Question : Let $p, q, r$ and $s$ be positive natural numbers having three exact factors including 1 and the number itself. If $q>p$ and both are two-digit numbers, and $r>s$ and both are one-digit numbers, then the value of the expression $\frac{p-q-1}{r-s}$ is:

Option 1: – s –1

Option 2: s – 1

Option 3: 1 – s

Option 4: s + 1

Question : If $\frac{1}{x^2+a^2}=x^2-a^2$, then the value of $x$ is:

Option 1: $\left(1-a^4\right)^\frac{1}{4}$

Option 2: $a$

Option 3: $\left(a^4-1\right)^\frac{1}{4}$

Option 4: $\left(a^4+1\right)^\frac{1}{4}$

Question : If $x_{1}x_{2}x_{3}=4(4+x_{1}+x_{2}+x_{3})$, then what is the value of $\left [ \frac{1}{(2+x_{1})} \right ]+\left [ \frac{1}{(2+x_{2})} \right ]+\left [ \frac{1}{(2+x_{3})} \right ]?$

Option 1: $1$

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

Option 3: $2$

Option 4: $\frac{1}{3}$

Question : If $\left(y+\frac{1}{y}\right)=4$, then find the value of $\left(y^6+\frac{1}{y^6}\right)$.

Option 1: 2702

Option 2: 2704

Option 3: 4096

Option 4: 2706

Question : Find the sum of $\left (1-\frac{1}{n+1} \right) + \left (1-\frac{2}{n+1} \right) + \left (1- \frac{3}{n+1} \right)+.....\left ( 1- \frac{n}{n+1} \right)$.

Option 1: $n$

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

Option 3: $(n+1)$

Option 4: $\frac{(n+1)}{2}$