Suppose x, y and z are variables taking positive real numbers as their possible values. It is given that y is directly proportional to x^2 and x is…

The correct option is B - 1/5.

Solution

Step 1: Establish the relationships between variables.
1. y is directly proportional to $x^{2}$x2: $y = k_1 \cdot x^{2}$y=k1​⋅x2
2. x is inversely proportional to z: $x = k_2 / z$x=k2​/z
Combining these, we get: $y = k_1 \cdot (k_2 / z)^{2} = K / z^{2}$y=k1​⋅(k2​/z)2=K/z2 (where K is a constant).

Step 2: Find the constant K using the given values.
Given: $z = 7/25$z=7/25, $x = 5$x=5, and $y = 50$y=50.
Using $y = K / z^{2}$y=K/z2:
$50 = K / (7/25)^{2}$50=K/(7/25)2
$50 = K / (49/625)$50=K/(49/625)
$K = 50 \cdot (49/625)$K=50⋅(49/625)
$K = (2 \cdot 25 \cdot 49) / (25 \cdot 25)$K=(2⋅25⋅49)/(25⋅25)
$K = (2 \cdot 49) / 25 = 98/25$K=(2⋅49)/25=98/25.

Step 3: Calculate z for the new value of y.
Given: $y = 98$y=98. Find z.
$y = K / z^{2}$y=K/z2
$98 = (98/25) / z^{2}$98=(98/25)/z2
Divide both sides by 98:
$1 = (1/25) / z^{2}$1=(1/25)/z2
$z^{2} = 1/25$z2=1/25
Since z is a positive real number:
$z = \sqrt{1/25} = 1/5$z=1/25​=1/5.

Why the other options are incorrect

• Option (a) - 1/7: This value would result if the relationship was $y \propto 1/z$y∝1/z instead of $y \propto 1/z^{2}$y∝1/z2, 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$7/25×5/1) and does not satisfy the inverse square relationship between y and z.
• Option (d) - 1: This value assumes a direct equality or a specific constant K that does not align with the provided data points ($z=7/25, y=50$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 ($y \propto x^{2}$y∝x2 and $x \propto 1/z$x∝1/z implies $y \propto 1/z^{2}$y∝1/z2).