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NCERT Solutions for miscellaneous exercise chapter 4 class 12 Determinants are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the latest syllabus and pattern of CBSE 2023-24. 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. Miscellaneous exercise class 12 chapter 4 are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise enumerated in NCERT Book together using the link provided below.
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Question:1 Prove that the determinant is independent of .
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 and are real numbers, and
Show that either or
Answer:
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
Answer:
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 .
Answer:
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 .
Answer:
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,
Answer:
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
Answer:
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
Answer:
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
Answer:
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
Answer:
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.
Answer:
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
Answer:
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
Answer:
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
Answer:
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
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 , 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;
Question:17 Choose the correct answer.
If are in A.P, then the determinant
is
(A) (B) (C) (D)
Answer:
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.
Question:18 Choose the correct answer.
If x, y, z are nonzero real numbers, then the inverse of matrix is
Answer:
Given Matrix ,
As we know,
So, we will find the ,
Determining its cofactor first,
Hence
Therefore the correct answer is (A)
Question:19 Choose the correct answer.
Let where . Then
(A) nbsp; (B)
(C) (D)
Answer:
Given determinant
Now, given the range of from
Therefore the .
Hence the correct answer is D.
The first 10 questions in the NCERT book Class 12 Maths chapter 4 miscellaneous exercise 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 NCERT textbook which you can solve to get conceptual clarity. Miscellaneous exercises questions are considered to be very important for board exams and for competitive exams. If you are preparing for engineering competitive exams, you must try to solve questions from this exercise. For good score in the CBSE board exam following NCERT syllabus will be helpful.
Also Read| Determinants Class 12 Chapter 4 Notes
Happy learning!!!
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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Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.
Possible steps:
Re-evaluate Your Study Strategies:
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Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.
I hope this information helps you.
Hi,
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hello mahima,
If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
Hello Akash,
If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.
You can get the Previous Year Questions (PYQs) on the official website of the respective board.
I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.
Thank you and wishing you all the best for your bright future.
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