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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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