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Imagine you are asked to solve different types of problems, like finding the area of a triangle using a determinant, checking if a system of equations has a solution, or finding the inverse of a matrix. How do you know which method to use? The Miscellaneous Exercise of NCERT Class 12 Maths Chapter 4 – Determinants brings together all the important topics from the chapter. It includes questions on: finding the value of determinants, using properties of determinants to simplify problems, calculating the area of a triangle using coordinates, solving systems of linear equations, finding the adjoint and inverse of a matrix. Miscellaneous Exercise of NCERT Class 12 Maths Chapter 4 helps you revise everything you have learned and tests your overall understanding. It is great for practice before exams. This article on the NCERT Solutions for miscellaneous exercise Class 12 Maths Chapter 4 - Determinant, offers clear and step-by-step solutions for the exercise problems to help the students understand the method and logic behind it. For syllabus, notes, and PDF, refer to this link: NCERT.
Question:1 Prove that the determinant
Answer:
Calculating the determinant value of
Clearly, the determinant is independent of
Question:2 Without expanding the determinant, prove that
Answer:
We have the
Multiplying rows with a, b, and c respectively.
we get;
= R.H.S.
Hence proved. L.H.S. =R.H.S.
Question 4: If
Answer:
We know from the identity that:
Then we can find easily,
Given
Given
Then we have to basically find the
So, Given matrix
Hence its inverse
Now, as we know that
So, calculating cofactors of B,
Now, We have both
Putting in the relation we know;
Question 5(i): Let
Answer:
Given that
So, let us assume that
Hence its inverse exists;
so, we now calculate the value of
Cofactors of A;
Finding the inverse of C;
Hence its inverse exists;
Now, finding the
or
Now, finding the R.H.S.
Cofactors of B;
Hence L.H.S. = R.H.S. proved.
Question 5(ii):Let
Answer:
Given that
So, let us assume that
Hence its inverse exists;
so, we now calculate the value of
Cofactors of A;
Finding the inverse of B ;
Hence its inverse exists;
Now, finding the
Hence proved L.H.S. =R.H.S..
Question:6 Evaluate
Answer:
We have determinant
Applying row transformations;
Taking out the common factor 2(x+y) from the row first.
Now, applying the column transformation;
Expanding the remaining determinant;
Question:7 Evaluate
Answer:
We have determinant
Applying row transformations;
Taking out the common factor -y from the row first.
Expanding the remaining determinant;
Question:8 Solve the system of equations
Answer:
We have a system of equations;
So, we will convert the given system of equations in a simple form to solve the problem by the matrix method;
Let us take,
Then we have the equations;
We can write it in the matrix form as
Now, Finding the determinant value of A;
Hence we can say that A is non-singular
Finding cofactors of A;
Now we will find the solutions by relation
Therefore we have the solutions
Or in terms of x, y, and z;
Question 9: Choose the correct answer.
If
is
(A)
(B)
(C)
(D)
Answer:
Given Matrix
As we know,
So, we will find the
Determining its cofactor first,
Hence
Therefore, the correct answer is (A)
Question:10 Choose the correct answer.
Let
(A)
(C)
Answer:
Given determinant
Now, given the range of
Therefore the
Hence, the correct answer is D.
Also read,
Here are the main topics covered in NCERT Class 12 Chapter 4, Determinants: Miscellaneous Exercise.
1. Evaluation of Determinants: Practice of expanding 3×3 determinants along any row or column using minors and cofactors.
2. Properties of Determinants: Apply key properties like row/column interchange, scalar multiplication, and linear combinations to simplify determinants.
3. Area of a Triangle Using Determinants: Use coordinate geometry and the determinant formula to find the area of a triangle formed by three points.
4. Consistency and Inconsistency of Systems: Determine whether systems of linear equations have unique solutions, no solutions, or infinitely many solutions based on determinant values.
5. Adjoint and Inverse of a Matrix: Calculate the adjoint and use it to find the inverse of a square matrix, then verify it through matrix multiplication.
6. Solving Linear Equations Using Matrix Inverse: Apply the formula
Also, read,
Given below are some useful links for subject-wise NCERT solutions of class 12.
Here are some links to subject-wise solutions for the NCERT exemplar class 12.
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
The value of the determinant is zero if any two rows of a determinant are identical.
The value of the determinant is zero If any two rows of a determinant are proportional.
If first row of matrix A is multiplied by 3 then the new value of |A| = 3x3 = 9.
No, singular matrices are non-invertible.
If the transpose of matrix A and matrix A are equal then such matrix is called symmetric matrix.
The matrix has all the non-diagonal elements zero is called a diagonal matrix.
Yes, all the identity matrices are diagonal matrices.
No, CBSE doesn't provide NCERT solutions for miscellaneous exercises.
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Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.
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