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

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Applied Mathematics 1
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If cos alpha cos beta=x/2, sinalpha sinbeta=y/2, prove that:

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

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

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)

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

If Z=tan^1 (x/y), where x=2t, y=1-t^2, "prove that" d_z/d_t=2/(1+t^2).

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

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

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

Evaluate : Lim_(x→0) (x)^(1/(1-x))

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

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

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

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

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

Prove that log [tan(pi/4+(ix)/2)]=i.tan^-1(sinhx)

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

If Z=x^2 tan-1y /x-y^2 tan -1 x/y del

Prove that (del^z z)/(del_ydel_x)=(x^2-y^2)/(x^2+y^2)

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

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

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

 Complex Numbers
Chapter:  Complex Numbers
Concept: Review of Complex Numbers‐Algebra of Complex Number

Show that the matrix A is unitary where A = [[alpha+igamma,-beta+idel],[beta+idel,alpha-igamma]] is unitary if alpha^2+beta^2+gamma^2+del^2=1

 Matrices
Chapter:  Matrices
Concept: Inverse of a Matrix

If z=tan(y-ax)+(y-ax)^(3/2) then show that (del^2z)/(delx^2)= a^2 (del^2z)/(dely^2)

 Partial Differentiation
Chapter:  Partial Differentiation
Concept: Partial Derivatives of First and Higher Order

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

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

If x=uv, y=u/v."prove that"  jj,=1

 Partial Differentiation
Chapter:  Partial Differentiation
Concept: Partial Derivatives of First and Higher Order

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

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

Find the stationary points of the function x3+3xy2-3x2-3y2+4 & also find maximum and minimum values of the function.

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

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

 Applications of Partial Differentiation , Expansion of Functions
Chapter:  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) 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

 Partial Differentiation
Chapter:  Partial Differentiation
Concept: Partial Derivatives of First and Higher Order

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

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

Show that sec h-1(sin θ) =log cot (theta/2 ).

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