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

Applied Mathematics 1 [CBCGS]
Date: December 2016

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

1
1.a

If cos alpha cos beta=x/2, sinalpha sinbeta=y/2, prove that:

sec(alpha -ibeta)+sec(alpha-ibeta)=(4x)/(x^2-y^2)

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter:  Complex Numbers
1.b

If z =log(e^x+e^y) "show that rt" - s^2 = 0  "where r"= (del^2z)/(delx^2),t=(del^2z)/(dely^2)"s"=(del^2z)/(delx dely)

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter:  Complex Numbers
1.c

If x = uv, y =(u+v)/(u-v).find (del(u,v))/(del(x,y)).

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter:  Complex Numbers
1.d

If y=2^xsin^2x cosx find y_n

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter:  Complex Numbers
1.e

Express the matrix as the sum of symmetric and skew symmetric matrices.

Chapter:  Matrices
1.f

Evaluat lim_(x->0) (e^(2x)-(1+x)^2)/(xlog(1+x)

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

Show that the roots of x5 =1 can be written as 1, alpha^1,alpha^2,alpha^3,alpha^4 .hence show that (1-alpha^1) (1-alpha^2) (1-alpha^3)(1-alpha^4)=5.

Concept: Powers and Roots of Trigonometric Functions
Chapter:  Complex Numbers
2.b

Reduce the following matrix to its normal form and hence find its rank.

Concept: Reduction to Normal Form
Chapter:  Matrices
2.c

Solve the following equation by Gauss-Seidel method upto four iterations

4x-2y-z=40, x-6y+2y=-28, x-2y+12z=-86.

Concept: Gauss Seidal Iteration Method
Chapter:  Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
3
3.a

Investigate for what values of μ and λ the equations x+y+z=6, x+2y+3z=10, x+2y+λz=μ has
1) No solution
2) A unique solution
3) Infinite number of solutions.

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

If u=x^2+y^2+z^2 where x=e^t, y=e^tsint,z=e^tcost

Prove that (du)/(dt)=4e^(2t)

Concept: Review of Complex Numbers‐Algebra of Complex Number
Chapter:  Complex Numbers
3.c

Show that sin(e^x-1)=x^1+x^2/2-(5x^4)/24+...................

Concept: Expansion of 𝑒^𝑥 , sin(x), cos(x), tan(x), sinh(x), cosh(x), tanh(x), log(1+x), 𝑠𝑖𝑛−1 (𝑥),𝑐𝑜𝑠−1 (𝑥),𝑡𝑎𝑛−1 (𝑥)
Chapter:  Applications of Partial Differentiation , Expansion of Functions
3.d

Expand 2x^3+7x^2+x-6 in powers of (x-2)

Concept: Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
Chapter:  Complex Numbers
4
4.a

If x = u+v+w, y = uv+vw+uw, z = uvw and φ is a function of x, y and z
Prove that Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)
Chapter:  Partial Differentiation
4.b

If tan(θ+iφ)=tanα+isecα
Prove that
1)e^(2varphi)=cot(varphi/2)
2) 2theta=npi+pi/2+alpha

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

Find the roots of the equation x^4+x^3 -7x^2-x+5 = 0 which lies between 2 and 2.1 correct to 3 places of decimals using Regula Falsi method.

Concept: Regula – Falsi Equation
Chapter:  Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
5
5.a

If y=(x+√x2-1 ,Prove that

(x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0

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

Using the encoding matrix [(1,1),(0,1)] encode and decode the messag I*LOVE*MUMBAI.

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

Considering only principal values separate into real and imaginary parts

i^((log)(i+1))

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
Chapter: [6.02] Logarithm of Complex Numbers
5.d

Show that ilog((x-i)/(x+i))=pi-2tan6-1x

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

Using De Moivre’s theorem prove that]

cos^6theta-sin^6theta=1/16(cos6theta+15cos2theta)

Concept: D’Moivre’S Theorem
Chapter:  Complex Numbers
6.b

If u =sin^(-1)((x^(1/3)+y^(1/3))/(x^(1/2)-y^(1/2))), Prove that

x^2(del^2u)/(delx^2)+2xy(del^2u)/(delxdely)+y^2(del^2u)/(dely^2)=tanu/144(tan^2u+13)

Concept: System of Homogeneous and Non – Homogeneous Equations
Chapter:  Matrices
6.c

Find the maxima and minima of x^3 y^2(1-x-y)

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

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