JEE Exam  >  JEE Notes  >  Mathematics (Maths) Class 12  >  NCERT Solutions Exercise 4.3: Determinants

NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q1: Write Minors and Cofactors of the elements of following determinants:
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans: (i) The given determinant is NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Minor of element aij is Mij.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q2: 
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans:
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q3: Using Cofactors of elements of second row, evaluate Δ =NCERT Solutions Class 12 Maths Chapter 4 - Determinants.
Ans: The given determinant is
NCERT Solutions Class 12 Maths Chapter 4 - Determinants .
We have:
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
We know that ∆ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
∆ = a21A21 + a22A22 + a23A23 = 2(7) + 0(7) + 1(−7) = 14 − 7 = 7.

Q4: Using Cofactors of elements of third column, evaluate Δ=  
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans: The given determinant is NCERT Solutions Class 12 Maths Chapter 4 - Determinants .
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
We know that Δ is equal to the sum of the product of the elements of the second row
with their corresponding cofactors.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Hence,

Q5: If ∆ = NCERT Solutions Class 12 Maths Chapter 4 - Determinantsand Aij is Cofactors of aij , then value of ∆ is given by
(A) a11 A31+ a12 A32 + a13 A33 
(B) a11 A11 + a12 A21 + a13 A31 
(C) a21 A11 + a22 A12 + a23 A13 
(D) a11 A11+ a21 A21 + a31 A
31
Ans: D

Given : Δ = NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Δ = Sum of products of elements of row (or column) with their corresponding cofactors.
Δ=a11 A11+ a21 A21 + a31 A31
So, option D is correct.


Old NCERT Questions


Q1: For the matrices A and B, verify that (AB)′ = B'A' where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans:
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Hence,

Q2: Examine the consistency of the system of equations.
 x + 2y = 2
 2x + 3y = 3

Ans:
The given system of equations is:
x + 2y = 2
2x + 3y = 3
The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants A is non-singular.
Therefore, A−1 exists.
Hence, the given system of equations is consistent.

Q3: Examine the consistency of the system of equations.
 2x − y = 5 x
 + y = 4

Ans: The given system of equations is:
2x − y = 5 x
+ y = 4
The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants A is non-singular.
Therefore, A−1 exists.
Hence, the given system of equations is consistent.

Q4: Examine the consistency of the system of equations.
 x + 3y = 5
 2x + 6y = 8

Ans: The given system of equations is:
x + 3y = 5
2x + 6y = 8
The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.

Q5: Examine the consistency of the system of equations.
 x + y + z = 1 

2x + 3y + 2z = 2 
ax + ay + 2az = 4
Ans: The given system of equations is:
x + y + z = 1 2x
+ 3y + 2z = 2 ax
+ ay + 2az = 4
This system of equations can be written in the form AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants A is non-singular.
Therefore, A−1 exists.
Hence, the given system of equations is consistent.

Q6: Examine the consistency of the system of equations.
 3x − y − 2z = 2
 2y − z = −1
 3x − 5y = 3

Ans: The given system of equations is:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
This system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
A is a singular matrix.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.

Q7: Examine the consistency of the system of equations.
 5x − y + 4z = 5
 2x + 3y + 5z = 2
 5x − 2y + 6z = −1

Ans: The given system of equations is:
5x − y + 4z = 5
2x + 3y + 5z = 2
5x − 2y + 6z = −1
This system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Therefore, A−1 exists.
Hence, the given system of equations is consistent.

Q8: Solve system of linear equations, using matrix method.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans: The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q9: Solve system of linear equations, using matrix method.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans:The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q10: Solve system of linear equations, using matrix method.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans: The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q11: Solve system of linear equations, using matrix method.
 5x + 2y = 3
 3x + 2y = 5

Ans: The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.

Q12: Solve system of linear equations, using matrix method.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Ans: The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q13: Solve system of linear equations, using matrix method.
 x − y + z = 4 

2x + y − 3z = 0 
x + y + z = 2
Ans: The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q14: Solve system of linear equations, using matrix method.
 2x + 3y + 3z = 5
 x − 2y + z = −4
 3x − y − 2z = 3

Ans: The given system of equations can be written in the form AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q15: Solve system of linear equations, using matrix method.
x − y + 2z = 7
3x + 4y − 5z = −5
2x − y + 3z = 12
Ans: The given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Thus, A is non-singular. Therefore, its inverse exists.
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q16: If  NCERT Solutions Class 12 Maths Chapter 4 - Determinants find A−1. Using A−1 solve the system of equations
Ans:
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Now, the given system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q17: 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.
Ans: Let the cost of onions, wheat, and rice per kg be Rs x, Rs y,and Rs z respectively.
Then, the given situation can be represented by a system of equations as:
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
This system of equations can be written in the form of AX = B, where
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Now,
X = A−1 B
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
Hence, the cost of onions is Rs 5 per kg, the cost of wheat is Rs 8 per kg, and the cost of rice is Rs 8 per kg.

The document NCERT Solutions Class 12 Maths Chapter 4 - Determinants is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on NCERT Solutions Class 12 Maths Chapter 4 - Determinants

1. What is a determinant in mathematics?
Ans. A determinant in mathematics is a scalar value that can be calculated from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible or singular.
2. How is the determinant of a 2x2 matrix calculated?
Ans. The determinant of a 2x2 matrix [a b; c d] is calculated as ad - bc.
3. What is the significance of the determinant of a matrix?
Ans. The determinant of a matrix is significant because it helps determine if the matrix is invertible. A matrix is invertible if and only if its determinant is non-zero.
4. Can determinants be negative?
Ans. Yes, determinants can be negative, positive, or zero, depending on the values of the matrix elements. The sign of the determinant does not affect its significance in terms of invertibility.
5. How are determinants used in solving systems of linear equations?
Ans. Determinants are used to solve systems of linear equations by representing the coefficients of the equations in matrix form and then using the determinant to determine whether a unique solution exists.
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