WWW - Manaresults.Co - In: I B. Tech I Semester Regular/Supplementary Examinations, Nov/Dec - 2017 Mathematics-I
WWW - Manaresults.Co - In: I B. Tech I Semester Regular/Supplementary Examinations, Nov/Dec - 2017 Mathematics-I
WWW - Manaresults.Co - In: I B. Tech I Semester Regular/Supplementary Examinations, Nov/Dec - 2017 Mathematics-I
∂u ∂u x3 y 3
e) Find x +y if u = 3 (2M)
∂x ∂y x + y3
f) From the partial differential equation by eliminating arbitrary constants from (2M)
z = ax + by + ab
∂2 u ∂2 u
g) Classify the nature of 2 + 3 = 0 the partial differential equations. (2M)
∂x 2 ∂y 2
PART -B
2. a) Find the charge and current in RC circuit if R = 20 ohms, c = 0.01 farad, (7M)
and E(t) = 20 sin 2t with q(0) = 0.
b) If 30% of radioactive substance disappears in 10 days, how long will it take for (7M)
90% of it to disappear?
3. a) At the end of three successive seconds, the distance of a point moving with simple (7M)
harmonic motion from its mean position, measured in the same direction are 1, 5,
5. Then find the complete oscillation.
b) Solve the D.E
+ 9 = 3 by the method of variation of parameters. (7M)
#
4. a) Show that
!
" = . (7M)
$
b) Find % &
(
7. a) Solve the PDE D 2 − 3 D − D1 + 3 D1 z = xy + e x + 2 y .
2
) (7M)
( )
b) Solve the PDE D 2 − DD1 z = sin x cos 2 y . (7M)
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Code No: R161102 R16 SET - 2
PART –A
12 3
e) Find 141516
for f(x,y,z) = exyz
(2M)
f) Form the partial differential equation by eliminating arbitrary constants from (2M)
2 2
z = ax + by + a + b .
2 1
(
g) Find the P.I of D − D z = cos( x + y ) .
2
) (2M)
PART -B
2. a) Find orthogonal trajectories of the Family of circles x2 + y2 + 2fy + 1 =0, f being the (7M)
parameter.
7
Show that ! " 893
4. a) ∞ (7M)
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Code No: R161102 R16 SET - 2
x, y , z
then find J r ,θ , φ .
#
b) Expand f(x,y) = xy2 +cos(xy) in powers of , 1 , . (7M)
@ B
6. a) Solve the PDE ? A ? A 1.
(7M)
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Code No: R161102 R16 SET - 3
PART –A
1. a) Write the Bernoulli’s equation in ‘y’. (2M)
b) Find the P.I of
1 . (2M)
c) Find % & , ' (2M)
C
D
FGIH FN⁄H
d) If % E GI J KL√ then find % E GI J
(2M)
# √
f) Form the partial differential equation by eliminating arbitrary constants from (2M)
z = ax + a 2 y 2 + b .
∂2 u ∂ 2u
g) Classify the nature of x + y = 0 if xy > 0 the partial differential (2M)
∂x 2 ∂y 2
equations.
PART -B
2. a) An RL circuit has an Emf given (in volts) by 4 sin t, a resistance of 100 (7M)
ohms, an inductance of 4 henries with no initial current. Find the current at
any time t.
b) Find the orthogonal trajectory of Q RS R0S . (7M)
1
3. a) Solve the D.E ( D + D ) y =
2
. (7M)
1 + ex
( )
b) Solve the D.E D3 + 1 y = cos ( 2 x −1) + x 2 e− x . (7M)
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Code No: R161102 R16 SET - 3
x2 y2 ∂u ∂u ∂ 2u ∂ 2u
5. a) If u = then find (i) x + y (ii ) x +y 2. (7M)
x+ y ∂x ∂y ∂x∂y ∂y
yz zx xy (7M)
Prove that JJ1 =1 if u = ,v = ,w = .
b) x y z
2
z 2 ( p + q2 ) = 1 .
6. a) Solve the PDE (7M)
(
7. a) Solve the PDE D 2 − 2 DD1 + D1 z = 2 x cos y .2
) (7M)
(
b) Solve the PDE D 2 + 2 DD1 − 8 D1 z = 2 x + 3 y .2
) (7M)
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Code No: R161102 R16 SET - 4
2 xy
e) Find lim 2 2
x →1 x + y + 1
. (2M)
y →1
f) Form the partial differential equation by eliminating arbitrary constants from (2M)
z = ax + by + a 2 + b2 .
(
g) Find the P.I of D 2 − D1 z = e x + y . 2
) (2M)
PART –B
b) The temperature of a cup of coffee is 920C, when freshly poured the room (7M)
temperature being 240C. In one minute it was coaled to 800C. How long a period
must elapse, before the temperature of the cup becomes 650C.
(
3. a) Solve the D.E D 2 + 2 D + 1 y = x .cos x . ) (7M)
b) Determine the charge on the capacitor at any time t > 0 in a series RLC circuit (7M)
having an E.M.F E(t) = 100 sin 60 t, a resistor of 2 ohms, an inductor of 0.1
1
henries and capacitor of farads, if the initial current in the inductor and
260
charge on the capacitor are both zero.
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Code No: R161102 R16 SET - 4
I F
]
4. a) Evaluate% &! "'.
4 (7M)
1/ 2
∂ 2u ∂ 2u 2
2 ∂ u
x+ y
2
if u = cos ec −1 1/3
x + y1/3
5. a) Find x 2
+ 2 xy + y 2
(7M)
∂x ∂x∂y ∂y
b) Find the extreme values of following using Lagrange’s multiplier method (7M)
2 2
xy subject to 3x + y = 6.
( )
7. a) Solve the PDE D 2 + DD 1 − 6 D12 z = x 2 sin( x + y ) . (7M)
( )
b) Solve the PDE D 3 − D13 z = x 3 y 3 . (7M)
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