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

# Mathematics-III-Engineering-Mumbai University-Dec2019

## 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(𝑡) = 2𝛼(1 + 𝑡𝑒 -t)2. 𝑒-2t where α is real constant. [5M]

b) Find the Fourier series for 𝑓(𝑥) = 𝑥 in (−3,3). [5M]

c) In what direction is the directional derivative of ∅(𝑥, 𝑦, 𝑧) = 2𝑥22𝑦 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) = (𝐴𝑥3 − 𝐵𝑥𝑦2 + 𝑠𝑖𝑛6𝑥. 𝑐𝑜𝑠ℎ6𝑦 + 𝐶. 𝑥) + 𝑖(𝐷𝑦𝑥2 − 9𝑦3 + 𝑐𝑜𝑠𝐸𝑥𝑠𝑖𝑛ℎ𝐹𝑦 + 101𝑦)is analytic . [5M]

Q.2 a)
Prove that  J_{1/2}(x)=\sqrt{\frac2{\mathrm{πx}}}\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(𝑥) = 𝑥 ,0 < 𝑥 < 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 ⃗ = (𝑥 + 2𝑦 + 2𝐿𝑧)𝑖 + (4𝑀𝑥 − 3𝑦 − 𝑧)𝑗 + (4𝑥 + 𝑁𝑦 + 2𝑧)𝑘is irrotational .Find the constants 𝐿, 𝑀, 𝑁 .Show that 𝐹 ⃗ 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.  [6M]

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

Q.4 a) Find the translation transformation using cross ratio property, which maps the points ∞ , −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 𝑥2 + 𝑦2 = 9 , 𝑥2 + 𝑦2 = 36. [6M]

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

Q.5 a) Define Orthogonal set of functions on (a,b) . If 𝑓(𝑥) = 𝑃1𝑓1(𝑥) + 𝑃2𝑓2(𝑥) +
𝑃3𝑓3(𝑥) , where 𝑃1, 𝑃2 , 𝑃3 are constants and 𝑓1(𝑥), 𝑓2(𝑥), 𝑓3(𝑥)are
orthogonal functions on (a,b) ,Then show that∫ [𝑓(𝑥)]2 𝑏
𝑎 . 𝑑𝑥 = 𝑃1
2𝑅1 +
𝑃2
2𝑅2 + 𝑃3
2𝑅3 where 𝑅𝑖
, are non zero for 𝑖 = 1,2,3.
6
b) Find the analytic function 𝑓(𝑧) = 𝑢 + 𝑖𝑣 in terms of Z if
𝑢 − 3𝑣 = 𝑥2 − 𝑦2 − 5𝑥 + 𝑦 + 2 .
6
c) Verify Green’s theorem for ∫ (𝑥2)𝑑𝑥 − (𝑥𝑦)𝑑𝑦 𝐶
, 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 𝑤 =
1
𝑧+𝑖
onto the W-plane .
6
b) Find Laplace transform of 𝑓(𝑡) = 𝑒−4𝑡 . 𝑐𝑜𝑠4𝑡. 𝑠𝑖𝑛4𝑡. 𝐻(𝑡 −
𝜋
2
) 6
c) Obtain Complex form of Fourier series for 𝑓(𝑥) = 𝑐𝑜𝑠ℎ2𝑥 + 𝑠𝑖𝑛ℎ2𝑥 in
(−2,2) .
8
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