Question-1 :- 

Solution :-

Question-2 :- 

Solution :-

Question-3 :- 

Solution :-

Question-4 :- 

Solution :-

Question-5 :- 

Solution :-

Question-6 :- 

Solution :-

Question-7 :- 

Solution :-

Question-8 :- 

Solution :-

(i)        
    

(ii)        
    

Question-9 :- 

Solution :-

Question-10 :- 

Solution :-

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

Solution :-

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

Solution :-

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

Solution :-

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

Solution :-

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

Solution :-

Question-16 :-  Solve the system of equations:

Solution :-

  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

Solution :-

  
  The correct answer is A.
    

Question-18 :- 

Solution :-

  
  The correct answer is A.
    

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]
  The correct answer is D.