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NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q1: 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.

Q2: Examine the consistency of the system of equations.
2x − y = 5 x
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.

Q3: 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.

Q4: 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.

Q5: 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.

Q6: 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.

Q7: 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

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
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants
NCERT Solutions Class 12 Maths Chapter 4 - Determinants

Q10: 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.

Q11: 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

Q12: 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

Q13: 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

Q14: 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

Q15: 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

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.
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) for JEE Main & Advanced.
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FAQs on NCERT Solutions Class 12 Maths Chapter 4 - Determinants

1. What are determinants in mathematics?
Ans. Determinants are mathematical objects that are used to solve systems of linear equations, calculate areas and volumes, and determine whether a matrix is invertible or not. They are denoted by vertical bars or double vertical bars enclosing a square matrix.
2. How do determinants help in solving systems of linear equations?
Ans. Determinants help in solving systems of linear equations by providing a criterion for the existence and uniqueness of solutions. If the determinant of the coefficient matrix is non-zero, then the system has a unique solution. If the determinant is zero, then either the system has infinitely many solutions or no solution at all.
3. Can determinants be negative?
Ans. Yes, determinants can be negative. The sign of the determinant depends on the number of row interchanges required to reduce the matrix to its upper triangular form. If the number of row interchanges is odd, the determinant is negative; if it is even, the determinant is positive.
4. How can determinants be used to calculate areas and volumes?
Ans. Determinants can be used to calculate areas and volumes by considering the coordinates of the vertices of a polygon or the vertices of a parallelepiped. By forming a matrix with the coordinates as rows or columns, taking its determinant gives the area of the polygon or the volume of the parallelepiped.
5. What is the importance of determinants in JEE exams?
Ans. Determinants are an important topic in the JEE exams as they form the basis for solving problems related to linear algebra and coordinate geometry. Questions related to determinants are frequently asked in the exams, and a good understanding of determinants is essential for scoring well in the mathematics section of JEE.
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