# NCERT Solutions for Miscellaneous Exercise Chapter 4 Class 12 - Determinants As the name suggests miscellaneous exercise consists of mixed questions from all other exercises of the chapter. In NCERT solutions for class 12 maths chapter 4 miscellaneous exercise, you will get questions like solving determinants using cofactor expansion, solving determinants using properties, solving system of linear equation, checking the consistency of the system of linear equations, etc. In this class 12 maths chapter 4 miscellaneous solutions, you will get some difficult questions as compared to previous exercises. So, If you are not able to solve these questions at first by yourself, you don't need to be a worry. Over 95% of the questions in the board exams are not asked from class 12 maths chapter 4 miscellaneous exercise. You can Check class 12 maths chapter 4 miscellaneous exercise solutions in this article. Check here for NCERT solutions.

Also, see

• NCERT solutions for class 12 maths chapter 4 Determinants Exercise 4.1
• NCERT solutions for class 12 maths chapter 4 Determinants Exercise 4.2
• NCERT solutions for class 12 maths chapter 4 Determinants Exercise 4.3
• NCERT solutions for class 12 maths chapter 4 Determinants Exercise 4.4
• NCERT solutions for class 12 maths chapter 4 Determinants Exercise 4.5
• NCERT solutions for class 12 maths chapter 4 Determinants Exercise 4.6

## Miscellaneous Exercise

Question:1 Prove that the determinant is independent of .

Calculating the determinant value of ;

Clearly, the determinant is independent of .

Question:2 Without expanding the determinant, prove that

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:3 Evaluate .

Given determinant ;

.

Question:4 If and are real numbers, and

Show that either or

We have given

Applying the row transformations; we have;

Taking out common factor 2(a+b+c) from the first row;

Now, applying the column transformations;

we have;

and given that the determinant is equal to zero. i.e., ;

So, either or .

we can write as;

are non-negative.

Hence .

we get then

Therefore, if given = 0 then either or .

Question:5 Solve the equation

Given determinant

Applying the row transformation; we have;

Taking common factor (3x+a) out from first row.

Now applying the column transformations; and .

we get;

as ,

or or

Question:6 Prove that .

Given matrix

Taking common factors a,b and c from the column respectively.

we have;

Applying , we have;

Then applying , we get;

Applying , we have;

Now, applying column transformation; , we have

So we can now expand the remaining determinant along we have;

Hence proved.

Question:7 If and , find .

We know from the identity that;

.

Then we can find easily,

Given and

Then we have to basically find the matrix.

So, Given matrix

Hence its inverse exists;

Now, as we know that

So, calculating cofactors of B,

Now, We have both as well as ;

Putting in the relation we know;

Question:8(i) Let . Verify that,

Given that ;

So, let us assume that matrix and then;

Hence its inverse exists;

or ;

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:8(ii) Let , Verify that

Given that ;

So, let us assume that

Hence its inverse exists;

or ;

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:9 Evaluate

We have determinant

Applying row transformations; , we have then;

Taking out the common factor 2(x+y) from the row first.

Now, applying the column transformation; and we have ;

Expanding the remaining determinant;

.

Question:10 Evaluate

We have determinant

Applying row transformations; and then we have then;

Taking out the common factor -y from the row first.

Expanding the remaining determinant;

Question:11 Using properties of determinants, prove that

Given determinant

Applying Row transformations; and , then we have;

Expanding the remaining determinant;

hence the given result is proved.

Question:12 Using properties of determinants, prove that

where p is any scalar.

Given the determinant

Applying the row transformations; and then we have;

Applying row transformation we have then;

Now we can expand the remaining determinant to get the result;

hence the given result is proved.

Question:13 Using properties of determinants, prove that

Given determinant

Applying the column transformation, we have then;

Taking common factor (a+b+c) out from the column first;

Applying and , we have then;

Now we can expand the remaining determinant along we have;

Hence proved.

Question:14 Using properties of determinants, prove that

Given determinant

Applying the row transformation; and we have then;

Now, applying another row transformation we have;

We can expand the remaining determinant along , we have;

Hence the result is proved.

Question:15 Using properties of determinants, prove that

Given determinant

Multiplying the first column by and the second column by , and expanding the third column, we get

Applying column transformation, we have then;

Here we can see that two columns are identical.

The determinant value is equal to zero.

Hence proved.

Question:16 Solve the system of equations

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 , where

Now, Finding the determinant value of A;

Hence we can say that A is non-singular its invers exists;

Finding cofactors of A;

, ,

, ,

, ,

as we know

Now we will find the solutions by relation .

Therefore we have the solutions

Or in terms of x, y, and z;

If are in A.P, then the determinantis

(A) (B) (C) (D)

Given determinant and given that a, b, c are in A.P.

That means , 2b =a+c

Applying the row transformations, and then we have;

Now, applying another row transformation, , we have

Clearly we have the determinant value equal to zero;

Hence the option (A) is correct.

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

Given Matrix ,

As we know,

So, we will find the ,

Determining its cofactor first,

Hence

Therefore the correct answer is (A)

Let where . Then

(A) nbsp; (B)

(C) (D)

Given determinant

Now, given the range of from

Therefore the .

Hence the correct answer is D.

More about NCERT Solutions for Class 12 Maths Chapter 4 Miscellaneous Exercise:-

The first 10 questions in the class 12 maths chapter 4 miscellaneous solutions are related to solving the determinants and the next five questions are related to solving determinants using properties of determinants. There are three multiple-choice types of questions in this exercise. Before this exercise, there are five solved examples given in the textbook which you can solve to get conceptual clarity. Miscellaneous exercises questions are considered to be very important for board exams but very important for competitive exams. If you are preparing for engineering competitive exams, you must try to solve questions from this exercise.

Benefits of NCERT Solutions for Class 12 Maths Chapter 4 Miscellaneous Exercise:-

• Class 12 maths chapter 4 miscellaneous solutions are designed in a very detailed manner which could be understood by an average student also.
• As miscellaneous exercise questions are difficult as compared to the previous exercise, you may not be able to these questions.
• You can take NCERT solutions for class 12 maths chapter 4 miscellaneous exercise for reference while solving miscellaneous questions.
• Miscellaneous exercise chapter 4 class 12 will check your understanding of this chapter.

Also see-

• NCERT solutions for class 12 maths chapter 4

• NCERT exemplar solutions class 12 maths chapter 4

## NCERT solutions of class 12 subject wise

• NCERT solutions for class 12 Maths

• NCERT solutions for class 12 Physics

• NCERT solutions for class 12 Chemistry

• NCERT solutions for class 12 Biology

Subject wise NCERT Exampler solutions

• NCERT Exemplar Solutions for Class 12th Maths

• NCERT Exemplar Solutions for Class 12th Physics

• NCERT Exemplar Solutions for Class 12th Chemistry

• NCERT Exemplar Solutions for Class 12th Biology

Happy learning!!!