BE Instrumentation Engineering Semester 1 (FE First Year)University of Mumbai
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Applied Mathematics 1 CBCGS 2017-2018 BE Instrumentation Engineering Semester 1 (FE First Year) Question Paper Solution

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Applied Mathematics 1
CBCGS
2017-2018 December
Marks: 80

[20]1
[3]1.1

Separate into real and imaginary parts of cos`"^-1((3i)/4)` 

 

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
Chapter: [6.02] Logarithm of Complex Numbers
[3]1.2

Show that the matrix A is unitary where A = `[[alpha+igamma,-beta+idel],[beta+idel,alpha-igamma]]` is unitary if `alpha^2+beta^2+gamma^2+del^2=1` 

Concept: Inverse of a Matrix
Chapter: [7] Matrices
[3]1.3

If `z=tan(y-ax)+(y-ax)^(3/2)` then show that `(del^2z)/(delx^2)= a^2 (del^2z)/(dely^2)`

Concept: Partial Derivatives of First and Higher Order
Chapter: [8] Partial Differentiation
[3]1.4

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

Concept: .Circular Functions of Complex Number
Chapter: [5] Complex Numbers
[3]1.5

Find the n^th derivative of `x^3/((x+1)(x-2))`

Concept: nth derivative of standard functions
Chapter: [6.01] Successive Differentiation
[3]1.6

Using the matrix A = `[[-1,2],[-1,1]]`decode the message of matrix C= `[[4,11,12,-2],[-4,4,9,-2]]`

Concept: Rank of a Matrix Using Echelon Forms
Chapter: [7] Matrices
[20]2
[6]2.1

`"If" sin^4θcos^3θ = acosθ + bcos3θ + ccos5θ + dcos7θ "then find"  a,b,c,d.` 

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[6]2.2

Using Newton Raphson method solve 3x – cosx – 1 = 0. Correct upto 3 decimal places. 

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[8]2.3

Find the stationary points of the function x3+3xy2-3x2-3y2+4 & also find maximum and minimum values of the function.

Concept: Maxima and Minima of a Function of Two Independent Variables
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[20]3
[6]3.1

Show that xcosecx = `1+x^2/6+(7x^4)/360+......` 

 

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[6]3.2

Reduce matrix to PAQ normal form and find 2 non-Singular matrices P & Q.

`[[1,2,-1,2],[2,5,.2,3],[1,2,1,2]]`

Concept: Reduction to Normal Form
Chapter: [7] Matrices
[8]3.3

If y= cos (msin_1 x).Prove that `(1-x^2)y_n+2-(2n+1)xy_(n+1)+(m^2-n^2)y_n=0`

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[20]4
[6]4.1

State and Prove Euler’s Theorem for three variables.

 

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
Chapter: [8] Partial Differentiation
[6]4.2

Show that all roots of `(x+1)^6+(x-1)^6=0` are given by -icot`((2k+1)n)/12`where k=0,1,2,3,4,5.

Concept: Powers and Roots of Trigonometric Functions
Chapter: [5] Complex Numbers
[8]4.3

Show that the following equations: -2x + y + z = a, x - 2y + z = b, x + y - 2z = c have no solutions unless a +b + c = 0 in which case they have infinitely many solutions. Find these solutions when a=1, b=1, c=-2. 

Concept: Rank of a Matrix Using Echelon Forms
Chapter: [7] Matrices
[20]5
[6]5.1

If Z=f(x.y). x=r cos θ, y=r sinθ. prove that `((delz)/(delx))^2+((delz)/(dely))^2=((delz)/(delr))^2+1/r^2((delz)/(delθ))^2`

Concept: Differentiation of Implicit Functions
Chapter: [8] Partial Differentiation
[6]5.2

If coshx = secθ prove that (i) x = log(secθ+tanθ). (ii) `θ=pi/2tan^-1(e^-x)`

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[8]5.3

Solve by Gauss Jacobi Iteration Method: 5x – y + z = 10, 2x + 4y = 12, x + y + 5z = -1. 

Concept: Gauss Jacobi Iteration Method
Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
[20]6
[6]6.1

Prove that `cos^-1tanh(log x)+ = π – 2(x-x^3/3+x^5/5.........)`

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[6]6.2

If` y= e^2x sin  x/2 cos   x/2 sin3x. "find"  y_n`

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[8]6.3
[4]6.3.1

Evaluate `Lim _(x→0) (cot x)^sinx.`

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
[4]6.3.2

Prove that log `[sin(x+iy)/sin(x-iy)]=2tan^-1 (cot x tanhy)`

Concept: Expansion of 𝑒^π‘₯ , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), π‘ π‘–π‘›βˆ’1 (π‘₯),π‘π‘œπ‘ βˆ’1 (π‘₯),π‘‘π‘Žπ‘›βˆ’1 (π‘₯)
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions

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