NCERT Solutions Exercise 4.2: Determinants - Notes | Study Mathematics (Maths) Class 12 - JEE

 Page 1


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
Page 2


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
Page 3


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
By applying C3 ? C3 + C2, we have: 
Here, two columns C1 and C3 are proportional. 
? = 0. 
Question 5:  
Using the property of determinants and without expanding, prove that: 
Answer 
Applying R2 ? R2 - R3, we have: 
Page 4


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
By applying C3 ? C3 + C2, we have: 
Here, two columns C1 and C3 are proportional. 
? = 0. 
Question 5:  
Using the property of determinants and without expanding, prove that: 
Answer 
Applying R2 ? R2 - R3, we have: 
Applying R1 ?R3 and R2 ?R3, we have: 
Applying R1 ? R1 - R3, we have: 
Applying R1 ?R2 and R2 ?R3, we have: 
From (1), (2), and (3), we have: 
Page 5


Exercise 4.2 
Question 1:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 2:  
Using the property of determinants and without expanding, prove that: 
Answer 
XII 
Here, the two rows R1 and R3 are identical. 
? = 0. 
Question 3:  
Using the property of determinants and without expanding, prove that: 
Answer 
Question 4:  
Using the property of determinants and without expanding, prove that: 
Answer 
By applying C3 ? C3 + C2, we have: 
Here, two columns C1 and C3 are proportional. 
? = 0. 
Question 5:  
Using the property of determinants and without expanding, prove that: 
Answer 
Applying R2 ? R2 - R3, we have: 
Applying R1 ?R3 and R2 ?R3, we have: 
Applying R1 ? R1 - R3, we have: 
Applying R1 ?R2 and R2 ?R3, we have: 
From (1), (2), and (3), we have: 
Hence, the given result is proved. 
Question 6:  
By using properties of determinants, show that: 
Answer 
We have, 
Here, the two rows R1 and R3 are identical. 
? = 0.