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TOPICS
Miscellaneous

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Question-11 :-  Using properties of determinants, prove that:

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Question-12 :-  Using properties of determinants, prove that:

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Question-13 :-  Using properties of determinants, prove that:

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Question-14 :-  Using properties of determinants, prove that:

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Question-15 :-  Using properties of determinants, prove that:

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Question-16 :-  Solve the system of equations:

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```  Let 1/x = a, 1/y = b, 1/z = c

2a + 3b + 10c = 4
4a – 6b + 5c = 1
6a + 9b – 20c = 2

a = 1/2; x = 2
b = 1/3; y = 3
c = 1/5; z = 5
```

Question-17 :-  If a, b, c, are in A.P, then the determinant (A) 0    (B) 1    (C) x    (D) 2x

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Question-19 :-  (A) Det(A) = 0   (B) Det(A) ∈ (2, ∞)   (C) Det(A) ∈ (2, 4)   (D) Det(A) ∈ [2, 4]

Solution :-
```  |A| = 1(1 + sin²θ) - sin θ(-sin θ + sin θ) + 1(sin²θ + 1)
= 1 + sin²θ + sin²θ + 1
= 2 + 2sin²θ
= 2(1 + sin²θ)
Now, 0 ≤ θ ≤ 2π
0 ≤ sin θ ≤ 1
0 ≤ sin²θ ≤ 1
1 ≤ 1+sin²θ ≤ 2
2 ≤ 2(1+sin²θ) ≤ 4
Det(A) ∈ [2, 4]
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