# Applied Mathematics 1 CBCGS 2017-2018 BE IT (Information Technology) Semester 1 (FE First Year) Question Paper Solution

Applied Mathematics 1 [CBCGS]
Date: June 2018

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

 1
 1.a

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

Concept: Inverse Hyperbolic Functions
Chapter:  Complex Numbers
 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)]

Concept: Transpose of a Matrix
Chapter:  Matrices
 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

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

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

Concept: Jacobian
Chapter:  Applications of Partial Differentiation , Expansion of Functions
 1.e

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

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

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

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

Solve  x^4-x^3+x^2-x+1=0.

Concept: D’Moivre’S Theorem
Chapter:  Complex Numbers
 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

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

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

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

Concept: consistency and solutions of homogeneous and non – homogeneous equations
Chapter:  Matrices
 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.

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

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

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

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

“ALL IS WELL” .

Concept: Application of Inverse of a Matrix to Coding Theory
Chapter:  Matrices
 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

Concept: Gauss Seidal Iteration Method
Chapter:  Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
 5
 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

Concept: Partial Derivatives of First and Higher Order
Chapter:  Partial Differentiation
 5.b

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

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

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

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

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

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

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

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

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

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

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

Concept: PAQ in normal form
Chapter:  Matrices
 6.c

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

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

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