mathematics ii midsem 25032016 (1)

8/16/2019 Mathematics II MIDSEM 25032016 (1)

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PARUL UNIVERSITY

FACULTY OF ENGINEERING AND TECHNOLOGY Second Semester- B.

Tec. !A"" Br#nces$ %&d Semester E'#m&n#t&on S()*ect Code+ ,/0 S()*ect N#me+

%#tem#t&cs- II

T&me+ 1 Ho(rs D#te+ 10t

%#rc2 1,3 Tot#" %#r4s+ 0,

Q.1Select a correct answer from the given options:

(i) The solution of the exact differential equation

(sec x tan x tan y - e x ')dx + sec xsec2 ydy = 0 is.

(!" secx tan y - e x = c (#" tan x sec$ % e x = c

(&" 2sec xtan y-e x = c ('" 2tan x tan y-e x = c

dy 2

(ii) The solution of differential equation (1 ) y ". *hich one of the followingdx

can +e a particular solution of this equation

(!" y = tan(x ) ," (#" x tan($ ) ,"

(&" y = tan x ) , ('" x tan y + ,

(iii)______________________________________________________________ -et T : R2 R2/

T (x/$" ($/x" then T° T (x/ y) = .

(!" (y2/ x2" (#" ($/ x"

(&" (2x/2 $" ('" (x/ $"

(iv) The new coordinate of the point (l/l" if orthogonal proection on the %axis is

followed +$ reflection a+out the line y = x/ is.

(!" (1/0" (#" (0/1"

(&" (1/1" ('" (1/%1"

a x

dydx-a 0

(!" 2a

(&" 0

3ind the +asis for range of T and 4ernel of T/ if T : R5

R6

/ T(X " AX where 7 8%1 2 0 69 % ,/ !lso find ;an< and nullit$ of T

.

(v"

Q.2(a"

!

(#" a ('" a2


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(ii" =valuate cos2 2x sin1 26xdx0

>;

0

1% ? 2 0 1 6

2 %9 2 6 5 1

2% @ 2 % 6 % 6 ?

30

(+" (i" &hec< the convergence of improper integral 57

comparison test.

6 x + 6

2 + q x using

(+" Arove that r 2 e x x2n 1e x x2n1 dx. Bse this relation to prove that =y[ft. ?0 2P a g e 1| 2


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Q., (a"

C y

(i" &hange the order of integration %%%%%%%%%%%%%%%dydx, hence evaluate the integral.0 x y n 2 2a

(+"

(ii" =valuate0 0

(i) =valuate (x ) y"2dxdy/ where ; is the parallelogram in the x$%plane R

with vertices (1/0"/(,/1"/(2/2" and (0/1"/ +$ appl$ing the transformation u = x + y

and v = x — 2 y and integrating over an appropriate region in the uv%plane.

(ii) =ratote xydydxD where ; is the regwn enclosed +$ pa.E +ola$ x Rlines x 1 and x%axis.

>;

Q, (a" (+"

R

2aV2

a

(ii" #$ changing into polar co%ordinates/ evaluate J J (x + $2

^)f dydx

0 0

Q.6(a"

(+"

! +od$ originall$ at 70F & cools down to 50F & in 20 minutes/ the temperature of the air +eing

60F &. 3ind the temperature of the +od$ after 60 minutes from the original.

dy G . G

Solve: ) 2 y tan x sin2x D $(0"1.dx

P a g e 2 |2

1 2 C x

&hange the order of integration xydydx0 x2

(i" =valuate (x2 : ) $2 ^jdxdy +$ changing the varia+les/ where ; is the regionl$ing in the first quadrant and +ounded +$ the h$per+olas x2 C y2 1

C y2 @/ xy = 2 and xy = 6.


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mathematics ii midsem 25032016 (1)

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