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JEE Main Previous Year Questions (2025): Determinants
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Page 1 JEE Main Previous Year Questions (2025): Determinants Q1: Let the system of equations ?? + ???? - ?? = ?? ???? + ???? - ???? = ?? ?????? + ?? + ?? ?? = ?? ?? , ?? ? R, have infinitely many solutions. Then the number of the solutions of this system, if ?? , ?? , ?? are integers and satisfy ?? = ?? + ?? + ?? = ???? , is : JEE Main 2025 (Online) 7th April Evening Shift Options: A. 4 B. 5 C. 3 D. 6 Ans: C Solution: For infinitely many solution ? = 0 | 1 5 -1 4 3 -3 24 1 ?? | = 0 ? 1(3?? + 3) - 5(4?? + 72) - 1(4 - 72) = 0 ? -17?? + 3 - 4 × 72 - 4 = 0 ? 17?? = -289 ? ?? = -17 ?1 = 0 ? | 1 5 -1 7 3 -3 ?? 1 -17 | = 0 ? 1(-51 + 3) - 5(-119 + 3?? ) - 1(7 - 3?? ) = 0 ? -48 + 595 - 15?? - 7 + 3?? = 0 ? 12?? = 540 ?? = 45 ?? + 5?? - ?? = 1 4?? + 3?? - 3?? = 7 24?? + ?? - 17?? = 45 Let ?? = 1 Page 2 JEE Main Previous Year Questions (2025): Determinants Q1: Let the system of equations ?? + ???? - ?? = ?? ???? + ???? - ???? = ?? ?????? + ?? + ?? ?? = ?? ?? , ?? ? R, have infinitely many solutions. Then the number of the solutions of this system, if ?? , ?? , ?? are integers and satisfy ?? = ?? + ?? + ?? = ???? , is : JEE Main 2025 (Online) 7th April Evening Shift Options: A. 4 B. 5 C. 3 D. 6 Ans: C Solution: For infinitely many solution ? = 0 | 1 5 -1 4 3 -3 24 1 ?? | = 0 ? 1(3?? + 3) - 5(4?? + 72) - 1(4 - 72) = 0 ? -17?? + 3 - 4 × 72 - 4 = 0 ? 17?? = -289 ? ?? = -17 ?1 = 0 ? | 1 5 -1 7 3 -3 ?? 1 -17 | = 0 ? 1(-51 + 3) - 5(-119 + 3?? ) - 1(7 - 3?? ) = 0 ? -48 + 595 - 15?? - 7 + 3?? = 0 ? 12?? = 540 ?? = 45 ?? + 5?? - ?? = 1 4?? + 3?? - 3?? = 7 24?? + ?? - 17?? = 45 Let ?? = 1 ?? + 5?? = 1 + ?? ] × 4 4?? + 3?? = 7 + 3?? 4x + 20y = 4 + 4?? -17y = 3 - ?? y = ?? - 3 17 , x = 1 + ?? - 5?? - 15 17 = 32 - 12?? 17 7 = ?? - 3 17 + 32 + 12?? 17 + ?? = 77 7 = 30?? + 29 17 = 77 3 = ?? = 42 ?? = 3,20,37 Q2: If the system of linear equations : ?? + ?? + ?? ?? = ?? ?? ?? + ?? ?? + ???? = ?? + ?? -?? - ?? ?? + ?? ?? = ?? ?? where ?? , ?? ? ?? , has infinitely many solutions, then ?? ?? + ?? ?? is equal to : JEE Main 2025 (Online) 22nd January Evening Shift Options: A. 12 B. 9 C. 22 D. 16 Ans: D Solution: We begin with the system: ?? + ?? + 2?? = 6 2?? + 3?? + ???? = ?? + 1 -?? - 3?? + ???? = 2?? Step 1. Solve the first equation for ?? : ?? = 6 - ?? - 2?? . Step 2. Substitute ?? = 6 - ?? - 2?? into the second equation: 2(6 - ?? - 2?? ) + 3?? + ???? = ?? + 1. Expanding and simplifying: 12 - 2?? - 4?? + 3?? + ???? = ?? + 1 ? ?? + (?? - 4)?? = ?? - 11. Call this Equation (I). Step 3. Substitute ?? = 6 - ?? - 2?? into the third equation: Page 3 JEE Main Previous Year Questions (2025): Determinants Q1: Let the system of equations ?? + ???? - ?? = ?? ???? + ???? - ???? = ?? ?????? + ?? + ?? ?? = ?? ?? , ?? ? R, have infinitely many solutions. Then the number of the solutions of this system, if ?? , ?? , ?? are integers and satisfy ?? = ?? + ?? + ?? = ???? , is : JEE Main 2025 (Online) 7th April Evening Shift Options: A. 4 B. 5 C. 3 D. 6 Ans: C Solution: For infinitely many solution ? = 0 | 1 5 -1 4 3 -3 24 1 ?? | = 0 ? 