Explanation
Solution
1. Total Outcomes:
First 10 primes ka set S={2,3,5,7,11,13,17,19,23,29} hai.
2×2 matrix [acamp;bamp;d] mein 4 positions hain aur har position ke liye 10 options hain.
2. Condition for Singular Matrix:
3. Counting Favorable Cases (ad=bc):
Kyuki saare numbers primes hain, toh ad=bc tabhi hoga jab dono sides ke prime factors same honge.
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Case 1: Saare entries same honge (a=b=c=d)
Total options = 10 (har prime ke liye ek matrix).
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Case 2: Do alag primes use honge (p1 aur p2)
Hame ad=bc chahiye. Agar hum do primes p1 aur p2 lete hain, toh combinations ye ho sakte hain:
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a=p1,d=p1 and b=p1,c=p1 (Ye Case 1 mein count ho gaya).
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a=p1,d=p2 and b=p1,c=p2
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a=p1,d=p2 and b=p2,c=p1
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a=p2,d=p1 and b=p1,c=p2
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a=p2,d=p1 and b=p2,c=p1
Har pair of primes ((210)=45 pairs) ke liye aise 4 unique matrices milenge.
4. Final Probability:
Total Favorable=10+180=190
P(Singular)=104190=10319
Correct Option: (C)
