Class 12 NCERT Solutions – Mathematics Part I – Chapter 4 Determinants – Exercise 4.1

Evaluate the determinants from the following Questions.

Question 1. 

Solution:

The determinant of a 2 x 2 matrix 

Hence,

Question 2. (i) 

Solution:

 from trigonometric identities

(ii) 

Solution:

Question 3. If  show that 

Solution:

LHS=>

Matrix, 

Hence, determinant, 

RHS=>

Determinant, 

Now, 

Hence, proved, LHS = RHS

Question 4. If  then show that |

Solution:

LHS=>

Matrix, 

Hence, determinant, 

RHS =>

Determinant, 

Now, 

Hence, proved, LHS = RHS

Question 5. Evaluate the determinants

(i) 

Solution:

Since the maximum number of zeroes are in the second row, we will expand the determinant along row 2.

(ii) 

Solution:

(iii) 

Solution:

Note: This matrix is skew symmetric i.e. 

For every skew symmetric matrix of “odd dimension”, the determinant vanishes i.e. determinant is zero.

(iv) 

Solution:

Since the maximum number of zeroes are in the second row, we will expand the determinant along row 2.

Question 6. If  find |A|

Solution:

Question 7. Find the values of x if

(i) 

Solution:

Solving determinants on both sides,

(ii) 

Solution:

Solving determinants on both sides

Question 8. If then x is equal to

(A) 6        (B)  ±6        (C) -6        (D) 0

Solution:

Solving determinants on both sides

Hence, Option (B)