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(C) Let the roots be $a, ar, ar^2, ar^3$ in $G$.$P$.From the first equation $x^2 - x + p = 0$, we have $\alpha + \beta = a + ar = 1$ and $\alpha\beta

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$34$

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(C) Let the roots be $a, ar, ar^2, ar^3$ in $G$.$P$.From the first equation $x^2 - x + p = 0$, we have $\alpha + \beta = a + ar = 1$ and $\alpha\beta = a(ar) = a^2r = p$.From the second equation $x^2 - 4x + q = 0$, we have $\gamma + \delta = ar^2 + ar^3 = r^2(a + ar) = 4$ and $\gamma\delta = (ar^2)(ar^3) = a^2r^5 = q$.Since $a + ar = 1$, we substitute this into the second sum equation: $r^2(1) = 4 \implies r^2 = 4 \implies r = \pm 2$.Case $1$: If $r = 2$, then $a(1 + 2) = 1 \implies a = 1/3$. Then $p = a^2r = (1/9)(2) = 2/9$, which is not an integer. This case is rejected.Case $2$: If $r = -2$, then $a(1 - 2) = 1 \implies -a = 1 \implies a = -1$.Now calculate $p$ and $q$: $p = a^2r = (-1)^2(-2) = -2$.$q = a^2r^5 = (-1)^2(-2)^5 = 1 	imes (-32) = -32$.Finally, $|p + q| = |-2 + (-32)| = |-34| = 34$.

Let $\alpha, \beta$ be the roots of the equation $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of the equation $x^2 - 4x + q = 0$, where $p, q \in Z$. If $\alpha, \beta, \gamma, \delta$ are in $G$.$P$.,then $|p + q|$ equals:

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