If `z(x+y)=(x^2+y^2)`, prove that `((delz)/(delx)-(delz)/(dely))^2=4(1-(delz)/(delx)-(delz)/(dely))`

If u = (x^2+y^2+z^2)^(-1/2) then prove that (del^2u)/(delx^2)+(del^2u)/(dely^2)+(del^2u)/(delz^2) =0

If 2^(x)=3^(y)=6^(-z) prove that (1)/(x)+(1)/(y)+(1)/(z)=0

z=x+iy , prove that ((z)/bar(z)+(bar(z))/(z))=2((x^(2)-y^(2))/(x^(2)+y^(2)))

a^(x)=b^(y)=c^(z) and b^(2)=ac then prove that (1)/(x)+(1)/(z)=(2)/(y)

If x+y+z=xyz prove that (2x)/(1-x^(2))+(2y)/(1-y^(2))+(2z)/(1-z^(2))=(2x)/(1-x^(2))(2y)/(1-y^(2))(2z)/(1-z^(2))

The Schrodinger wave equation for H-atom is nabla^(2) Psi = (8pi^(2)m)/(h^(2)) (E-V) Psi = 0 Where nabla^(2) = (del^(2))/(delx^(2)) +(del^(2))/(dely^(2)) +(del^(2))/(delz^(2)) E = Total energy and V=potential energy wave function Psi_(((r, theta,phi)))R_((r))Theta_((theta))Phi_((phi)) R is radial wave function which is function of ''r'' only, where r is the distance from nucleus. Theta and Phi are angular wave function. R^(2) is known as radial probability density and 4pir^(2)R^(2)dr is known as radial probability function i.e., the probability of finding the electron is spherical shell of thickness dr. Number of radial node =n -l - 1 Number of angular node = l For hydrogen atom, wave function for 1s and 2s-orbitals are: Psi_(1s) = sqrt((1)/(pia_(0)^(a)))e^(-z_(r)//a_(0)) Psi_(2s) = ((Z)/(2a_(0)))^(½) (1-(Zr)/(a_(0)))e^(-(Zr)/(a_(0))) The plot of radial probability function 4pir^(2)R^(2) aganist r will be: Answer the following questions: The following graph is plotted for ns-orbitals The value of 'n' will be:

If x+y+z=xzy , prove that : (2x)/(1-x^2) + (2y)/(1-y^2) + (2z)/(1-z^2) = (2x)/(1-x^2) (2y)/(1-y^2) (2z)/(1-z^2)

if x^(2) + y^(2) = z^(2) then prove that log_(y)(z+x) + log_(y) (z-x)=2

For conservative force field, vec(F)=-(delU)/(delx)hat(i)-(delU)/(dy)hat(j)-(delU)/(delz)hat(k) where Frarr Maginitude of Force. Urarr Potential energy and (delU)/(delx)= Differentiation of U w.r.t. x keeping y and z constant and so on. Choose incorrect matching :-

If xy+yz+xz=1, then prove that (x)/(1-x^(2))+(y)/(1-y^(2))+(z)/(1-z^(2))=(4xyz)/((1-x^(2))(1-y^(2))(1-z^(2)))