1(3?? + 3) - 5(4?? + 72) - 1(4 - 72) = 0 ? -17?? + 3 - 4 × 72 - 4 = 0 ? 17?? = -289 ? ?? = -17 ?1 = 0 ? | 1 5 -1 7 3 -3 ?? 1 -17 | = 0 ? 1(-51 + 3) - 5(-119 + 3?? ) - 1(7 - 3?? ) = 0 ? -48 + 595 - 15?? - 7 + 3?? = 0 ? 12?? = 540 ?? = 45 ?? + 5?? - ?? = 1 4?? + 3?? - 3?? = 7 24?? + ?? - 17?? = 45 Let ?? = 1 ?? + 5?? = 1 + ?? ] × 4 4?? + 3?? = 7 + 3?? 4x + 20y = 4 + 4?? -17y = 3 - ?? y = ?? - 3 17 , x = 1 + ?? - 5?? - 15 17 = 32 - 12?? 17 7 = ?? - 3 17 + 32 + 12?? 17 + ?? = 77 7 = 30?? + 29 17 = 77 3 = ?? = 42 ?? = 3,20,37 Q2: If the system of linear equations : ?? + ?? + ?? ?? = ?? ?? ?? + ?? ?? + ???? = ?? + ?? -?? - ?? ?? + ?? ?? = ?? ?? where ?? , ?? ? ?? , has infinitely many solutions, then ?? ?? + ?? ?? is equal to : JEE Main 2025 (Online) 22nd January Evening Shift Options: A. 12 B. 9 C. 22 D. 16 Ans: D Solution: We begin with the system: ?? + ?? + 2?? = 6 2?? + 3?? + ???? = ?? + 1 -?? - 3?? + ???? = 2?? Step 1. Solve the first equation for ?? : ?? = 6 - ?? - 2?? . Step 2. Substitute ?? = 6 - ?? - 2?? into the second equation: 2(6 - ?? - 2?? ) + 3?? + ???? = ?? + 1. Expanding and simplifying: 12 - 2?? - 4?? + 3?? + ???? = ?? + 1 ? ?? + (?? - 4)?? = ?? - 11. Call this Equation (I). Step 3. Substitute ?? = 6 - ?? - 2?? into the third equation: -(6 - ?? - 2?? ) - 3?? + ???? = 2?? . Expanding and simplifying: -6 + ?? + 2?? - 3?? + ???? = 2?? ? -2?? + (?? + 2)?? = 2?? + 6. Call this Equation (II). Step 4. For the system to have infinitely many solutions, the two equations in ?? and ?? must be dependent-that is, one must be a constant multiple of the other. Assume there exists a constant ?? such that -2 = ?? · 1 ? ?? = -2. Apply this to the coefficient of ?? and the constant term. For the ?? -coefficient in Equations (I) and (II): ?? + 2 = ?? (?? - 4) = -2(?? - 4) = -2?? + 8. Thus, ?? = -2?? + 6. For the constant term: 2?? + 6 = ?? (?? - 11) = -2(?? - 11) = -2?? + 22. Substitute ?? = -2?? + 6 into this equation: 2(-2?? + 6) + 6 = -2?? + 22 ? -4?? + 12 + 6 = -2?? + 22. Simplify: -4?? + 18 = -2?? + 22. Solve for ?? : -4?? + 18 + 4?? = -2?? + 22 + 4?? ? 