NCERT Solutions for Class 12 Maths Maths Part-1 Chapter 4

Determinants Class 12

Exercise 4.1 : Solutions of Questions on Page Number : 108

Q1 :

Evaluate the determinants in Exercises 1 and 2.

= 2( - 1) - 4( - 5) = - 2 + 20 = 18

Q2 :

Evaluate the determinants in Exercises 1 and 2.

(i) (ii)

(i) = (cos θ)(cos θ) - ( - sin θ)(sin θ) = cos2 θ+ sin2 θ = 1

(ii)

= (x2 - x + 1)(x + 1) - (x - 1)(x + 1)

= x3 - x2 + x + x2 - x + 1 - (x2 - 1)

= x3 + 1 - x2 + 1

= x3 - x2 + 2

Q3 :

If, then show that

Q4 :

If, then show that

Q5 :

Evaluate the determinants

(i) (iii)

(ii) (iv)

Q6 :

Find values of x, if

(i) (ii)

Q7 :

If, then x is equal to

(A) 6 (B) ±6 (C) - 6 (D) 0

Exercise 4.2 : Solutions of Questions on Page Number : 119

Q1 :

Using the property of determinants and without expanding, prove that:

Q2 :

Using the property of determinants and without expanding, prove that:

Q3 :

Using the property of determinants and without expanding, prove that:

Q4 :

Using the property of determinants and without expanding, prove that:

Q5 :

Using the property of determinants and without expanding, prove that:

Q6 :

By using properties of determinants, show that:

Q7 :

By using properties of determinants, show that:

Q8 :

By using properties of determinants, show that:

(i)

(ii)

Q9 :

By using properties of determinants, show that:

Q10 :

By using properties of determinants, show that:

(i)

(ii)

Q11 :

By using properties of determinants, show that:

(i)

(ii)

Q12 :

By using properties of determinants, show that:

Q13 :

By using properties of determinants, show that:

Q14 :

By using properties of determinants, show that:

Q15 :

Let A be a square matrix of order 3 × 3, then is equal to

A. B. C. D.

Q16 :

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

Exercise 4.3 : Solutions of Questions on Page Number : 122

Q1 :

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)

Q2 :

Show that points

are collinear

Q3 :

Find values of k if area of triangle is 4 square units and vertices are

(i) (k, 0), (4, 0), (0, 2) (ii) (-2, 0), (0, 4), (0, k)

Q4 :

(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

Q5 :

If area of triangle is 35 square units with vertices (2, -6), (5, 4), and (k, 4). Then k is

A. 12 B. -2 C. -12, -2 D. 12, -2

Exercise 4.4 : Solutions of Questions on Page Number : 126

Q1 :

Write Minors and Cofactors of the elements of following determinants:

(i) (ii)

Q2 :

(i) (ii)

Q3 :

Using Cofactors of elements of third column, evaluate

Q4 :

If and Aij is Cofactors of aij, then value of Δ is given by

Exercise 4.5 : Solutions of Questions on Page Number : 131

Q1 :

Find adjoint of each of the matrices.

Q2 :

Find adjoint of each of the matrices.

Q3 :

Q4 :

Q5 :

Find the inverse of each of the matrices (if it exists).

Q6 :

Find the inverse of each of the matrices (if it exists).

Q7 :

Find the inverse of each of the matrices (if it exists).

Q8 :

Find the inverse of each of the matrices (if it exists).

Q9 :

Find the inverse of each of the matrices (if it exists).

Q10 :

Find the inverse of each of the matrices (if it exists).

.

Q11 :

Find the inverse of each of the matrices (if it exists).

Q12 :

Let and. Verify that

Q13 :

If, show that. Hence find.

Q14 :

For the matrix, find the numbers a and b such that A2 + aA + bI = O.

Q15 :

For the matrixshow that A3 - 6A2 + 5A + 11 I = O. Hence, find A - 1.

Q16 :

If verify that A3 - 6A2 + 9A - 4I = O and hence find A - 1

Q17 :

Let A be a nonsingular square matrix of order 3 × 3. Then is equal to

A. B. C. D.

Q18 :

If A is an invertible matrix of order 2, then det (A - 1) is equal to

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

Exercise 4.6 : Solutions of Questions on Page Number : 136

Q1 :

Examine the consistency of the system of equations.

x + 2y = 2

2x + 3y = 3

Q2 :

Examine the consistency of the system of equations.

2x - y = 5

x + y = 4

Q3 :

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Q4 :

Examine the consistency of the system of equations.

x +y + z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Q5 :

Examine the consistency of the system of equations.

3x - y - 2z = 2

2y - z = -1

3x - 5y = 3

Q6 :

Examine the consistency of the system of equations.

5x - y + 4z = 5

2x + 3y + 5z = 2

5x - 2y + 6z = -1

Q7 :

Solve system of linear equations, using matrix method.

Q8 :

Solve system of linear equations, using matrix method.

Q9 :

Solve system of linear equations, using matrix method.

Q10 :

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Q11 :

Solve system of linear equations, using matrix method.

Q12 :

Solve system of linear equations, using matrix method.

x - y + z = 4

2x + y - 3z = 0

x + y + z = 2

Q13 :

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x - 2y + z = -4

3x - y - 2z = 3

Q14 :

Solve system of linear equations, using matrix method.

x - y + 2z = 7

3x + 4y - 5z = -5

2x - y + 3z = 12

Q15 :

If, find A - 1. Using A - 1 solve the system of equations

Q16 :

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.

Exercise Miscellaneous : Solutions of Questions on Page Number : 141

Q1 :

Prove that the determinant is independent of θ.

Q2 :

Without expanding the determinant, prove that

Q3 :

Evaluate

Q4 :

If a, b and c are real numbers, and,

Show that either a + b + c = 0 or a = b = c.

Q5 :

Solve the equations

Q6 :

Prove that

Q7 :

If

Q8 :

Let verify that

(i)

(ii)

Q9 :

Evaluate

Q10 :

Evaluate

Q11 :

Using properties of determinants, prove that:

Q12 :

Using properties of determinants, prove that:

Q13 :

Using properties of determinants, prove that:

Q14 :

Using properties of determinants, prove that:

Q15 :

Using properties of determinants, prove that:

Q16 :

Solve the system of the following equations

Q17 :

If a, b, c, are in A.P., then the determinant

A. 0 B. 1 C. x D. 2x

Q18 :

If x, y, z are nonzero real numbers, then the inverse of matrix is

A. B.

C. D.

Q19 :

Let, where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4]