JEE Advanced 2022 · Paper 2, Mathematics question 11

Polar form. Let $z=re^{i\theta}$. Then
$$(\bar z)^2+\frac1{z^2}=\left(r^2+\frac1{r^2}\right)e^{-2i\theta}.$$
Its real and imaginary parts are integers $p,q$ exactly when
$$\left(r^2+\frac1{r^2}\right)^2=p^2+q^2$$
is an integer $m$ that is a sum of two squares (and some $\theta$ then works).

In terms of $s=r^4$: $s+\frac1s=m-2$.

Test the options. Each has the form $s=\frac{a+\sqrt D}{k}$, and in each case $\frac1s$ is the conjugate $\frac{a-\sqrt D}{k}$, so $s+\frac1s=\frac{2a}k$.

• (A) $s+\frac1s=43$, so $m=45=6^2+3^2$. Possible.
• (B) $s+\frac1s=\frac72$: $m$ is not an integer.
• (C) $s+\frac1s=\frac92$: $m$ is not an integer.
• (D) $s+\frac1s=\frac73$: $m$ is not an integer.

(A).