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Question Paper Solutions for Applied Mathematics 1 CBCGS 2016-2017 BE Chemical Engineering Semester 1 (FE First Year)

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Applied Mathematics 1
CBCGS
2016-2017 December
Marks: 80

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


[20]1
[3]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)`

Chapter: [5] Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number
[3]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)`

Chapter: [5] Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number
[3]1.c

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

Chapter: [5] Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number
[3]1.d

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

Chapter: [5] Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number
[4]1.e

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

Chapter: [7] Matrices
Concept: Addition of a Matrix
[4]1.f

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

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.`

Chapter: [5] Complex Numbers
Concept: Powers and Roots of Trigonometric Functions
[6]2.b

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

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

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
Concept: Gauss Seidal Iteration Method
[20]3
[6]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. 

Chapter: [7] Matrices
Concept: consistency and solutions of homogeneous and non – homogeneous equations
[6]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)`

Chapter: [5] Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number
[4]3.c

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

Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
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)
[4]3.d

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

Chapter: [5] Complex Numbers
Concept: Expansion of sinnθ, cosnθ in powers of sinθ, cosθ
[20]4
[6]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

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

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

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

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations
Concept: Regula – Falsi Equation
[20]5
[6]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`

Chapter: [6.01] Successive Differentiation
Concept: Leibnitz’S Theorem (Without Proof) and Problems
[6]5.b

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

Chapter: [7] Matrices
Concept: Application of Inverse of a Matrix to Coding Theory
[4]5.c

Considering only principal values separate into real and imaginary parts

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

Chapter: [6.02] Logarithm of Complex Numbers
Concept: Separation of Real and Imaginary Parts of Logarithmic Functions
[4]5.d

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

Chapter: [6.02] Logarithm of Complex Numbers
Concept: Logarithmic Functions
[20]6
[6]6.a

Using De Moivre’s theorem prove that]

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

Chapter: [5] Complex Numbers
Concept: D’Moivre’S Theorem
[6]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)`

Chapter: [7] Matrices
Concept: System of Homogeneous and Non – Homogeneous Equations
[8]6.c

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

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

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