18 = 2?? + 22, 2?? = 18 - 22 = -4 ? ?? = -2. Substitute ?? = -2 into ?? = -2?? + 6 : ?? = -2(-2) + 6 = 4 + 6 = 10. Step 5. We now compute 7?? + 3?? = 7(-2) + 3(10) = -14 + 30 = 16. Thus, the value of 7?? + 3?? is 16. Q3: If the system of linear equations ?? ?? + ?? + ???? = ?? ?? ?? + ???? - ?? = -?? ?? + ?? ?? + ?? = ?? has infinitely many solutions, then the value of ???? ?? - ?? ?? is : JEE Main 2025 (Online) 2nd April Morning Shift Options: A. 31 B. 37 C. 43 D. 49 Ans: A Page 4 JEE Main Previous Year Questions (2025): Determinants Q1: Let the system of equations ?? + ???? - ?? = ?? ???? + ???? - ???? = ?? ?????? + ?? + ?? ?? = ?? ?? , ?? ? R, have infinitely many solutions. Then the number of the solutions of this system, if ?? , ?? , ?? are integers and satisfy ?? = ?? + ?? + ?? = ???? , is : JEE Main 2025 (Online) 7th April Evening Shift Options: A. 4 B. 5 C. 3 D. 6 Ans: C Solution: For infinitely many solution ? = 0 | 1 5 -1 4 3 -3 24 1 ?? | = 0 ? 1(3?? + 3) - 5(4?? + 72) - 1(4 - 72) = 0 ? -17?? + 3 - 4 × 72 - 4 = 0 ? 17?? = -289 ? ?? = -17 ?1 = 0 ? | 1 5 -1 7 3 -3 ?? 1 -17 | = 0 ? 1(-51 + 3) - 5(-119 + 3?? ) - 1(7 - 3?? ) = 0 ? -48 + 595 - 15?? - 7 + 3?? = 0 ? 12?? = 540 ?? = 45 ?? + 5?? - ?? = 1 4?? + 3?? - 3?? = 7 24?? + ?? - 17?? = 45 Let ?? = 1 ?? + 5?? = 1 + ?? ] × 4 4?? + 3?? = 7 + 3?? 4x + 20y = 4 + 4?? -17y = 3 - ?? y = ?? - 3 17 , x = 1 + ?? - 5?? - 15 17 = 32 - 12?? 17 7 = ?? - 3 17 + 32 + 12?? 17 + ?? = 77 7 = 30?? + 29 17 = 77 3 = ?? = 42 ?? = 3,20,37 Q2: If the system of linear equations : ?? + ?? + ?? ?? = ?? ?? ?? + ?? ?? + ???? = ?? + ?? -?? - ?? ?? + ?? ?? = ?? ?? where ?? , ?? ? ?? , has infinitely many solutions, then ?? ?? + ?? ?? is equal to : JEE Main 2025 (Online) 22nd January Evening Shift Options: A. 12 B. 9 C. 22 D. 16 Ans: D Solution: We begin with the system: ?? + ?? + 2?? = 6 2?? + 3?? + ???? = ?? + 1 -?? - 3?? + ???? = 2?? Step 1. Solve the first equation for ?? : ?? = 6 - ?? - 2?? . Step 2. Substitute ?? = 6 - ?? - 2?? into the second equation: 2(6 - ?? - 2?? ) + 3?? + ???? = ?? + 1. Expanding and simplifying: 12 - 2?? - 4?? + 3?? + ???? = ?? + 1 ? ?? + (?? - 4)?? = ?? - 11. Call this Equation (I). Step 3. Substitute ?? = 6 - ?? - 2?? into the third equation: -(6 - ?? - 2?? ) - 3?? + ???? = 2?? . Expanding and simplifying: -6 + ?? + 2?? - 3?? + ???? = 2?? ? -2?? + (?? + 2)?? = 2?? + 6. Call this Equation (II). Step 4. For the system to have infinitely many solutions, the two equations in ?? and ?? must be dependent-that is, one must be a constant multiple of the other. Assume there exists a constant ?? such that -2 = ?? · 1 ? ?? = -2. Apply this to the coefficient of ?? and the constant term. For the ?? -coefficient in Equations (I) and (II): ?? + 2 = ?? (?? - 4) = -2(?? - 4) = -2?? + 8. Thus, ?? = -2?? + 6. For the constant term: 2?? + 6 = ?? (?? - 11) = -2(?? - 11) = -2?? + 22. Substitute ?? = -2?? + 6 into this equation: 2(-2?? + 6) + 6 = -2?? + 22 ? -4?? + 12 + 6 = -2?? + 22. Simplify: -4?? + 18 = -2?? + 22. Solve for ?? : -4?? + 18 + 4?? = -2?? + 22 + 4?? ? 