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Question Bank Solutions for BE Biotechnology Semester 1 (FE First Year) - University of Mumbai - Applied Mathematics 1

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Applied Mathematics 1
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Find maximum and minimum values of x3 +3xy2 -15x2-15y2+72x.

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

If U = `e^(xyz) f((xy)/z)` prove that `x(delu)/(delx)+z(delu)/(delx)2xyzu` and `y(delu)/(delx)+z(delu)/(delz)=2xyzu` and hence show that `x(del^2u)/(delzdelx)=y(del^2u)/(delzdely)`

[6.01] Successive Differentiation
Chapter: [6.01] Successive Differentiation
Concept: Successive Differentiation

If y = log `[tan(pi/4+x/2)]`Prove that

I. tan h`y/2 = tan  pi/2`
II. cos hy cos x = 1

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

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

[7] Matrices
Chapter: [7] Matrices
Concept: Addition of a Matrix

Find the maximum and minimum values of `f(x,y)=x^3+3xy^2-15x^2-15y^2+72x`

[8] Partial Differentiation
Chapter: [8] Partial Differentiation
Concept: Total Differentials

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

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

Obtain tan 5𝜽 in terms of tan 𝜽 & show that `1-10tan^2  x/10+5tan^4  x/10=0`

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

If y=etan_1x. prove that `(1+x^2)yn+2[2(n+1)x-1]y_n+1+n(n+1)y_n=0`

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

Find tanhx if 5sinhx-coshx = 5 

 

 

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

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

 

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

Considering only principal values separate into real and imaginary parts

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

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

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

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

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

[9] Applications of Partial Differentiation , Expansion of Functions
Chapter: [9] Applications of Partial Differentiation , Expansion of Functions
Concept: Jacobian

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

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

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

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

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)]`

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

Evaluate `lim_(x->0) sinx log x.`

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

Find the nth derivative of y=eax cos2 x sin x.

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

Find nth derivative of `1/(x^2+a^2.`

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

Find all values of `(1+i)^(1/3)` & show that their continued
Product is (1+i).

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