2.If `u=f(r,s),r=x+y,s=x-y` prove that `((delu)/(delx)+(delu)/(dely))=2(delu)/(delr)`

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

Select the correct statements for ideal gases: 1. ((delH)/(delT))_(P)-((delU)/(delT))_(V)=R 2. ((delH)/(delT))_(P)-((delU)/(delT))_(P)-((delU)/(delT))_(V) =work done 3. ((delU)/(delV))_(T)=+ve 4. ((delT)/(delP))_(H)=+ve

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 u = log ((x ^ (2) + y ^ (2)) / (x + y)), prove that x (u) / (x) + y (u) / (y) = 1

If x+y=1, prove that sum_(r=0)^(n)nC_(r)x^(r)y^(n-r)

13.If s(r)=a+at+at++a, Then prove that lim s(r)=f(a)

If (p+iq)/(r+is) = x + iy , prove that (p-iq)/(r-is) = x -iy and (p^(2)+q^(2))/(r^(2)+s^(2))=x^(2)+y^(2) ,where p,q,r,s,x,y,inR

If y_(1)(x) and y_(2)(x) are two solutions of (dy)/(dx)+f(x)y=r(x), then y_(1)(x)+y_(2)(x) is solution of : (A) (dy)/(dx)+f(x)y=0 (B) (dy)/(dx)+2f(x)y=r(x)(C)(dy)/(dx)+f(x)y=2r(x)(D)(dy)/(dx)+2f(x)y=2r(x)

If f(x+y)=f(x)+f(y),AAx ,\ y in R then prove that f(k x)=kf(x)for\ AAk ,\ x in Rdot

A function f : R to R Satisfies the following conditions (i) f (x) ne 0 AA x in R (ii) f(x +y)= f(x) f(y) AA x, y, in R (iii) f(x) is differentiable (iv ) f'(0) =2 The derivative of f(x) satisfies the equation