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

Applied Mathematics 1 [CBCGS]
Marks: 80 University of Mumbai
BE Civil Engineering
BE Computer Engineering
BE Mechanical Engineering
BE Biotechnology
BE Marine Engineering
BE Printing and Packaging Technology
BE Production Engineering
BE IT (Information Technology)
BE Electrical Engineering
BE Electronics and Telecommunication Engineering
BE Instrumentation Engineering
BE Electronics Engineering
BE Chemical Engineering
BE Construction Engineering
BE Biomedical Engineering
BE Automobile Engineering

Date: June 2017

[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
[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
[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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