Question

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0, 2)$ that can be written as fractions $\dfrac{n}{d}$ where $n$ and $d$ are integers with $1 \le d \le 5.$ What is the probability that $(\cos(a\pi) + i\sin(b\pi))^4$ is a real number?

$\dfrac{3}{50}$

$\dfrac{4}{25}$

$\dfrac{41}{200}$

$\dfrac{6}{25}$

$\dfrac{13}{50}$

Solution:

There are $20$ possible values for each of $a$ and $b,$ namely the reduced fractions in $[0, 2)$ with denominator dividing into $1 \le d \le 5.$

Writing $x = \cos(a\pi)$ and $y = \sin(b\pi),$ the fourth power $(x + iy)^4$ is real if and only if $x = 0,$ $y = 0,$ or $x = \pm y.$

The case $x = 0$ means $a \in \left\{\dfrac12, \dfrac32\right\},$ giving $2\cdot 20 = 40$ pairs; the case $y = 0$ means $b \in \{0, 1\},$ giving another $40$ pairs, of which $4$ were already counted. The remaining condition $\cos(a\pi) = \pm\sin(b\pi)$ with neither zero contributes $20$ more pairs.

In all there are $40 + 40 - 4 + 20 = 96$ valid pairs out of $400,$ so the probability is $\dfrac{96}{400} = \dfrac{6}{25}.$

Thus, the correct answer is D.

2015 AMC 12A Problem 24

Solution:

Problem 24 in Other Years