NCERT MCQ Solutions for Class 12 Mathematics Chapter 4 Determinants to Enhance your understanding of Determinants. This chapter covers essential MCQs like properties of determinants, applications and problem-solving techniques. Practice with Multiple Choice Questions to strengthen your grasp on key concepts and improve your problem-solving speed. Ace your exams with these well-structured MCQs designed for concept clarity and success.
Class 12 Maths Chapter 4 NCERT Solutions
Class 12 Maths Chapter-wise MCQ
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Core Idea of Determinants in Class 12 Mathematics
Determinants form a crucial part of higher-level mathematics, especially when dealing with systems of linear equations and matrices. In NCERT Chapter 4, you are introduced to how determinants are calculated for square matrices and why these calculations matter in broader mathematical and practical contexts. One way to think about a determinant is that it provides a single numeric value that summarizes specific properties of a square matrix. For instance, a determinant reveals important information about whether a system of linear equations has a unique solution, infinite solutions, or no solution at all. Although there is a defined procedure to calculate determinants, the core idea rests on understanding how each element in a matrix contributes to this special numeric value. Learning determinants is not just an exercise in computation but also a gateway to comprehending advanced concepts like inverse matrices and eigenvalues. By grasping these foundational ideas, students can more easily transition to other topics where determinants play a vital role, such as vector calculus and transformations in coordinate geometry.
Q1. Given that A is a square matrix of order 3 and |A| = – 4, then |adj. A| is equal to:
Q2. Given that A = [aᵢⱼ] is a square matrix of order 3 x 3 and |A| = -7, then the value of ∑³₁ ₌ ᵢ aᵢ₂ A ᵢ₂, where Aᵢⱼdemotes the cofactor of element a ᵢⱼis:
Expanding Determinants and Key Terminologies
In class 12th mathematics chapter 4, you will learn terms like minor, cofactor and adjoint, which help in understanding how determinants can be expanded. A minor involves temporarily ignoring the row and column of a specific element in a matrix to focus on a smaller matrix. A cofactor adjusts the sign of a minor based on the position of the element in the original matrix. By methodically multiplying these cofactors with their respective elements and summing all results, you can determine the value of a matrix’s determinant. The idea is systematic, and practicing these expansions develops a strong logical and structured approach to problem-solving. While you study these concepts, you start seeing patterns such as certain rows or columns making the calculation simpler if they contain zeros. Hence, class 12 mathematics chapter 4 encourages strategic thinking – selecting the best row or column for expansion – and fosters an appreciation for how neatly mathematics can handle seemingly complex operations.
Q3. Which of the following is correct?
Q4. If A is any square matrix of order 3 x 3 such that |A| = 3, then the value of |adj. A| is
Determinants Properties that Simplify Calculations
A significant part of the chapter 4 in 12th mathematics focuses on various properties of determinants that help simplify the computation process. You learn about how swapping two rows or columns affects the sign of the determinant and how multiplying a row or column by a constant scales the determinant accordingly. Another important property explains how adding or subtracting a multiple of one row (or column) to another row (or column) does not alter the determinant’s overall value. By applying these properties cleverly, students can simplify matrices before calculating determinants, saving time and reducing the chance of arithmetic mistakes. You discover the relationship between determinants and certain matrix operations, such as transposition and the creation of block matrices. Through these topics, Chapter 4 demonstrates that determinants are not just numbers but powerful tools that reveal a matrix’s intrinsic characteristics, making them indispensable in many areas of applied mathematics.
Q5. If A is a square matrix such that |A| = 5, the value of |AAᵀ| is
Q6. If A is any square matrix of order 3 x 3 such that |adj. A| = 25 and |A| is non-positive, then the value of |A| is
Linking Determinants to System of Equations
One of the central applications of determinants in 12th Maths NCERT Chapter 4 is solving a system of linear equations. By associating a system of equations with its coefficient matrix, the determinant acts as a test for the uniqueness of the solution. If the determinant is not zero, the system has a unique solution. If the determinant is zero, the system either has infinitely many solutions or none at all, depending on additional conditions. While you are spared from memorizing or using complicated formulas in this discussion, the conceptual understanding is clear: a non-zero determinant implies that the matrix is invertible, allowing you to find precise values for each variable. This aspect of determinants extends beyond textbook problems, as real-world scenarios—from engineering to economics — often involve analyzing conditions under which certain processes or models have unique outcomes. Thus, determinants become a tool for both theoretical insight and practical decision-making.
