If v = 2xy, then the analytic function¦(z) = u+ iv is
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Solution
φ(x, y) =\(\frac{\partial v}{\partial x}=2y\)
φ(z, 0) =2 × 0 = 0
φ(x, y) =\(\frac{\partial v}{\partial y}=2y\)
φ(0,z) =2z
By Milne’s Method
f(z) =∫φ(0,z)+iφ(z, 0)dz
=∫(2z+i0)dz
= z2+ c
Solution of differential equation (2D+ 1)2 y=4 e\(^{\frac{-x}{2}}\)is
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Solution
For C.F.,
(2D + 1)2 y= 0(2D + 1)2 = 0 ⇒ D= -1⁄2,-1⁄2
C.F. = (C1+ C2x)e\(^{-\frac{1}{2}x}\)
P.I. =\(\frac{1}{(2D+1)^{2}}.4e^{-\frac{x}{2}}\)
=\(4.\frac{1}{(2D+1)^{2}}e^{-\frac{x}{2}}\)
=\(4x\frac{1.e^{-\frac{x}{2}}}{2\times 2(2D+1)}\)
=\(\frac{x}{(2D+1)}e^{-\frac{x}{2}}\)
=\(\frac{xe^{-\frac{x}{2}}}{2}\)
y= C.F. + P.I.
= (C1+ C2x)\(e^{-\frac{1}{2}x}+\frac{xe^{-x/2}}{2}\)
y = (C1+ C2x +x⁄2)
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Solution
\(\int \frac{dx}{\sqrt{2x^{2}+3x+4}}\)
=\(\frac{1}{\sqrt{2}}\int \frac{dx}{\sqrt{\left ( x+\frac{3}{4} \right )^{2}+\left ( \frac{\sqrt{23}}{4} \right )^{2}}}\)
=\(\frac{1}{\sqrt{2}}sin\, h^{-1}\frac{x+\frac{3}{4}}{\frac{\sqrt{23}}{4}}\)
=\(\frac{1}{\sqrt{2}}sin\, h^{-1}\left ( \frac{4x+3}{\sqrt{23}} \right )\)
Let f(x) = x (x+ 3)e\(^{-\frac{x}{2}}\), –3 ≤x ≤ 0. Let C ↔ ] – 3, 0[such that f(C) = 0. Then,the value of C is _______
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Solution
f(x) = \(x(x+3)e^{-\frac{x}{2}}=(x^{2}+3x)e^{-\frac{x}{2}}\)
f'(x) = (2x+ 3)e\(^{-\frac{x}{2}}-\frac{1}{2}(x2+ 3x)e^{-\frac{x}{3}}\)
By Rolle's theorem,
f'(C) = 0
(2C+ 3)e\(^{-\frac{C}{2}}-\frac{1}{2}(C^{2}+ 3C)e^{-\frac{C}{2}}=0\)
e\(^{-\frac{C}{2}}\neq 0\)
(2C+ 3) – 1⁄2(C2 + 3C) = 0
4C+ 6– C22 – 3C = 0
– C2+ C + 6 = 0
C2– C –6 = 0
C = 3, – 2
– 2 ∈(– 3, 0)
If A and B are square matrices of order 4 × 4 such that A= 5B and |A| = α.|B|, then a is ________ .
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Solution
As,|KA|= Kn|A|
A= 5B
|A|= |5B| = 54|B|= 625|B|
So,α = 625
The variation of saturation pressure with saturation temperature for a liquid is 0.1 bar/k at 400K. The specific volumes of saturated liquid and dry saturated vapour at 400 K are 0.251 and 0.001 m3/ kg. Then the value of latent heat of vaporization using Clausius clayper on equation will be _________
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Solution
From Clausius clayperyon equation,
\(\frac{dp}{dT}=\frac{h_{g}}{T(V_{g}-V_{f})}\)
⇒0.1×102=\(\frac{h_{g}}{400(0.251-0.001)}\)
hg= 10 × 100 = 1000 kJ/kg
Match List-I with List-II:
List-I List-II
A.Radiation heat 1.Biots numbertransfer
B.Conduction heat 2.View factortransfer
C.Forced convection 3.Fourier’s Law
D.Transient heat flow 4.Stanton number
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Solution
Radiation heat transfer = View factor
Conduction heat transfer = Fourier’s Law
Forced convection = stanton number
Transient heat flow = Biots number
For an air-conditioned space, RTH= 100 kW, RSHF =0.75, Volume flow rate = 100 m3/min and indoor design specific humidity is 0.01 kg/Hg of dry air.Then the specific humidity of supply air will be _______
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Solution
We know that,
RSHF=\(\frac{RSH}{RTH}\)
where, RTH= RSH+ RLH
Given, RSHF= 0.75, RTH= 100 kW
RSH= RSHF× RTH= 0.75 × 100 = 75 kW
A centrifugal pump needs 1000 W of power when operting at 1500 rpm. If the speed of pump is in creased to 3000 rpm, then the power requirement will be _______
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Solution
Given P1= 1000 W, P2= 1,N1= 1500 rpm,
N2= 3000 rpm
Using the following relation,
\(\frac{p_{1}}{p_{2}}=\left ( \frac{N_{1}}{N_{2}} \right )^{3}\)
\(p_{2}=\left ( \frac{N_{2}}{N_{1}} \right )^{3}p_{1}=\left ( \frac{3000}{1500} \right )^{3}\times 1000=8000W\)
For maximum blade efficiency of a single stage impulse turbine, the blade speed ratio, (α is the angle made by absolute velocity at inlet) should be,
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Solution
For maximum blade efficiency of single-stage impulse turbine, blade speed ratio = cos α/2