18 = 2?? + 22, 2?? = 18 - 22 = -4 ? ?? = -2. Substitute ?? = -2 into ?? = -2?? + 6 : ?? = -2(-2) + 6 = 4 + 6 = 10. Step 5. We now compute 7?? + 3?? = 7(-2) + 3(10) = -14 + 30 = 16. Thus, the value of 7?? + 3?? is 16. Q3: If the system of linear equations ?? ?? + ?? + ???? = ?? ?? ?? + ???? - ?? = -?? ?? + ?? ?? + ?? = ?? has infinitely many solutions, then the value of ???? ?? - ?? ?? is : JEE Main 2025 (Online) 2nd April Morning Shift Options: A. 31 B. 37 C. 43 D. 49 Ans: A Solution: 3?? + ?? + ???? = 3 2?? + ???? - ?? = -3 ?? + 2?? + ?? = 4 has infinite solution ? ? = 0, ? 1 = ? 2 = ? 3 ? = 0 ? | 3 1 ?? 2 ?? -1 1 2 1 | = 0 ? 2 = 0 ? | 3 3 ?? 2 -3 -1 1 4 1 | = 0 ? 3(-3 + 4) - 3(2 + 1) + ?? (8 + 3) = 0 ? 3 - 9 + 11?? = 0 ? ?? = 6 11 ? 3 = 0 ? | 3 1 3 2 ?? -3 1 2 4 | = 0 ? 3(4?? + 6) - 1(8 + 3) + 3(4 - ?? ) = 0 12?? + 18 - 11 + 12 - 3?? = 0 9?? = -19 ?? = -19 9 ? 22?? - 9?? = 31 Q4: If the system of equations ?? ?? + ???? + ?? ?? = ?? ?? ?? + ?? ?? - ?? = ?? ?? ?? + ?? ?? + ???? = ?? has infinitely many solutions, then (?? ?? + ?? ?? ) is equal to : JEE Main 2025 (Online) 2nd April Evening Shift Options: A. 30 B. 26 C. 22 D. 18 Ans: B Solution: Page 5 JEE Main Previous Year Questions (2025): Determinants Q1: Let the system of equations ?? + ???? - ?? = ?? ???? + ???? - ???? = ?? ?????? + ?? + ?? ?? = ?? ?? , ?? ? R, have infinitely many solutions. Then the number of the solutions of this system, if ?? , ?? , ?? are integers and satisfy ?? = ?? + ?? + ?? = ???? , is : JEE Main 2025 (Online) 7th April Evening Shift Options: A. 4 B. 5 C. 3 D. 6 Ans: C Solution: For infinitely many solution ? = 0 | 1 5 -1 4 3 -3 24 1 ?? | = 0 ? 1(3?? + 3) - 5(4?? + 72) - 1(4 - 72) = 0 ? -17?? + 3 - 4 × 72 - 4 = 0 ? 17?? = -289 ? ?? = -17 ?1 = 0 ? | 1 5 -1 7 3 -3 ?? 1 -17 | = 0 ? 1(-51 + 3) - 5(-119 + 3?? ) - 1(7 - 3?? ) = 0 ? -48 + 595 - 15?? - 7 + 3?? = 0 ? 12?? = 540 ?? = 45 ?? + 5?? - ?? = 1 4?? + 3?? - 3?? = 7 24?? + ?? - 17?? = 45 Let ?? = 1 ?? + 5?? = 1 + ?? ] × 4 4?? + 3?? = 7 + 3?? 4x + 20y = 4 + 4?? -17y = 3 - ?? y = ?? - 3 17 , x = 1 + ?? - 5?? - 15 17 = 32 - 12?? 17 7 = ?? - 3 17 + 32 + 12?? 17 + ?? = 77 7 = 30?? + 29 17 = 77 3 = ?? = 42 ?? = 3,20,37 Q2: If the system of linear equations : ?? + ?? + ?? ?? = ?? ?? ?? + ?? ?? + ???? = ?? + ?? -?? - ?? ?? + ?? ?? = ?? ?? where ?? , ?? ? ?? , has infinitely many solutions, then ?? ?? + ?? ?? is equal to : JEE Main 2025 (Online) 22nd January Evening Shift Options: A. 12 B. 9 C. 22 D. 16 Ans: D Solution: We begin with the system: ?? + ?? + 2?? = 6 2?? + 3?? + ???? = ?? + 1 -?? - 3?? + ???? = 2?? Step 1. Solve the first equation for ?? : ?? = 6 - ?? - 2?? . Step 2. Substitute ?? = 6 - ?? - 2?? into the second equation: 2(6 - ?? - 2?? ) + 3?? + ???? = ?? + 1. Expanding and simplifying: 12 - 2?? - 4?? + 3?? + ???? = ?? + 1 ? ?? + (?? - 4)?? = ?? - 