NOTE on Examination Data Defect: As printed in the official question paper, the variances are given as $Var(X_1) = 5$ and $Var(X_2) = 9$. With these printed values, $Cov(3X_1 + 4X_2, 3X_1 - X_2) = 9 Var(X_1) - 4 Var(X_2) = 9(5) - 4(9) = 9 \ne 0$, yielding a correlation coefficient $r \approx 0.089$. Consequently, with the printed data, NO listed option is mathematically correct. However, the question setter intended $Var(X_1) = 4$ and $Var(X_2) = 9$, under which $Cov = 9(4) - 4(9) = 0$ and $r = 0$. The official NTA Final Answer Key awards Option 1 ($0$). In adherence to the historical exam record and official key sovereignty, the question text is preserved verbatim and Option 1 is retained.
Option analysis:
(A) Standard derivation assuming intended variance $Var(X_1) = 4$: $Cov(Y_1, Y_2) = 9(4) - 4(9) = 0$, giving $r = 0$.
Hence, Option 1 ($0$) is the keyed answer.
नोट: आधिकारिक प्रश्न पत्र में प्रसरण $Var(X_1) = 5$ और $Var(X_2) = 9$ छपा था। इन मानों के साथ $r \approx 0.089$ आता है, जिससे दिए गए विकल्पों में से कोई भी विकल्प गणितीय रूप से सही नहीं है। परंतु परीक्षक का आशय $Var(X_1) = 4$ था जिससे $r = 0$ आता है। NTA की अंतिम उत्तर कुंजी में विकल्प 1 ($0$) मान्य है।
