Question : A person distributes his pens among four friends A, B, C and D in ratio $\frac{1}{3}:\frac{1}{4}:\frac{1}{5}:\frac{1}{6}$. What is the minimum number of pens that the person should have?

Option 1: 57

Option 2: 65

Option 3: 75

Option 4: 45

Question : If $a+b+c+d=4$, then find the value of $\frac{1}{(1-a)(1-b)(1-c)}+\frac{1}{(1-b)(1-c)(1-d)}+\frac{1}{(1-c)(1-d)(1-a)}+\frac{1}{(1-d)(1-a)(1-b)}$?

Option 1: 0

Option 2: 5

Option 3: 1

Option 4: 4

Question : The simplified value of the following is:
$\left (\frac{3}{15}a^{5}b^{6}c^{3}\times \frac{5}{9}ab^{5}c^{4} \right )\div \frac{10}{27}a^{2}bc^{3}$.

Option 1: $\frac{9a^{2}bc^{4}}{10}$

Option 2: $\frac{3ab^{4}c^{3}}{10}$

Option 3: $\frac{3a^{4}b^{10}c^{4}}{10}$

Option 4: $\frac{1a^{4}b^{4}c^{10}}{10}$

Question : If $\frac{x}{xa+yb+zc}=\frac{y}{ya+zb+xc}=\frac{z}{za+xb+yc}$ and $x+y+z\neq 0$, then each ratio is:

Option 1: $\frac{1}{a-b-c}$

Option 2: $\frac{1}{a+b-c}$

Option 3: $\frac{1}{a-b+c}$

Option 4: $\frac{1}{a+b+c}$

Question : If $a+b+c=13$ and $a b+b c+c a=45$, find $a^2+b^2+c^2$.

Option 1: 79

Option 2: 65

Option 3: 57

Option 4: 85

Question : The value of the expression $\frac{(a-b)^{2}}{(b-c)(c-a)}+\frac{(b-c)^{2}}{(a-b)(c-a)}+\frac{(c-a)^{2}}{(a-b)(b-c)}$ is:

Option 1: $0$

Option 2: $3$

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

Option 4: $2$