Question : Ram and Shyam are partners sharing profits and losses in the ratio of $\frac{7}{12}: \frac{5}{12}$ They admit Gopi as a new partner for 1/6^th share, which he acquires equally from Ram and Shyam, the new profit sharing ratios of the partners will be

Option 1: 7:5:6

Option 2: 1:2:3

Option 3: 3:2:1

Option 4: 5:3:2

Question : The value of $\frac{(x-y)^3+(y-z)^3+(z-x)^3}{6(x-y)(y-z)(z-x)}$, where $x \neq y \neq z$, is equal to:

Option 1: $\frac{1}{4}$

Option 2: $\frac{1}{2}$

Option 3: $\frac{1}{3}$

Option 4: $\frac{1}{9}$

Question : Simplify the given expression. $\frac{x^3+y^3+z^3-3 x y z}{(x-y)^2+(y-z)^2+(z-x)^2}$

Option 1: $\frac{1}{3}(x+y+z)$

Option 2: $(x+y+z)$

Option 3: $\frac{1}{4}(x+y+z)$

Option 4: $\frac{1}{2}(x+y+z)$

Question : If $\frac{1}{x+2}=\frac{3}{y+3}=\frac{1331}{z+1331}=\frac{1}{3}$, then what is the value of $\frac{x}{x+1}+\frac{y}{y+6}+\frac{z}{z+2662}$?

Option 1: $0$

Option 2: $1$

Option 3: $\frac{3}{2}$

Option 4: $3$

Question : A, B and C were partners in a firm sharing profits in a 3: 3: 2 ratio. They admitted D as a new partner for 4/7 profit. D acquired his share 2/7 from A, 1 / 7 from B and 1 / 7 from C. The new profit sharing ratio will be

Option 1: 5:13:6:32

Option 2: 5:13:6:4

Option 3: 3: 3: 2: 4

Option 4: None of the above