मान लीजिए x, y और z चर हैं, जो अपने संभाव्य मानों के रूप में धनात्मक वास्तविक संख्याएं प्राप्त करते हैं। यह दिया गया है कि y, के अनुक्रमानुपाती है और x, z के व्युत्क्रमानुपाती है। z = 7/25 के लिए x और y के मान क्रमशः 5 और 50 हैं। यदि y = 98 है, तो z किसके बराबर है?
सही उत्तर: (a) 1/7
दिया गया है कि y, x^2 के अनुक्रमानुपाती है (y = k1 x x^2) और x, z के व्युत्क्रमानुपाती है (xz = k2)। z = 7/25 के लिए x = 5 है, अतः xz = 5 x (7/25) = 7/5। y = 50 और x = 5 के लिए, k1 = y / (x^2) = 50 / 25 = 2, अर्थात y = 2 x^2। जब y = 98 है, तो 2 x^2 = 98 => x^2 = 49 => x = 7। चूँकि xz = 7/5, अतः 7 x z = 7/5 जिससे z = 1/5 प्राप्त होता है। अतः विकल्प (b) सही है।
In English (Question & Model Answer)
Suppose x, y and z are variables taking positive real numbers as their possible values. It is given that y is directly proportional to 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?
Correct Option: (a) 1/7
Given y is directly proportional to x^2, so y = k1 x x^2. Also, x is inversely proportional to z, so x = k2 / z. Combining these, y = k x (k2/z)^2 = K / z^2. Given for z = 7/25, x = 5 and y = 50. Let's check the relation: y = k x^2 / z^2. Substitute the given values: 50 = k x (5^2) / (7/25)^2 = k x 25 / (49/625). Alternatively, use the proportionality constants: y / x^2 = constant1 = 50 / (5^2) = 50 / 25 = 2. So y = 2 x^2. Also, x x z = constant2 = 5 x (7/25) = 7/5. So xz = 7/5, which means x = (7/5) / z. Now we are given y = 98. Since y = 2 x^2, 98 = 2 x^2 => x^2 = 49 => x = 7. Using xz = 7/5, we get 7 x z = 7/5 => z = 1/5? Wait, let's recalculate: 7 x z = 7/5 => z = 1/5. Let's re-verify: if z = 1/5, x = (7/5)/(1/5) = 7, and y = 2 x^2 = 2 x 49 = 98. Thus, z = 1/5? Wait, let's check option (a) 1/7, (b) 1/5. Let's check if z = 1/7: x = (7/5) / (1/7) = 49/5, y = 2 x (49/5)^2 = 98 x 2401 / 25. Let's re-read carefully: y is directly proportional to x^2 (y = k1 x^2) and x is inversely proportional to z (xz = k2). For z = 7/25, x = 5, so constant k2 = 5 x (7/25) = 7/5. For y = 50 and x = 5, k1 = y / x^2 = 50 / 25 = 2. When y = 98, 2 x^2 = 98 => x^2 = 49 => x = 7. Since xz = 7/5, 7 x z = 7/5 => z = 1/5. Wait, option (b) is 1/5. Let's check why the answer is 1/5.