If x,y,z∈R, x+y+z=5, x2+y2+z2=9𝑥,𝑦,𝑧∈𝑅, 𝑥+𝑦+𝑧=5, 𝑥2+𝑦2+𝑧2=9, then length of interval in which x𝑥 lies is

If x,y,z∈R, x+y+z=5, x2+y2+z2=9𝑥,𝑦,𝑧∈𝑅, 𝑥+𝑦+𝑧=5, 𝑥2+𝑦2+𝑧2=9, then length of interval in which x𝑥 lies is

The curl of F⃗ =(z−x−y)i⃗ +(x−y−z)j⃗ +(y−z−x)k⃗ 𝐹→=(𝑧−𝑥−𝑦)𝑖→+(𝑥−𝑦−𝑧)𝑗→+(𝑦−𝑧−𝑥)𝑘→ is

he complex number z1,z2𝑧1,𝑧2 satisfies the equation z+1+8 i=|z|(1+i)𝑧+1+8 𝑖=𝑧1+𝑖, where i=−1−−−√𝑖=-1 then the equation whose roots are |z1|𝑧1 and |z2|𝑧2 is

If 𝑢 = 𝑥2 tan−1 𝑦𝑥 − 𝑦2 tan−1 𝑥𝑦 show that 𝜕2𝑢𝜕𝑥𝜕𝑦 = 𝑥2−𝑦2𝑥2+𝑦2 and 𝜕2𝑢𝜕𝑥𝜕𝑦 = 𝜕2𝑢𝜕𝑦𝜕𝑥.

If 𝑣 = 𝑟𝑚, where 𝑟2 = 𝑥2 + 𝑦2 + 𝑧2, then prove that 𝑣𝑥𝑥 + 𝑣𝑦𝑦 + 𝑣𝑧𝑧 = 𝑚(𝑚 + 1)𝑟𝑚−2.