11. Call this Equation (I). Step 3. Substitute ?? = 6 - ?? - 2?? into the third equation: -(6 - ?? - 2?? ) - 3?? + ???? = 2?? . Expanding and simplifying: -6 + ?? + 2?? - 3?? + ???? = 2?? ? -2?? + (?? + 2)?? = 2?? + 6. Call this Equation (II). Step 4. For the system to have infinitely many solutions, the two equations in ?? and ?? must be dependent-that is, one must be a constant multiple of the other. Assume there exists a constant ?? such that -2 = ?? · 1 ? ?? = -2. Apply this to the coefficient of ?? and the constant term. For the ?? -coefficient in Equations (I) and (II): ?? + 2 = ?? (?? - 4) = -2(?? - 4) = -2?? + 8. Thus, ?? = -2?? + 6. For the constant term: 2?? + 6 = ?? (?? - 11) = -2(?? - 11) = -2?? + 22. Substitute ?? = -2?? + 6 into this equation: 2(-2?? + 6) + 6 = -2?? + 22 ? -4?? + 12 + 6 = -2?? + 22. Simplify: -4?? + 18 = -2?? + 22. Solve for ?? : -4?? + 18 + 4?? = -2?? + 22 + 4?? ? 18 = 2?? + 22, 2?? = 18 - 22 = -4 ? ?? = -2. Substitute ?? = -2 into ?? = -2?? + 6 : ?? = -2(-2) + 6 = 4 + 6 = 10. Step 5. We now compute 7?? + 3?? = 7(-2) + 3(10) = -14 + 30 = 16. Thus, the value of 7?? + 3?? is 16. Q3: If the system of linear equations ?? ?? + ?? + ???? = ?? ?? ?? + ???? - ?? = -?? ?? + ?? ?? + ?? = ?? has infinitely many solutions, then the value of ???? ?? - ?? ?? is : JEE Main 2025 (Online) 2nd April Morning Shift Options: A. 31 B. 37 C. 43 D. 49 Ans: A Solution: 3?? + ?? + ???? = 3 2?? + ???? - ?? = -3 ?? + 2?? + ?? = 4 has infinite solution ? ? = 0, ? 1 = ? 2 = ? 3 ? = 0 ? | 3 1 ?? 2 ?? -1 1 2 1 | = 0 ? 2 = 0 ? | 3 3 ?? 2 -3 -1 1 4 1 | = 0 ? 3(-3 + 4) - 3(2 + 1) + ?? (8 + 3) = 0 ? 3 - 9 + 11?? = 0 ? ?? = 6 11 ? 3 = 0 ? | 3 1 3 2 ?? -3 1 2 4 | = 0 ? 3(4?? + 6) - 1(8 + 3) + 3(4 - ?? ) = 0 12?? + 18 - 11 + 12 - 3?? = 0 9?? = -19 ?? = -19 9 ? 22?? - 9?? = 31 Q4: If the system of equations ?? ?? + ???? + ?? ?? = ?? ?? ?? + ?? ?? - ?? = ?? ?? ?? + ?? ?? + ???? = ?? has infinitely many solutions, then (?? ?? + ?? ?? ) is equal to : JEE Main 2025 (Online) 2nd April Evening Shift Options: A. 30 B. 26 C. 22 D. 18 Ans: B Solution: 2?? + ???? + 3?? = 5 3?? + 2?? - ?? = 7 4?? + 5?? + ???? = 9 For infinite solutions ? ? = 0 = ? 1 = ? 2 = ? 3 ? = | 2 ?? 3 3 2 -1 4 5 ?? | = 0 ? -4?? - 3???? + 4?? + 31 = 0 ? 1 = | 5 ?? 3 7 2 -1 9 5 ?? | = 0 ? -9?? - 7???? + 10?? + 76 = 0 ? 2 = | 2 3 5 3 -1 7 4 ?? 9 | = 0 ? ?? + 5 = 0 ? ?? = -5 ? 3 = | 2 ?? 5 3 2 7 4 5 9 | = 0 ? ?? + 1 = 0 ? ?? = -1 ? For infinite solution ?? = -5 and ?? = -1 Now ?? 2 + ?? 2 = 25 + 1 = 26 Q5: Let the system of equations : ?? ?? + ?? ?? + ?? ?? = ?? ?? ?? + ?? ?? - ?? ?? = ?? ???? ?? + ?? ?? - (?? + ?? )?? = ???? - ?? have infinitely many solutions. Then the radius of the circle centred at ( ?? , ?? ) and touching the line ?? ?? = ?? ?? is : JEE Main 2025 (Online) 7th April Morning Shift Options: A. 7 5 B. 21 5 C. 7 D. 17 5 Ans: ARead More
