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Roll No. .........
B. Tech.- P-' Mid Semester Er:amination-
Paper: Finl
Mathematics -I
IT - Branch
MTII-SlfJ/
·1 I:\I E- 1: 50 hrs.
I. Vind th e shortest distanc e fi' om th e point (1 , 2, - l) to the sphere
~ " 2 ) ) ) J uu cu
2. Jfu = log(x - +y- x-y -xy ). Th enshow that -=,- +-:,_- = --. - CT 0 1 x+y
(/ +v2 r .y X
3. '/( u(x,y) -= ~ - 2n·· -
- · + xj( ) +g(-) , where }.· o are orhitran ' func:ti nns. prm·e
. .!J,\ ·- ) X _v ,..,..
. 1· I. I I a
usin g : 11 er s t 1eorem t ?lit (x -::;- 1 y "' )· 11 (x. y ) (x • ·' )".
ox (,)'_
4 If
. " + y X' I' d I I 1· C(lt. I') A d 1·. II -. 11 · - und r O^ , ,- , 1
111 I 1e va ue o --. ' re II on ,, un c /1 0 11 0 l'
x )' (x · y ) · ?( x.y ) ·
rela!ed! /{so. find th e relalionship.
5. Tra ce th e curve: y ' (a+x) - x' (3u - x).
6. /:·valuate JI ydxdy over th e area hounded by th e ellipse.
Roll No ..........
B. Tech.- P' End Semester Examination-
Paper: First
Mathematics-I
MTH-SJOJ
M.M.-50 IT-Branclt TIME-3:00^ hrs.
vi: Prove that every convergent series is bounded and have unique limit.
At 1
.)('Test the convergence of the series ~ n((l 0 gi)" p>
A ~"' /_. ({/'!;if/ JJ
1 '. (x, yr,t(0, 0)
J, Test the continuity ofith efufic tion f(x,y) ={ (h[l.
4~\Jf rp(x ,y, z) = 0, t' b,,,~ ~~-~ t/>')}z )i,~ )z .-1. "·.
":t!... ".ii' az 8 ..\i"•itf ~ ; ....
~ ' ' .._ "~A'.i>A &. t:"":~.i' .Y~;.
\·"~' ' .,,.,,,, ~"<;.~. ,,-rit: -:t~ o/;
@1'race the curve ~
2
y_
2 =;?--~~.. :-:~·-: ~ / I~
,i, ~. ..
_f· Find the first six ter11Js of th e.expans ions oftfze function e" log(l + y) in a
\
I '< ' 'I;);· • , < ;, • .,. : ' \ I.
T 1
. b h~ ,... '!¾ "~ -
ay or _senes a out t e or;gm. \1!-J~ 'r ,. • ,,:·.. •. i"
;7, The tem'Pe~ ture Tat ~ y-~) oint l ,z). ~··1~;~;l{=40('xyi. Find the ~-- ~ r,_ ~ ,... '= ,,..,. j/ """ ~ ""' ~ •.• ,. ,,.:''
highest temperatwe at the surfa ce ·ofa unit 'iph_e re x2 +y2 +z2 = 1.
. \•· .. ·, ., ~ ,.;. ' ,(.
J 2-x
8. Change the order of int~gf,;a tion in +~J ~ydydx and hence evaluate the same.
-~\ •.'' o)j
~,- ',";:f' , --..,~ "'
I
9.. Evaluate the integral f x
5
(1-.x3)1° dx
0
10. Evaluate Jf fxy2Sin(t+y+z)dxdyd;
subject to the condition x + y + z ::; n.
2
over all positive values of variables
'
I
! •
University Institute of Engineering and Technology
END SEMESTER EXAMINATION
B.Tech. I Scm (CHE), (MTH -lOl)(Dec.-2016)
Time: 3 Hrs Max. Marks : 50
Note. Attempt all questions.
. ' Sei;tion-V • 2 each
...,./1. Find the volume of the solid generated by the revolution of r = 2acos0 ab out the
~ ~!line.
V uetermine the radius of convergence for the powL ,:;rries
I+ -x + -x· + -,-,-, .,.. 1. ...
2
3J5J 3-5-7 -
3 If
2
, y (^) 2. , x fi d. ifu
. u = x tan - - y tan r- , in - ·-. X .. ✓ - y ox8y
/F,ind the value of a if the vector (ax
2 y + yz )i + ( xl - xz
2 )i + (2xyz- 2x
2 l )k is
~ mct the directional derivative of¢ -=-5x
2
y - 5 y2 : + 2.5.:::
1 x at the po int P( I , I. I) in
h d
.. f h 1 · x - I v - 3 t e 1rect1on o t e me - = =- --- - = z.
6/1=1nd the work done in moving a particle in the force field ·•
V F:::3x
2 i+(2xz-y)J+zkalol)g the straight line from (0 ,0,0) to (2,1,3).
