Question : If $\left (2a-1 \right )^{2}+\left (4b-3 \right)^{2}+\left (4c+5 \right)^{2}=0$, then the value of $\frac{a^{3}+b^{3}+c^{3}-3abc}{a^{2}+b^{2}+c^{2}}$ is:

Option 1: $1\tfrac{3}{8}$

Option 2: $2\tfrac{3}{8}$

Option 3: $3\tfrac{3}{8}$

Option 4: $0$

Question : If $a^{2}+b^{2}+c^{2}-ab-bc-ca=0$, then

Option 1: $a=b=c$

Option 2: $a\neq b=c$

Option 3: $a=b\neq c$

Option 4: $a\neq b\neq c$

Question : If $\small x=a\left (b-c \right),\; y=b\left (c-a \right) ,\; z=c\left (a-b \right)$, then the value of $\left (\frac{x}{a} \right)^{3}+\left (\frac{y}{b} \right)^{3}+\left (\frac{z}{c} \right)^{3}$ is:

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

Option 2: $\frac{xyz}{abc}$

Option 3: $0$

Option 4: $\frac{3xyz}{abc}$

Question : If $a+b :\sqrt{ab} = 4:1 $ where $ a > b > 0$, then $ a:b$ is:

Option 1: $\left (2+\sqrt{3} \right):\left (2-\sqrt{3} \right)$

Option 2: $\left (2-\sqrt{3} \right):\left (2+\sqrt{3} \right)$

Option 3: $\left (3+\sqrt{2} \right):\left (3-\sqrt{2} \right)$

Option 4: $\left (3-\sqrt{2} \right):\left (3+\sqrt{2} \right)$

Question : If $\small c+\frac{1}{c}=3$, then the value of $\left (c-3 \right )^{7}+\frac{1}{c^{7}}$ is:

Option 1: 2

Option 2: 0

Option 3: 3

Option 4: 1