4. DETERMINANTS

Class 12 Mathematics Chapter 4. DETERMINANTS

Chapter 4. Determinants

Class 12 Chapter 4. Determinants Exercise 4.1

Evaluate the determinants in Exercises 1 and 2 .

Solution: We have,

2. (i) (ii)

Solution: (i) We have

3. If , then show that .

Solution: Given

Again,

Proved.

(ii) We have,

4. If , then show that .

Solution: We have,

Again,

Therefore, Proved

5. Evaluate the determinants :

(i) (ii) (iii) (iv)

Solution: (i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

6. If , find .

Solution: We have,

7. Find values of , if

(i) (ii)

Solution: (i) We have,

(ii) We have,

8. If , then is equal to

(A) 6 (B) (C) – 6 (D) 0

Solution: We have,

The correct answer (B) ± 6 .

Class 12 Chapter 4. Determinants Exercise 4.2

1. Find area of the triangle with vertices at the point given in each of the following :
(i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8)
(iii) (–2, –3), (3, 2), (–1, –8)

Solution: We know that the area of the triangle whose vertices are , and is given by

(i) (1, 0), (6, 0), (4, 3)

Here, , , , , ,

We have,

Square units.

(ii) (2, 7), (1, 1), (10, 8)
Here, , , , , ,

We have,

(iii) (–2, –3), (3, 2), (–1, –8)

Here, , , , , ,

We have,

Square units.

2. Show that points A (a, b + c), B (b, c + a), C (c, a + b) are collinear.

Solution: Here, , , , , ,

We have,

Therefore, the points A (), B (), C () are collinear.

3. Find values of k if area of triangle is 4 sq. units and vertices are :
(i) (k, 0), (4, 0), (0, 2) (ii) (–2, 0), (0, 4), (0, k)

Solution: (i) (k, 0), (4, 0), (0, 2)

Here, , , , , ,

We know that ,

Therefore, the value of K are 0 and 8 .

(ii) (–2, 0), (0, 4), (0, k)

Here, , , , , ,

We know that,

Therefore, The value of K are 0 and 8 .

4. (i) Find equation of line joining (1, 2) and (3, 6) using determinants.
(ii) Find equation of line joining (3, 1) and (9, 3) using determinants.

Solution: (i) Let be any point on the line joining (1,2) and (3,6) .

Here, , , , , ,

We know that,

Therefore, the equation is .

(ii) Let be any point on the line joining (3,1) and (9,3) .

Here, , , , , ,

We know that,

Therefore, the equation is .

5. If area of triangle is 35 sq units with vertices (2, – 6), (5, 4) and (k, 4). Then k is
(A) 12 (B) –2 (C) –12, –2 (D) 12, –2

Solution: Here, , , , , ,

We know that,

Therefore, the value of k are – 2 and 12 .

Class 12 Chapter 4. Determinants Exercise 4.3

Write Minors and Cofactors of the elements of following determinants :

1. (i) (ii)

Solution: (i)

Minor of an element is .

Now, Cofactor of is .

So,

(ii)

Minor of an element is .

Now, Cofactor of is .

So,

2. (i) (ii)

Solution: (i) We have,

Minor of an element is .

Now, Cofactor of is .

So,

(ii) We have,

Minor of an element is .

Now, Cofactor of is .

So,

3. Using Cofactors of elements of second row , evaluate .

Solution: We have,

Here, , ,

The cofactor of

4. Using Cofactors of elements of second row , evaluate .

Solution: Here, , ,

The cofactor of

5. If and is Cofactors of , then value of is given by

(A)

(B)

(C)

(D)

Class 12 Chapter 4. Determinants Exercise 4.4

Find adjoint of each of the matrices in Exercises 1 and 2 .

1. 2.

Verify in Exercise 3 and 4

3. 4.

Find the inverse of each of the matrices (if it exists) given in Exercises 5 to 11.

5. 6. 7. 8. 9. 10.

11.

12. Let and B . Verify that .

13. If A , Show that . Hence find .

14. For the matrix A , find the numbers and such that .

15. For the matrix A . Show that . Hence , find .

16. If A . Verify that and hence find .

17. Let A be a non-singular square matrix of order 3×3 . Then is equal to

(A) (B) (C) (D)

18. If A is an invertible matrix of order 2 , then is equal to :

(A) (B) (C) 1 (D) 0

Class 12 Chapter 4. Determinants Exercise 4.5

Examine the consistency of the system of equations in Exercises 1 to 6.
1. x + 2y = 2 ; 2x + 3y = 3

2. 2x – y = 5 ; x + y = 4

3. x + 3y = 5 ; 2x + 6y = 8

4. x + y + z = 1 , 2x + 3y + 2z = 2 , ax + ay + 2az = 4

5. 3x–y – 2z = 2 , 2y – z = –1 , 3x – 5y = 3

6. 5x – y + 4z = 5 , 2x + 3y + 5z = 2 , 5x – 2y + 6z = –1
Solve system of linear equations, using matrix method, in Exercises 7 to 14.
7. 5x + 2y = 4 ; 7x + 3y = 5

8. 2x – y = –2 ; 3x + 4y = 3

9. 4x – 3y = 3 ; 3x – 5y = 7
10. 5x + 2y = 3 , 3x + 2y = 5

11. 2x + y + z = 1 , x – 2y – z = , 3y – 5z = 9

12. x – y + z = 4 ; 2x + y – 3z = 0 , x + y + z = 2
13. 2x + 3y +3 z = 5 , x – 2y + z = – 4 , 3x – y – 2z = 3

14. x – y + 2z = 7 , 3x + 4y – 5z = – 5 , 2x – y + 3z = 12

15. If A , find . Using solve the system of equations
                     2x – 3y + 5z = 11
                     3x + 2y – 4z = – 5
                       x + y – 2z = – 3
16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70. Find cost of each item per kg by matrix method.

Class 12 Chapter 4. Determinants Miscellaneous Exercises

1. Prove that the determinant is independent of .

2. Without expanding the determinant , prove that .

3. Evaluate:

4. If and are real numbers and , Show that either or .

5. Solve the equation .

6. Prove that:

7. If and , find .

8. Let A . Verify that (i) (ii)

9. Evaluate :

10. Evaluate:

Using properties of determinant in Exercises 11 to 15 , Prove that :

11.

12.

13.

14.

15.

16. Solve the system of the following equations

Choose the correct answer in Exercise 17 to 19 .

17. If are in A.P. then the determinant is

(A) 0 (B) 1 (C) (D)

18. If are non-zero real numbers, then the inverse of matrix is

(A) (B)

19. Let A , where . Then

(A) (B) (C) (D)