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(06 Marks)

(07 Marks)

(07 Marks)

(06lVlarks)

and c =-41+5J+i[-(07 Marks)

(07 Marks)

Fourth Semester B.E. Degree Examination, Dec.2013/Jan.2$t4

Advanced Mathematics - llTime: 3 hrs. Max. Marks:100

Note: Answer any FIVEfull questions.

I a. Prove that sin' cr + sin' B + sin' y = 2. (06 Marks)

b. If 11; ml, n1 and 1.2, mz, fl2 are direction cosines of two lines then pxove that the angle

between,,them is cos 0: l-J-zt m1m2 * nrflz. : ,,ir (07 Marks)

c. Find the equation of the plane through the interaction of the pl4nes'2x + 3y - z : 5 andx-2y -32= - 8, also perpendicularto the plane x +y -z:2. (07Marks)

d = 2il2 j_+ k jrLb = 1- 2i

12kaa = 4r+ 6j+ 2k, b=3i +10j +5k

()oo*g.=

oo!

Eg

cEe76-o0 I

coo.: an

tsooE-o

;:rUO

b0iccd-o>!a6

'od3AOE

^Xtro.o."o -_I

e 'ii3oaLE

LO)E

>- (!-^oeooo=sotr>^-o5eU<- a'.1

OoZ

!oE

2 a. Prove that the equation of the plane in the intercept form is I + {*i=,b. Find the equation of the plane through the points (l; -2,2) (-3, l, -2) and perpendicular to

theplane2x-y-z+6:0"c. Find the angle between the following lines:

x-2:,u-l _r-3 end x+i_>-3_z-1t22*1 0

a' Find the sine of the angle between

b. Find the value of ), if the vectors

are coplanar.c. Prove the following: , '

i) (34 - 26) x 1-la + 2b-; = l4(A + 6)

ii) (2a+ 36) x (a +,::,45) = 5(a + 6)

a. A particle rr,lclVes along the curve i = (t' -4O1+(t2 +at)j+(8t2-.3t3;[. nina the velocityand acceleration at t : 1 and also find their magnitude. ., (06 Marks)

b. nind,ihe unit normal vector to the surface xy3 z2 :4 at the point (- 1 , -1 , 2). :_ (07 Marks)c. Findthedirectionalderivative ofx2yz3 at(1, 1, 1)inthedirectionof i+j+2k (07Marks)

rr {,;a' Find div F and curl F, where F = xti + ytj + zt[ . (06 Marks),,"'lb. Prove that curl grad Q

: 0. (oiiMarks)c. Find the constants a,b, c such that the vectorF =(x* y +az)i+(x+ cy +22)k+(bx+ 2y -r)j

is irrotational. (07 Marks)

a.

b.c.

d.

Find the Laplace transform of the following:sin 4t cos3tcos hat

t e-t sin t1 - cost

20 Marks)

GENTH'alLlElgi&RY

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MATDIP4Ol

7 Find the inverse Laplace transform of- /s+1\a' '"1r r.,J (06 Marks)

- s+lb. __-:_ (07 Marks)s'+2s+2

., t' G+1)G{2X,L:5

(o7ilIarks;

",{}1* , ,,.,.

8 a. tr;'@g Laplce transforms, solve the differential equation {furgl. 6y =5e"dt'- dtsubjectEffi-the conditions y(0) : y'(0) :0. , (10 Marks)

b. Solve the siffilppeous equations **r=sint, gy-*x

=9oF$, using Laplace transforms.Lfl./ dt dt ,.ftrI*Given that x : 1, yffiwhen t : 0. (lo Marks)

4 d,.

, **{<i<{<,"

" '

t:: ti'

... q i: ::: ::

; "'*

,,*;t - I '!i;-,::'

i"::{;lli};,..,

::::.

::" _r*!#U' .{ }.

'' ',,,"""""",,,,':1 .

, ,,,-, =.=

,

.::::

t- :\,,1'qtl, '{+,t

ta..""':':""""' '"'

-.::::- '::::,

' u-

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