Question

If $p,q,r$ are positive integers, and
$\Delta =12\left|\begin{array}{ccc}{}^p{C}_1& {}^p{C}_2& {}^p{C}_3\\ {}^q{C}_1& {}^q{C}_2& {}^q{C}_3\\ {}^r{C}_1& {}^r{C}_2& {}^r{C}_3\end{array}\right|$
then

• A. $\Delta$ is an even integer
• B. $\Delta$ is divisible by 12
• C. $\Delta$ is an rational number
• D. None of these

Answer 1

Answer: (A), (C)

$\Delta$ is an even integer
$\Delta$ is an rational number

We have
${}^p{C}_1=p{,}^p{C}_2=\frac{1}{2}p(p-1)and{}^p{C}_3=\frac{1}{6}p(p-1)(p-2)$
Thus,
$\Delta =pqr\left|\begin{array}{ccc}1& p-1& (p-1)(p-2)\\ 1& q-1& (q-1)(q-2)\\ 1& r-1& (r-1)(r-2)\end{array}\right|$
Using $R_3\to R_3-R_{2}and{R}_2\to R_2-R_1$ we get
$\Delta =pqr\left|\begin{array}{ccc}1& p-1& (p-1)(p-2)\\ 0& q-p& (q-p)(q+p-3)\\ 0& r-q& (r-q)(r+q-3)\end{array}\right|$
$=pqr(q-p)(r-q)\left|\begin{array}{cc}1& q+p-3\\ 1& r+q-3\end{array}\right|$
$=pqr(q-p)(r-q)(r-p)$
Now, use at least one of these six factors is even.