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Applied Mathematics 1 CBCGS 2017-2018 BE Electronics Engineering Semester 1 (FE First Year) Question Paper Solution

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Applied Mathematics 1
CBCGS
2017-2018 June
Marks: 80

(1) Question no. 1 is compulsory.
(2) Attempt any 3 questions from remaining five questions.


[20]1
[3]1.a

If `tan(x/2)=tanh(u/2),"show that" u = log[(tan(pi/4+x/2))] `

Chapter: [5] Complex Numbers
Concept: Inverse Hyperbolic Functions
[3]1.b

Prove that the following matrix is orthogonal & hence find 𝑨−𝟏.

A`=1/3[(-2,1,2),(2,2,1),(1,-2,2)]`

Chapter: [7] Matrices
Concept: Transpose of a Matrix
[3]1.c

State Euler’s theorem on homogeneous function of two variables and if `u=(x+y)/(x^2+y^2)` then evaluate `x(delu)/(delx)+y(delu)/(dely`

Chapter: [8] Partial Differentiation
Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
[3]1.d

If `u=r^2cos2theta, v=r^2sin2theta. "find"(del(u,v))/(del(r,theta))`

Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
Concept: Jacobian
[4]1.e

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

Chapter: [6.01] Successive Differentiation
Concept: nth derivative of standard functions
[4]1.f

Evaluate : `lim_(x->0)((2x+1)/(x+1))^(1/x)`

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
Concept: L‐ Hospital Rule
[20]2
[6]2.a

Solve  `x^4-x^3+x^2-x+1=0.`

Chapter: [5] Complex Numbers
Concept: D’Moivre’S Theorem
[6]2.b

If `y=e^(tan^(-1)x)`.Prove that

`(1+x^2)y_(n+2)+[2(n+1)x-1]y_(n+1)+n(n+1)y_n=0`

Chapter: [6.01] Successive Differentiation
Concept: Leibnitz’S Theorem (Without Proof) and Problems
[8]2.c

Examine the function `f(x,y)=xy(3-x-y)` for extreme values & find maximum and minimum values of `f(x,y).`

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

Investigate for what values of 𝝁 𝒂𝒏𝒅 𝝀 the equation x+y+z=6; x+2y+3z=10; x+2y+𝜆z=𝝁 have
(i)no solution,
(ii) a unique solution,
(iii) infinite no. of solution.

Chapter: [7] Matrices
Concept: consistency and solutions of homogeneous and non – homogeneous equations
[6]3.b

If u =`f((y-x)/(xy),(z-x)/(xz)),` show that `x^2(delu)/(delx)+y^2(delu)/(dely)+z^2(delu)/(delz)=0`.

Chapter: [8] Partial Differentiation
Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
[8]3.c

Prove that `log((a+ib)/(a-ib))=2itan^(-1)  b/a      &    cos[ilog((a+ib)/(a-ib))=(a^2-b^2)/(a^2+b^2)]`

Chapter: [6.02] Logarithm of Complex Numbers
Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
[20]4
[6]4.a

If `u=sin^(-1)((x+y)/(sqrtx+sqrty))`,Prove that

`x^2u_(x x)+2xyu_(xy)+y^2u_(yy)=(-sinu.cos2u)/(4cos^3u)`

Chapter: [8] Partial Differentiation
Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
[6]4.b

Using encoding matrix `[(1,1),(0,1)]`encode and decode the message

“ALL IS WELL” .

Chapter: [7] Matrices
Concept: Application of Inverse of a Matrix to Coding Theory
[8]4.c

Solve the following equation by Gauss Seidal method:

`10x_1+x_2+x_3=12`
`2x_1+10x_2+x_3-13`
`2x_1+2x_2+10x_3=14`

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
Concept: Gauss Seidal Iteration Method
[20]5
[6]5.a

If `u=e^(xyz)f((xy)/z)` where `f((xy)/z)` is an arbitrary function of `(xy)/z.`

Prove that: `x(delu)/(delx)+z(delu)/(delz)=y(delu)/(dely)+z(delu)/(delz)=2xyz.u`

Chapter: [8] Partial Differentiation
Concept: Partial Derivatives of First and Higher Order
[6]5.b

Prove that `sin^5theta=1/16[sin5theta-5sin3theta+10sintheta]`

Chapter: [5] Complex Numbers
Concept: Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
[8]5.c
[4]5.c.i

Prove that `log(secx)=1/2x^2+1/12x^4+.........`

Chapter: [6.02] Logarithm of Complex Numbers
Concept: Logarithmic Functions
[4]5.c.ii

Expand `2x^3+7x^2+x-1` in powers of x - 2

Chapter: [5] Complex Numbers
Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ

Prove that `sin^(-1)(cosec  theta)=pi/2+i.log(cot  theta/2)`

Chapter: [5] Complex Numbers
Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ
[20]6
[6]6.a

Prove that `sin^(-1)(cosec  theta)=pi/2+i.log(cot  theta/2)`

Chapter: [5] Complex Numbers
Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ
[6]6.b

Find non singular matrices P and Q such that A = `[(1,2,3,2),(2,3,5,1),(1,3,4,5)]`

Chapter: [7] Matrices
Concept: PAQ in normal form
[8]6.c

Obtain the root of 𝒙𝟑−𝒙−𝟏=𝟎 by Regula Falsi Method
(Take three iteration).

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
Concept: Regula – Falsi Equation

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