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Q7. If A is any square matrix of order 2 x 2 such that |A| = -7, then the value of |adj. A| is
Q8. If A is any square matrix of order 3 x 3 such that |A| = 4, then the value of |A⁻¹| is
Strategies for Mastering Determinants
To excel in NCERT Class 12 Mathematics Chapter 4, it is vital to practice a variety of problems that range from straightforward expansions to more involved applications. Begin by reinforcing your knowledge of how to identify minors and cofactors, then gradually incorporate the properties to make calculations efficient. Focus on understanding each step rather than just completing the mechanical process of expansion, as it will deepen your insight into why determinants behave as they do. Look for patterns in the problems — notice which rows or columns are simpler to work with and recognize when certain operations will simplify a matrix quickly. In addition, pay attention to conceptual relationships, like how determinant properties carry over to advanced concepts in linear algebra. By systematically developing both speed and understanding, you will be well-prepared for exam questions that test your computational skills and your ability to interpret the meaning behind the calculations.
What is the best way to begin tackling multiple-choice questions on determinants?
A helpful first step is to revisit key definitions and fundamental properties before jumping into the multiple-choice format. Familiarize yourself with terms like minor, cofactor and the impact of row and column operations on the determinant. Once you are comfortable with these concepts, practice a few basic examples and observe how each step logically follows the last. When you move on to actual MCQs, make a habit of carefully reading each question and identifying any keywords that might hint at specific properties. If a question seems complicated, try simplifying the matrix or identifying patterns—such as a row with several zero elements—before choosing your answer. This systematic approach ensures that you don’t rush into wrong choices and that you develop a consistent technique for addressing determinant-based MCQs.
How do I avoid careless mistakes in determinant-based MCQs?
Most careless mistakes stem from rushing through steps or overlooking sign changes, particularly when working with cofactors or swapping rows. To counter this, it is good practice to outline each step, even if only briefly, rather than doing everything mentally. For instance, list the row or column you plan to expand along with the signs that each cofactor will take. Another pitfall is misreading the question, so always double-check what is being asked—whether it’s the value of the determinant, the effect of a specific operation or the rank implications of having a zero determinant. If time permits, verify results by using more than one method or by plugging in simpler numbers to see if they lead to the same outcome. This attention to detail helps you spot inconsistencies and choose the correct option more reliably.
How can I manage time effectively while solving MCQs on Chapter 4 in an exam?
Time management in determinant-based MCQs often comes down to strategic question selection. Quickly skim the set of problems and identify those you can solve in minimal steps — perhaps they involve matrices with zeros in convenient places or clear patterns in the rows and columns. Tackle those first to secure quick marks and build confidence. For lengthier questions, leverage determinant properties that allow you to add or subtract multiples of rows and columns to simplify calculations. This reduces the amount of expansion needed. Keep track of the clock and if a question becomes too time-consuming, consider marking it and returning later if time remains. By systematically optimizing which questions to address first and how to approach each calculation, you can maintain a steady pace and avoid last-minute anxiety.
How do I handle tricky conceptual questions that don’t seem purely computational?
Many MCQs in NCERT Maths Chapter 4 require deeper conceptual understanding rather than straightforward number crunching. These might test your knowledge about when a determinant is zero or non-zero and how that relates to the existence of solutions in a system of equations. In such cases, focus on the underlying principle rather than jumping straight into expansion. If the question references a property—like the effect of interchanging two rows—revisit that property in your mind and see how it influences the options given. Think about how each choice aligns or conflicts with what you know about determinants. Reflecting on these conceptual links helps you quickly eliminate incorrect options and choose the most logical response, even if a detailed expansion isn’t required.
What role do practice and mock tests play in mastering MCQs on determinants?
Regular practice is vital because it helps you identify recurring patterns and commonly tested properties, such as how the determinant changes when you factor out elements from rows or columns. Begin with standard questions to develop confidence, then gradually work up to mixed-difficulty sets that challenge both your computational and conceptual skills. Mock tests simulate the pressure of an actual exam environment, requiring you to balance accuracy and speed. Reviewing answers after a mock test is just as important as taking it. Check the solutions to each problem — whether you got it right or wrong—to understand the logic behind the correct choice. With consistent practice under timed conditions and thorough post-test reviews, you can significantly refine your skills in choosing the correct MCQ responses and reduce the likelihood of repeated errors.
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