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Let Tn = det([n-1, n-2, n-6; 2n-4, 2n-6, 2n-11; 3n-9, 3n-12, 3n-18]) where n is a natural number. Which of the following is/are true? (A) Product of T1 to Tn = 6ⁿ (B) Product of T1 to T10 = 60 (C) T(n+1) / Tn = 2 (D) T100 * T101 = 48
- (A) ∏(n=1 to n) Tn = 6ⁿ
- (B) ∏(n=1 to 10) Tn = 60
- (C) T(n+1)/Tn = 2
- (D) T100 · T101 = 48
Correct answer: (A) ∏(n=1 to n) Tn = 6ⁿ
Solution
After column operations C2->C2-C1 and C3->C3-C1, then row operations R2->R2-2*R1 and R3->R3-3*R1, the determinant reduces to expanding along column 2 with only one non-zero entry, yielding Tn = 6 for all n. Therefore the product of n terms = 6ⁿ (option A). Options B (product of 10 terms = 6¹⁰, not 60), C (ratio = 1, not 2), and D (product = 36, not 48) are all false.
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