7. Discuss the maxima and minima of the function 11 = - -;· .!'.:'...{'-....
x· I· 2y· I 6
JS~how that the series t( - lf;' ' - ~ is convergent but not absolutel y.
yE'v~luate the integral JJJ1/•Y" dxdydz taken over the positive octant such lh.1i
x+ y 1- z ~ I with the help.of Liouvillc ' s Theorem.
1
n/4 ,. Section-ff _ 4 each
J· rind the larger of the two areas into which the circle x
2
➔ l - ~64a : is divided by
the parabola l = 12ax. •.
~il'!d the centre of mass of the area of the cardioid r=a(l+cos0).
'
11 Tkhe convergence of the series r~' ~ =- x" (ii) f-- L2 .t.;'7. · - x"
~.,,' v' f-1 '1/n' (^) 1 I ,, , 7.10 .. \Jn t 4)
. ~xamine the convergence of ine,,,._
/-
yz xz yx a(u,v,w)
=- 2 , v = 2 ,w= 2 , find - ( ) ' X y Z O X,y,z
! 4. If-u be a homogeneous function of degree n, ·then prove that
., a:i.u 0
2 u , a~u
x· --+ 2xy--+ 11 - -- -=n(n 1).
ox
2
axay. a/
- If y=sinlog(x
2 +2x+l),prove that (x+I/ Y,, .; +(2n+l)(x+l)y,,_ 1 +(n
1 +4)y,, =0.
Hence using Maclaurin 's theorem expand yin ascending powers of x.
JV-Verify Stoke ' s theorem for the ve·ctor field F = (2x - y )i - yz
2 J--l=k, o ve r the
upper half surface of x2 + l + :1 °^0 I. bou~ded by its projection on the xy - pl a ne.
- Evaluate th e integ ral r· J; ·.I ( x~ + / )dxdy. Als o, evaluate th e same by changing
◄ a
the order of integration.
.
U•I. E- T., C · !::> • c-l ' l ' l · V ' l'-fll'Hvr-...
MI'D SEMESTER.. Ex A Ml NATION - 2ol
MTH- S \0 I
... 'B;51011cR- IT
Ti rn-e : I l tou:n.
O Te2.l J:lle Con v~ e hce of .H.e ...SenieA
M-M-
o(. -ac-2.. ~
\ + ~ -+ I +-;,c_ 2- t I+ a3 -t - - - - ~ > o
2- ~{ cJ ~ 0in-'.c)2" .B,12-1 CvJuale
ll_- .,z-) d n -1,:,.. - c_ 2' n .-) <)(.a'H· 1 - Y11...cf .,, = O ·
O'f\J ~€Y1Ce .~-\J drf
0 )
'3. S-Rour ..±Ro.± ~e iuY\c+i'o~
. ~
...t)( ~ ~ f~,d) =p- (o 1 o)
-t (!( J ~) ::; / o'-1.+ 2-f- ~
. L (^) O ~, d) -_ Lo I^ o)
1..1> -no-:- CoY1b Y\O~ Qt Co I
a) Ju± 1 ..ls
~a,rl-iol' d-e:n i vo±iveJ ":f d(. £. -ta ex iSl:s al CO , a)
½.~l a.le 2..-- .,PJ10 i,e.. , CJ en '...>., .::-.ffi eo.J1 e m.. io .:)i_
. -~'mOa € neo~ f Unc-\i<h,_, · I
QI) ~ u =- .Y., i Y-1 Co?+ cf) ls- H,en.. ~;'f\d
cJ:- o '1..Lt o 1- 4. '2.... o '2.. lt.
d~2.. t _')...'d(o Oac.dc) -+. ~. O<)'.L
S • 'ffie }em..}e;i °-:1Ll-'Je T a+ Cl'Yl(} Jo i nl Co!, ~, 3}
~ S~re is (t.&-<> ~ T ~ Yoo 6C <} z;2- '#ind
Wie 'l.~ ut 1{!31~u~~. al: ffie ,t A w,::J'Qre_
~ ~ eJ\e '<).'._ + t~ z. = I
C"3J
\
u: I. E. T I C .s. c--. \Yj. u.. < AN? l) R.
f:NJ) 5EMGS TtR. [xRl'1INA"iION- 2.
'BR.~NC\i- JT
~v'l'TH- SIOI
f\1 · M. -5o
C'3J
I
9 ' ~i..8c~ .,t.he on..,L1Y\u1__d O+ ➔ ?I-, o ::. ~ ·a..1·+~J2-
i.? c· L: ·.14 1> -t> ( 4) [ a ~r
0 1a1° ·
'2- ~=o,
0
-:::o
oJ: ~e 001 t". dJ,, LOA,k!,,t,11,.L<M , i::3]
lo · f,nJ J:Re YoJume ~ J:Re .l;c_2·& ?jen€J!aled.. J,y. · \
n
O,...v v-
"1evo.x.v1n9 ~e X\eo.. UY\d€Jt ~e ·cLOJve Y =- o:. ¼
o a '2(.1-·-t ~
Aech6h - 13 C:3]
~ Ybii~J ~Jlee'' A "ffieo:-m ~jf@a!-- 8<J)d21+(Y)J- 611-~JJJ]
~€)le C I~ ~e !oundetllo o1. ffte a:nea. enJo.sed
id Cwwv.. ~ • Ji, Clnd · d ~ ?f.?-. C 5]
[SJ
[SJ
tvJi(ale
i
Cs]
__UN...... 1 _V:...'E...,'R.__S1...1JJ.NSllZ'(!.T.lJ_QE ENGINEERIIYG AND
l.EClf~NOLO(j.Y
Mid-Semester /Jxaminution-
Course : B. Tech. (f' Sem.)
