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Applied Mathematics 1 CBCGS 2016-2017 BE Marine Engineering Semester 1 (FE First Year) Question Paper Solution

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Applied Mathematics 1 [CBCGS]
Marks: 80 Academic Year: 2016-2017
Date: June 2017
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[20] 1
[3] 1.1

Prove that 𝒕𝒂𝒏𝒉−𝟏(𝒔𝒊𝒏 𝜽) = 𝒄𝒐𝒔𝒉−𝟏(𝒔𝒆𝒄 𝜽) 

Concept: .Circular Functions of Complex Number
Chapter: [5] Complex Numbers
[3] 1.2

Prove that the matrix `1/sqrt3`  `[[ 1,1+i1],[1-i,-1]]` is unitary. 

Concept: .Circular Functions of Complex Number
Chapter: [5] Complex Numbers
[3] 1.3

`"If"  x=uv & y=u/v "prove that"  jj^1=1`

Concept: .Circular Functions of Complex Number
Chapter: [5] Complex Numbers
[3] 1.4

If Z=tan^1 (x/y), where` x=2t, y=1-t^2, "prove that" d_z/d_t=2/(1+t^2).` 

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter: [5] Complex Numbers
[4] 1.5

Find the nth derivative of cos 5x.cos 3x.cos x. 

 

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter: [5] Complex Numbers
[4] 1.6

Evaluate : `Lim_(x→0) (x)^(1/(1-x))`

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter: [5] Complex Numbers
[20] 2
[6] 2.1

Find all values of `(1+i)^(1/3)` & show that their continued
Product is (1+i).

Concept: D’Moivre’S Theorem
Chapter: [5] Complex Numbers
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[6] 2.2

Find non singular matrices P & Q such that PAQ is in normal form where A `[[2,-2,3],[3,-1,2],[1,2,-1]]`

Concept: Reduction to Normal Form
Chapter: [7] Matrices
[8] 2.3

Find the maximum and minimum values of `f(x,y)=x^3+3xy^2-15x^2-15y^2+72x`

Concept: Total Differentials
Chapter: [8] Partial Differentiation
[20] 3
[6] 3.1

If u=`f((y-x)/(xy),(z-x)/(xz)),"show that"  x^2 (del_u)/(del_x)+y^2 (del_u)/(del_y)+x^2 del_u/del_z=0`

Concept: .Circular Functions of Complex Number
Chapter: [5] Complex Numbers
[6] 3.2

Using encoding matrix `[[1,1],[0,1]]` ,encode & decode the message "MUMBAI" 

Concept: Rank of a Matrix Using Echelon Forms
Chapter: [7] Matrices
[8] 3.3

Prove that log `[tan(pi/4+(ix)/2)]=i.tan^-1(sinhx)`

Concept: Logarithmic Functions
Chapter: [6.02] Logarithm of Complex Numbers
[20] 4
[6] 4.1

Obtain tan 5𝜽 in terms of tan 𝜽 & show that `1-10tan^2  x/10+5tan^4  x/10=0`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
Chapter: [6.02] Logarithm of Complex Numbers
[6] 4.2

If y=etan_1x. prove that `(1+x^2)yn+2[2(n+1)x-1]y_n+1+n(n+1)y_n=0`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
Chapter: [6.02] Logarithm of Complex Numbers
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[8] 4.3
[4] 4.3.1

Express `(2x^3+3x^2-8x+7)` in terms of (x-2) using taylor'r series. 

 

Concept: Taylor’S Theorem (Statement Only)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[4] 4.3.2

Prove that `tan_1 x=x-x^3/3+x^5/5+.............`

Concept: Taylor’S Theorem (Statement Only)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[20] 5
[6] 5.1

If `Z=x^2 tan-1y /x-y^2 tan -1 x/y del` 

Prove that `(del^z z)/(del_ydel_x)=(x^2-y^2)/(x^2+y^2)`

Concept: Logarithmic Functions
Chapter: [6.02] Logarithm of Complex Numbers
[6] 5.2

Investigate for what values of 𝝁 "𝒂𝒏𝒅" 𝝀 the equations : `2x+3y+5z=9`

`7x+3y-2z=8`

`2x+3y+λz=μ`

Have (i) no solution (ii) unique solution (iii) Infinite value 

Concept: Reduction to Normal Form
Chapter: [7] Matrices
[8] 5.3

Obtain the root of `x^3-x-1=0` by Newton Raphson Method` (upto three decimal places). 

Concept: Reduction to Normal Form
Chapter: [7] Matrices
[20] 6
[6] 6.1

Find tanhx if 5sinhx-coshx = 5 

 

 

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
Chapter: [6.02] Logarithm of Complex Numbers
[6] 6.2

If u= `sin^-1 ((x+y)/(sqrtx+sqrty)), " prove that ""`i.xu_x+yu_y=1/2 tanu`

ii. `x^2uxx+2xyu_xy+y^2u_(y y)=(-sinu.cos2u)/(4cos^3u)`

Concept: System of Homogeneous and Non – Homogeneous Equations
Chapter: [7] Matrices
[8] 6.3

Solve the following system of equation by Gauss Siedal Method,20x+y-2z=17
             3x+20y-z =-18
             2x-3y+20z=𝟐𝟓

Concept: Gauss Seidal Iteration Method
Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
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