# NCERT Solutions for class 12th Mathematics Chapter 4 Determinants

#### Exercise 4.1

Question 1. Evaluate the following determinant :

Question 2.

Question 3.

Question 4.

Question 5. Evaluate the determinants :

Question 6.

Question 7. Find the value of x, if

Question 8.

#### Exercise 4.2

Using the property of determinants and without expanding in Questions 1 to 5, prove that

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.

By using properties of determinants, in Questions 6 to 14, show that

Question 6.

Question 7.

Question 8.

Question 9.

Question 10.

Note: In case of determinants, we can use both elementary row operations and elementary column operations to arrive at the result, as you have seen in the above question.

Question 11.

Question 12.

Question 13.

Question 14.

Question 15. If A be a square matrix of order 3 Ã— 3, then | kA | is equal to

Question 16. Which of the following is correct :

(a) Determinant is a square matrix

(b) Determinant is a number associated to a matrix

(c) Determinant is a number associated to a square matrix

(d) None of these

Sol. (c) Since we know that, determinant of any matrix is defined if and only if Ais a square matrix.\ determinant is a number associated to a square matrix.

#### Exercise 4.3

Question 1. Find the 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) (â€“2, â€“ 3), (3, 2), (â€“1, â€“ 8)

Note: Since area is positive, we always use the absolute value of the determinant (Which is used for the area).

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

Hint: Use area of triangle

Note: Area of a triangle formed by three collinear points is zero.

Question 3. Find the value of k if area of triangle is 4 square units and vertices are (i) (k, 0), (4, 0), (0, 2) (ii) (â€“2, 0), (0, 4), (0, k)

Note: If area of a triangle is given, we use both positive and negative values of the determinant for calculations.

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

(ii) Find the equation of line joining (3, 1), (9, 3) using determinants.

Question 5. If area of triangle is 35 sq. units with vertices (2, â€“ 6), (5, 4) and (k, 4). Then k is (a) 12 (b) â€“ 2 (c) â€“ 12, â€“ 2 (d) 12, â€“ 2

#### Exercise 4.4

Question 1. Write the minors and cofactors of the elements of following determinants :

Question 2. Write Minors and Cofactor of elements of the determinant

Question 3. Using cofactors of elements of second row, evaluate Del =

Question 4. Using Cofactors of elements of third column, evaluate

Question 5.

#### Exercise 4.5

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

Question. Verify A (adj A) = (adj A) Ã— A = |A| I in Questions 3 and 4

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

Question 12.

Question 13.

Question 14.

Question 15.

Question 16.

Question 17. Let A be a non-singular square matrix of order 3 Ã— 3. Then | Adj A | is equal to :

Question 18. If A is an invertible matrix of order 2, then det. (Aâ€“1) is equal to :

#### Exercise 4.6

Question. Examine the consistency of the system of equations in Questions 1 to 6 :

Question : Solve the following of linear equations using matrix method in Exercises 7 to 14 :

Question 15.

Question 16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is rs. 69. The cost of 2 kg onion, 4 kg wheat and 6 kg rice is rs. 90. The cost of 6 kg onion, 2 kg wheatand 3 kg rice is rs.70. Find the cost of each item per kg by matrix method.

#### Miscellaneous Exercise

Question 1. Prove that the determinant

Question 2. Without expanding the determinant, prove that

Question 3. Evaluate

Question 4.

Question 5. Solve the equation

Question 6.

Question 7.

Question 8.

Question 9.

Question 10. Evaluate

Using properties of determinants in exercises 11 to 15, prove that :

Question 11.

Question 12.

Question 13.

Question 14.

Question 15.

Question 16. Solve the system of the following equations

Question . Choose the correct answer in Questions 17 to 19

Question 18. If x, y, z are nonzero real numbers, then the inverse of matrix

Question 19.

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