Date: June 2017

Prove that 𝒕𝒂𝒏𝒉−𝟏(𝒔𝒊𝒏 𝜽) = 𝒄𝒐𝒔𝒉−𝟏(𝒔𝒆𝒄 𝜽)

Chapter: [5] Complex Numbers

Prove that the matrix `1/sqrt3` `[[ 1,1+i1],[1-i,-1]]` is unitary.

Chapter: [5] Complex Numbers

`"If" x=uv & y=u/v "prove that" jj^1=1`

Chapter: [5] Complex Numbers

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

Chapter: [5] Complex Numbers

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

Chapter: [5] Complex Numbers

Evaluate : `Lim_(x→0) (x)^(1/(1-x))`

Chapter: [5] Complex Numbers

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

Product is (1+i).

Chapter: [5] Complex Numbers

Find non singular matrices P & Q such that PAQ is in normal form where A `[[2,-2,3],[3,-1,2],[1,2,-1]]`

Chapter: [7] Matrices

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

Chapter: [8] Partial Differentiation

If u=`f((y-x)/(xy),(z-x)/(xz)),"show that" x^2 (del_u)/(del_x)+y^2 (del_u)/(del_y)+x^2 del_u/del_z=0`

Chapter: [5] Complex Numbers

Using encoding matrix `[[1,1],[0,1]]` ,encode & decode the message "MUMBAI"

Chapter: [7] Matrices

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

Chapter: [6.02] Logarithm of Complex Numbers

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

Chapter: [6.02] Logarithm of Complex Numbers

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`

Chapter: [6.02] Logarithm of Complex Numbers

Express `(2x^3+3x^2-8x+7)` in terms of (x-2) using taylor'r series.

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

Prove that `tan_1 x=x-x^3/3+x^5/5+.............`

Chapter: [9] Applications of Partial Differentiation , Expansion of 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)`

Chapter: [6.02] Logarithm of Complex Numbers

Investigate for what values of 𝝁 "𝒂𝒏𝒅" 𝝀 the equations : `2x+3y+5z=9`

`7x+3y-2z=8`

`2x+3y+λz=μ`

Have (i) no solution (ii) unique solution (iii) Infinite value

Chapter: [7] Matrices

Obtain the root of `x^3-x-1=0` by Newton Raphson Method` (upto three decimal places).

Chapter: [7] Matrices

Find tanhx if 5sinhx-coshx = 5

Chapter: [6.02] Logarithm of Complex Numbers

If u= `sin^-1 ((x+y)/(sqrtx+sqrty)), " prove that ""`i.xu_x+yu_y=1/2 tanu`

ii. `x^2uxx+2xyu_xy+y^2u_(y y)=(-sinu.cos2u)/(4cos^3u)`

Chapter: [7] Matrices

Solve the following system of equation by Gauss Siedal Method,20x+y-2z=17

3x+20y-z =-18

2x-3y+20z=𝟐𝟓

Chapter: [10] Indeterminate Forms, Numerical Solutions of Transcendental Equations and System of Linear Equations

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