FAQs on JEE Main Previous Year Questions (2025): Determinants
| 1. What is a determinant, and why is it important in mathematics? |  |
Ans. A determinant is a scalar value that is calculated from the elements of a square matrix. It provides important information about the matrix, such as whether the matrix is invertible (non-singular) or not (singular). Determinants are widely used in various fields of mathematics, including linear algebra, calculus, and differential equations, as they play a key role in solving systems of linear equations, finding eigenvalues, and analyzing transformations.
| 2. How do you calculate the determinant of a 2x2 matrix? | |
Ans. The determinant of a 2x2 matrix of the form | a b | | c d | is calculated using the formula: det(A) = ad - bc. This means you multiply the elements on the main diagonal (a and d) and subtract the product of the elements on the other diagonal (b and c).
| 3. What are some properties of determinants that are useful for calculations? | |
Ans. Several key properties of determinants include: 1. The determinant of a product of matrices is the product of their determinants: det(AB) = det(A) * det(B). 2. If a matrix has two identical rows or columns, its determinant is zero. 3. The determinant changes sign if two rows or columns are swapped. 4. Multiplying a row or column by a scalar multiplies the determinant by that scalar. These properties can simplify the process of calculating determinants, especially for larger matrices.
| 4. Can you explain how to find the determinant of a 3x3 matrix? | |
Ans. The determinant of a 3x3 matrix | a b c | | d e f | | g h i | can be calculated using the rule of Sarrus or cofactor expansion. Using cofactor expansion along the first row, the determinant is given by: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg). This involves calculating the determinants of the 2x2 matrices formed by removing the row and column of each element in the first row.
| 5. What applications do determinants have in real-world problems? | |
Ans. Determinants have numerous applications in real-world problems, including: 1. In physics, they are used to solve systems of linear equations that can describe physical systems, such as electrical circuits or mechanical systems. 2. In computer graphics, determinants help in transformations and rendering processes, including rotations and scaling. 3. In economics, they can be used to analyze equilibrium in supply and demand models. 4. In engineering, determinants are essential for stability analysis and control systems design. These applications demonstrate the significance of understanding determinants in various fields.
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