrm oia curve is A) r = f(e) e) 0 = f(y) C) r = f(x) D) None ofthese

i!) Therateatwhichthccurveisbendingcalled_,A)Radiusofcurvaturel B)Curvature: C) Circle ofcurvaturei D) Evaluale.

,lf'aluate ;;61

s'n t l .\ .01 \

Find the angles ofintersection ofthe f-ollor\ ing pairs ofcurves. r = aO(t+ 0): r=a/(l+e2).Find the radius ofcurvature at (3al.2, 3a12) on xr +yr =3axy.Choose your answers for the following :

i) II' u = x' I )' rhen (i"u) (Cxf,) is equal to A)2 B)0 C) 2x D\ 2y

ii) If z= f(x, y) where x = u-vand y= uv then (u+v)(62l6x)isA) u(Azlev\-v?zlAi B)u(62l6\r)+v(dzl6v) q Azleu+Azttu D1 dzldu-dztdv

iii) If x : r cos 0, y: r sin O then [6(r,0)]/[d(x, y)] is A) r B) th C) .1 D) -liv) In errors and approximations ax /x. fo/y, 6flf are called

A) relative error B) percentage error C) error in x, y and f D)-none ofthese

lf x'yt "' =c.showthat A'.z-l 6x4, = -[xlogex]

r,whenx=y=2.

Obtain the Jacobian of 6(x. y, z)/flr. 0. $) lbr change of coordinate liomspherical polar coordinates.In estimating the cost ofa pile ol bricks measured as 2mx l5mx 1.2m. the tape is stretched +l7o beyond the standard l€ngth.If the count is 450 bricks to I cu.cm and bricks cost of 530 per 1000, tind the approximate error in the cost. (04 Mark)

F- (r -) +az)i .(bx I 2) -z)j , (x r cy r 2z)k is imotational.(06 Marks).

Derive an expression lor V .A in orthogonal curvilinear coordinates. Deduce V . A is rectangular coordinates.

1.,t2 (04 n{arks)

(04 Marks)

c) (s tog 3f " e5' or(stog3)"e-"

(04 Mark)

D)2B) -2 C) 2 D)3

C) x+y+z D) x y

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5a.

.c.

d.

6a.

b.

c.

d.

7a.

c.

d.

8a.

PART - BChoose your answers for the following :

i.1 The valueof"J"-.,r, is A) leJ- -'

ii) An integrating factor for ydx - xdy = 0 is A) /y B) y/x C) l(xy)iii) The differential equation satisrying the relation x : A cos (mt o.) is

A)(dvdo= I -t' B) (d'?x/de) - d2x C) (d']x/df) = m2x

iv) The orthogonal tnljectories ofthe system given by r = a0 is

IOMATI I(0,1Mark)

D) -llu

D) 18/35

l \ l ,\Solve (l-y2)-(x-e'd )Jdy dx=0.Prove that the system ofparabola f = 4a (x + a) is selforthogonal.

Choose your answers for the following :

find the inverse tran sform ati on s.

TR o ,lDiagonalizethematri*. n_l-o z 01.

l, 4 r]Reduce the quadratic form, xf +Zxl-lxl -rr,riitJii, into sum ofsquares.

2 ot2

A) AX : l.x B) i,(A-x)=0 C) xA- A1. = o D) la-rrlx=oiu) Two square maffices Aand B are similar if, A) A=B; B)B:P-rAP; C)A'=B';D) A-r =B-

Show that the transformation, yt=2x1-2x2-x1, y, =-4xr +5x2 +3xt, yl -xt -x2 -x., is, regular and

(06 M{rks)

(06 Mirks)

(01 Marks)

(01 iUarks)

D) cos (y/x) : cD) l/(x':+f )

D) (dx/dt) : -m:x

D)12:1"-e'(06 Mark)

(06 Marks)

(04 Mar|(s)

(01 lt{ arks)

(06 Marks)

(06 Nlarks)

(04 Marks)

A)r2: keo B)r=keo c)l e-8'=pSolve (x cos(y / x) + ysin(y/ x))y - (ysin(y/ x) - x cos(y/ x)) x (dy / dx) = 0.

B) - l/e C) l/a

tt2ii) The valueofthe integral fsinTxdxis A) 35/16 B)16/35 C)-16135)0

iii) The volume generated by revolving the cardioid r = a(1 + cos 0) about the initial line is

A) (3ra2)/8 B)(37ra3)/8 C)(2ra'1) 19 D)Noneiv) The area ofthe loop ofthe curve r = a sin 30 is _ A) a'?l12 ; B) nll2: C) na'?ll2 ; D) None

r/lb. By app lying d i fferentia I underlhe integral sign evaluate 'ilo8l I+),srn)rro^.

;srnY

Evaluale of fsin n \ dx where n is any integer.J

findthe leng:h ofthe arch ofthe cycloid x=a(0-sin0); y:a (1 - cos0); 0<0 (2r.Choose your answers for the following :

i) The goneral solution ofthe differential equation (dy/dx)=(y/x)+tan(y/x) is

A) sin (y/x) : c B) sin (y/x) : cx C) cos (y/x) : cx

[] -r 2li) Find the rank of I '. .' ^1, A} 3 B) 2 C) 4 D) I-,'-|l-l I 2l

ii) The exact solution of the system of equation lOx + y + z : 12. x + 10y + z = 12, x + y + l0z = 12 byinspection is equal to A) (1, l, l) ; B) (1, l, 1) ; C) Cl, -1, -l) ; D) None

iiD If the given system of lillear equations in 'n' variables is consistent then the number of linearlyindependent solution is given by A)n ; B)n-l ; C)r n ; D)n r

iv) The trivial solution for the given system ofequations 9x-y+42=0,4x 2y+32=0,5x+y 6z:0isA) (1,2, 0) B) (0,4, l) c) (0,0, 0)

?],1, ,r;9,Using elementary hansformation reduce each offollowing matrices to the normal arr,

| ] ,, , ,

| ,* ,,.*r

l, ,,r]Test for consistency and solve the system. 2x + y + z=10,3x +2y +32-18, x + 4y +gz=16. (06 Mark)

Appii Gauss-Jordan method to solve the system of equations, 2x+5y+72=52, 2x+y-z=0,)\+y+z=9 (04 Mirks)

Choose your answers for the following : (04 Mark)

i) A square matrix A is called orthogonal if, A) A=42 B)A=A-r C) AA-r =l D)None

iD The eigen values ofthe matrix. I u, -,t t,l ,r" A) 2. 3. 8 B) 2,3,9 C) 2,2, 8 D) None

| , -, ,iii) The eigen vector X ofthe matrix A corresponding to eigen value 1" and satisry lhe equation,

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