Question

Suppose that the number of elements in set $A$ is $p$, the number of elements in set $B$ is $q$ and the number of elements in $A\times B$ is $7$. Then $p^2+q^2=$

• A. $42$
• B. $49$
• C. $50$
• D. $51$

Answer 1

The correct option is C

$50$

Find the value of $p^2+q^2$:

Step-1 : Finding the values of $p$and $q$

Given that $n\left(A\right)=p$, $n\left(B\right)=q$ and $n(A\times B)=7$.

We know that $n(A\times B)=n\left(A\right)n\left(B\right)$. So, we must have $pq=7$. Since, $p$ and $q$ are positive integers, the only possible values will be $p=1,q=7$ or $p=7,q=1$.

Step-2 : Finding the value of $p^2+q^2$

For $p=1,q=7$,

$$\begin{gathered} p^2+q^2=1^2+7^2 \\ =50 \end{gathered}$$

and for $p=7,q=1$,

$$\begin{gathered} p^2+q^2=7^2+1^2 \\ =50 \end{gathered}$$

Hence, option (C) is the correct answer.