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Question Bank Solutions for BE Electrical 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)`

[5] Complex Numbers
Chapter: [5] 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)`

[5] Complex Numbers
Chapter: [5] 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).` 

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

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

 

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

Evaluate : `Lim_(x→0) (x)^(1/(1-x))`

[5] Complex Numbers
Chapter: [5] 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))`.

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

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

[5] Complex Numbers
Chapter: [5] 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)`

[5] Complex Numbers
Chapter: [5] 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` 

[7] Matrices
Chapter: [7] 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)`

[8] Partial Differentiation
Chapter: [8] 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`

[8] Partial Differentiation
Chapter: [8] Partial Differentiation
Concept: Partial Derivatives of First and Higher Order

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

[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

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

[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

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

[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)` 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`

[8] Partial Differentiation
Chapter: [8] 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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Question Bank Solutions for BE Electrical Engineering Semester 1 (FE First Year) University of Mumbai Applied Mathematics 1. You can further filter Question Bank Solutions by subjects and topics. Solutions for most of the questions for University of Mumbai can be found here on Shaalaa.com. You can use these solutions to prepare for your studies and ace in exams. Solving questions is a great way to practice and with Shaalaa.com, you can answer a question and then also check your answer with the solutions provided.
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