If `cos alpha cos beta=x/2, sinalpha sinbeta=y/2`, prove that:

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

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)`

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).`

Concept: Review of Complex Numbers‐Algebra of Complex Number

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

Concept: Review of Complex Numbers‐Algebra of Complex Number

Evaluate : `Lim_(x→0) (x)^(1/(1-x))`

Concept: Review of Complex Numbers‐Algebra of Complex Number

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

Concept: Review of Complex Numbers‐Algebra of Complex Number

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

Concept: Review of Complex Numbers‐Algebra of Complex Number

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

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)`

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)`

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`

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)`

Concept: Partial Derivatives of First and Higher Order

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

Concept: Logarithmic Functions

If `x=uv, y=u/v."prove that" jj,=1`

Concept: Partial Derivatives of First and Higher Order

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

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.

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).`

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`

Concept: Partial Derivatives of First and Higher Order

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

Concept: Logarithmic Functions

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

Concept: Logarithmic Functions