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 inversely proportional to z. For z = 7/25, the values of x and y are 5 and 50, respectively. If y = 98, what is z equal to?
- (a) 1/7
- (b) 1/5 ✓ UPSC's answer
- (c) 5/7
- (d) 1
Why the answer is (b)
• Since y is directly proportional to x^2, we have y = k1 * x^2. Using the given values x = 5 and y = 50, we find k1 = 50 / 25 = 2, so y = 2x^2.
• Since x is inversely proportional to z, we have x = k2 / z, or xz = k2. Using x = 5 and z = 7/25, we find k2 = 5 * (7/25) = 7/5.
• We are given y = 98. Using y = 2x^2, we solve for x: 98 = 2x^2, which gives x^2 = 49, so x = 7 (since x is positive).
• Now, using the relationship xz = 7/5 with x = 7, we have 7z = 7/5.
• Solving for z gives z = (7/5) / 7 = 1/5.
• Therefore, the value of z is 1/5, which corresponds to option (b).
Why the other options are wrong
- (a) 1/7
- Option (a) 1/7 is incorrect because it does not satisfy the derived relationship xz = 7/5 when x = 7.
- (c) 5/7
- Option (c) 5/7 is incorrect because it results from an error in calculating the constant of proportionality or solving for z.
- (d) 1
- Option (d) 1 is incorrect because substituting z = 1 would imply x = 7/5, which contradicts the calculated x = 7.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2026, held on 24 May 2026. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.