The correct option is B - 1/5.

[as per provisional answerkey]

Solution

Step 1: Establish the relationships between variables.
1. y is directly proportional to $x^{2}$ : $y = k_{1} \cdot x^{2}$
2. x is inversely proportional to z: $x = k_{2} / z$
Combining these, we get: $y = k_{1} \cdot (k_{2} / z)^{2} = K / z^{2}$ (where K is a constant).

Step 2: Find the constant K using the given values.
Given: $z = 7 / 25$ , $x = 5$ , and $y = 50$ .
Using $y = K / z^{2}$ :
$50 = K / (7 / 25)^{2}$
$50 = K / (49 / 625)$
$K = 50 \cdot (49 / 625)$
$K = (2 \cdot 25 \cdot 49) / (25 \cdot 25)$
$K = (2 \cdot 49) / 25 = 98 / 25$ .

Step 3: Calculate z for the new value of y.
Given: $y = 98$ . Find z.
$y = K / z^{2}$
$98 = (98 / 25) / z^{2}$
Divide both sides by 98:
$1 = (1 / 25) / z^{2}$
$z^{2} = 1 / 25$
Since z is a positive real number:
$z = \sqrt{1 / 25} = 1 / 5$ .

Why the other options are incorrect

Option (a) - 1/7: This value would result if the relationship was $y \propto 1 / z$ instead of $y \propto 1 / z^{2}$ , or if a calculation error occurred while squaring the fraction.
Option (c) - 5/7: This value is the reciprocal of the ratio used in the initial conditions ( $7 / 25 \times 5 / 1$ ) and does not satisfy the inverse square relationship between y and z.
Option (d) - 1: This value assumes a direct equality or a specific constant K that does not align with the provided data points ( $z = 7 / 25, y = 50$ ).

Key Concept

The principle of combined variation, where the relationship between the first and last variable is derived by substituting the intermediate variable ( $y \propto x^{2}$ and $x \propto 1 / z$ implies $y \propto 1 / z^{2}$ ).