Section: CHE.
TIME : 1: 30 hours
Subject :CALCULUS
Subject Code: J,fIH-SJ OI
MM: 30
vY.'Evaluate (i) lim _Z/ Y_ ; x :;t O, y -:1, o. and (ii) Jim :-Y 2
; x :t: O, y :.c 0.
¥ ➔ 0 X + y ,-.1 X + y
y-+O y ➔ O
... 2 ,.. ·, ,,._"I
2
.l" 0 ...,,I) (JU (~ · 1, ( :· 11
.! X·t·y
0
=Le cos¢nn<l x-y ::. ,:.,;e cos¢ tI-enshowthat-:---+-:··-= 4.9-·-- ·
i iJ0 2 c-'/ &i.: 'f', •
Y, The sum of thret: numbers is constant. Prove that 1 ·heir prodlict is
iJiaximnm when they· are equal..
✓ E:xparnt the fon,:tion f(x ,y) _; / in powers of (x -1 ) and (y-1).
. • F. ' tt ➔ , 1 l f' 2 • 1·
v°' ma ,1e C{lns i: as:Js m m'.t. n svcn t 1 ,i'~ ue sur are 1:.~ -1 .'?)i: = (m + 4)x 'N! t c-e
orthug01Hl.1 Lu the surfa..::e 4x ::y+z
3 ,,,4 at ~-h,;; point (1,-1, 2).
/show that di :·(gradr") ,·, ,1:'.(n+ l)r '' -
2 wht.!re r=Jx
2
2 and hence show
t
11'-'l l ..... 0 l ", :f (--:• -·.
r
';J( "' ,,,._ '7 If 1· ,- . ·•) ... 0 • :> l' !i ,n()I ')-· 0 t-i-,,.,.,,., ,,;-, .., _ tl1at (_ d((J a;, aJ O(p , ,. , .~, )'. - • ,.:~h, •,. .. , - ·- t i! , ,.u. • , ••\ J n • 7 ---- =: --- -- - •
C!-' <JZ iU 1.:Jx ~V
I. I
/. If y;; + y. :;; co 2x 1 then .::how that 1>_·, --l)Y11+:1 +(2n ·f· lh"Yn+ l + (n
2
--n/)y, == 0.
_.y. State and prove ~r's theoreni.
f · · -' ,
... UNIVERSITY _ ...____ ----..-...... INSTITUTE - _...., ____ OF ENG/NEERT!j(, ~ ANI!
TECHENOLOGY
.'-'!id-Semester Examination-
Course : B. Tech. (f' Sem.)
SectiQ11: CHE.
, Subject :CALCULUS
Subject Code: MfH-SJOI
11ME: 1: 30 hours MM:
- Evaluate (i) Jim ..:..;J..; x .;co. y :;co. and (ii) tim .::.:.Y • 1
x "o, y * o.
<· ➔ OX. + V x ➔•) X2 ..:- yl
y · ➔ O • y ➔ O
. o o
1 u a
2
u i/ u
2. If x+ y=2e"cos¢and. x-y=2ze cos¢ then show that
- The sum of three numbers is constant. Prove that their product 1s
ma,-x.imum when th-ey are equal.
4. Expai--id the function f(x,y) == ·yx in powers of (x-1) and (y-1).
5. Find the consl3.nts rn ar.d n such that the surfac <:' 11 ·1x
2
-2~\ S' :.~ (:n ~- 4)x will be
rtl
I 1 .f.". • 7. } , ( ' l "' '
(1 :i(1gona1 to t.ne sur.i:&ct :1.c y -:- z ·. =l} :it t 1e r:ornt .. 1 :·- , .::. ,.
r----- -·
- Sho ,v that div(gradrn)::::n(r:1-l)rn••
2 where r= - .j x; +y
2 +z
2 and hence shO\V
th~t (^) 6 1<.!.> = 0.
r
~ · c, d ·y··
'"'/ lf r ) r ~ ... -· 0 th h ., ·tl· t OJ U({, z (!.. orp
. -' ,.(x,y =0 anu ,n(,,-. .. )= ens 0.\ !.a·- ·--- -·--=- -- --.
·• · r. oyozdx &8y
I I
8. If y:; + y-:,; =1x, then show that (x
2 -l)Yn+2+(2n+l)xyn~l +(n
·-m
2 )y,, :.:0.
- State and prove Fubr'3 theorem.
