Pi rationals

Number Theory Level 3

Let $p,q,r$ be rational numbers, where p is not equal to zero and $\left| \begin{matrix} p & \pi \\ r & q \end{matrix} \right| =0$ , then $q^{3}+r^{2}$ equals:

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Sualeh Asif
Apr 27, 2015

The determinant of a square 2×2 matrix is $ad-bc$ . In this case it is $pq-r\pi =0 \longrightarrow pq=r\pi$

Now since $\pi$ is irrational the RHS is irrational (if r is non-zero).Thus the LHS should have an irrational number but that isn't the case since p,q,r are rational. However this isn't possible and we revert to the special case:

The LHS=RHS=0 . Then since p is non-zero q and r must be equal to 0 .

Thus, $q^3 +r^3 = 0^3 +0^3 =\boxed {0}$

Yes as simple as that...

Nishu sharma - 6 years, 1 month ago

Pi rationals

This problem is a part of the set advanced is basic

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