Mathematics-III-Engineering-Mumbai University-Dec2019 - Grad Plus

Mathematics-III-Engineering-Mumbai University-Dec2019

MUMBAI UNIVERSITY

Subject: Mathematics-III

Semester: 3

[Total Time: 3 Hours]
[Total Marks: 80 M]
N.B.
1) Question No. 1 is compulsory .
2) Attempt any three questions out of the remaining five questions .
3) Figures to the right indicate full marks


Q.1 a) Find the Laplace transform of 𝑓(𝑑) = 2𝛼(1 + 𝑑𝑒 -t)2. 𝑒-2t where Ξ± is real constant. [5M]

b) Find the Fourier series for 𝑓(π‘₯) = π‘₯ in (βˆ’3,3). [5M]

c) In what direction is the directional derivative of βˆ…(π‘₯, 𝑦, 𝑧) = 2π‘₯22𝑦 2(8𝑧 4) at (1,-1,-2) is maximum ? Find its magnitude . [5M]

d) Determined the constants A,B,C,D,E & F if 𝑓(𝑧) = (𝐴π‘₯3 βˆ’ 𝐡π‘₯𝑦2 + 𝑠𝑖𝑛6π‘₯. π‘π‘œπ‘ β„Ž6𝑦 + 𝐢. π‘₯) + 𝑖(𝐷𝑦π‘₯2 βˆ’ 9𝑦3 + π‘π‘œπ‘ πΈπ‘₯π‘ π‘–π‘›β„ŽπΉπ‘¦ + 101𝑦)is analytic . [5M]

Q.2 a)
Prove thatΒ  J_{1/2}(x)=\sqrt{\frac2{\mathrm{Ο€x}}}\sin x [6M]

b) Evaluate \int_0^\infty e^{-8t}\;\left\{\int_0^t\int_0^t\int_0^tx.\;\sin4x.\cos4x.{(dx)}^3\right\}.dt [6M]

c) Obtain half range cosine series for 𝑓(π‘₯) = π‘₯ ,0 < π‘₯ < 1 and hence prove that the value of \frac{\pi^4}{96}={\textstyle\sum_{n=1}^\infty}\frac1{\left(2n-1\right)^4} using Parseval’s identity . [8M]

Q.3 a) If 𝐹 βƒ— = (π‘₯ + 2𝑦 + 2𝐿𝑧)𝑖 + (4𝑀π‘₯ βˆ’ 3𝑦 βˆ’ 𝑧)𝑗 + (4π‘₯ + 𝑁𝑦 + 2𝑧)π‘˜is irrotational .Find the constants 𝐿, 𝑀, 𝑁 .Show that 𝐹 βƒ— can be expressed as the gradient of the scalar function . [6M]

b) Find Fourier series for the following function.
f(x)=\left\{\begin{array}{l}{(x-\mathrm\pi)}^2\;0\leq\mathrm x\leq\mathrm\pi\\0\;\mathrm\pi\leq\mathrm x\leq2\mathrm\pi\end{array}\right.Β  [6M]

c) Solve using Laplace transform (𝐷2 + 25)𝑦 = (𝐾 + 6). 𝑑 ,if 𝑦(0) = 0 , 𝑦′(0) = 0and find the value of the constant K if 𝑦(πœ‹) = 1. [8M]

Q.4 a) Find the translation transformation using cross ratio property, which maps the points ∞ , βˆ’1 ,1of Z-plane onto the points ∞, 3 ,2 of W-plane . [6M]

b) By using Stokes theorem evaluate \int_c\overrightarrow F.\overrightarrow{dr}\;where\;\overrightarrow F\;=(x^2+2y^2)i+(2x^2-y^2)j and C is the boundary of the region enclosed by circle π‘₯2 + 𝑦2 = 9 , π‘₯2 + 𝑦2 = 36. [6M]

c) Find Inverse Laplace transform of [8M]
𝑖) {s+∝/s2+16}and find the constant , if f(Ο€/8)=1
ii) {s+6/(s-5)2121}

Q.5 a) Define Orthogonal set of functions on (a,b) . If 𝑓(π‘₯) = 𝑃1𝑓1(π‘₯) + 𝑃2𝑓2(π‘₯) +
𝑃3𝑓3(π‘₯) , where 𝑃1, 𝑃2 , 𝑃3 are constants and 𝑓1(π‘₯), 𝑓2(π‘₯), 𝑓3(π‘₯)are
orthogonal functions on (a,b) ,Then show that∫ [𝑓(π‘₯)]2 𝑏
π‘Ž . 𝑑π‘₯ = 𝑃1
2𝑅1 +
𝑃2
2𝑅2 + 𝑃3
2𝑅3 where 𝑅𝑖
, are non zero for 𝑖 = 1,2,3.
6
b) Find the analytic function 𝑓(𝑧) = 𝑒 + 𝑖𝑣 in terms of Z if
𝑒 βˆ’ 3𝑣 = π‘₯2 βˆ’ 𝑦2 βˆ’ 5π‘₯ + 𝑦 + 2 .
6
c) Verify Green’s theorem for ∫ (π‘₯2)𝑑π‘₯ βˆ’ (π‘₯𝑦)𝑑𝑦 𝐢
, C is a triangle whose
vertices are A(0,2) , B(2,0) , C(4,2) in the XY-plane .
8
Q.6 a) Find the image of the real axis of the Z-plane under the transformation 𝑀 =
1
𝑧+𝑖
onto the W-plane .
6
b) Find Laplace transform of 𝑓(𝑑) = π‘’βˆ’4𝑑 . π‘π‘œπ‘ 4𝑑. 𝑠𝑖𝑛4𝑑. 𝐻(𝑑 βˆ’
πœ‹
2
) 6
c) Obtain Complex form of Fourier series for 𝑓(π‘₯) = π‘π‘œπ‘ β„Ž2π‘₯ + π‘ π‘–π‘›β„Ž2π‘₯ in
(βˆ’2,2) .
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