Exams › JEE Advanced › Maths

Which of the following determinants is equal to zero (vanishes)?

1. | 1 bc bc(b+c); 1 ca ca(c+a); 1 ab ab(a+b) |
2. | 1 ab 1/a + 1/b; 1 bc 1/b + 1/c; 1 ca 1/c + 1/a |
3. | 0 a-b a-c; b-a 0 b-c; c-a c-b 0 |
4. | logₓ(xyz) logₓ(y) logₓ(z); log_y(xyz) 1 log_y(z); log_z(xyz) log_z(y) 1 |

Correct answer: | 1 ab 1/a + 1/b; 1 bc 1/b + 1/c; 1 ca 1/c + 1/a |

Solution

In option (B), multiplying column 3 by abc turns its entries (c+...) into expressions that are linear combinations of the other columns, so the columns are dependent and the determinant vanishes.

Related JEE Advanced Maths questions

• What is the nature of the solution for the equations x + y + z = 3, 2x + y + 2z = 5, and x − y + 3z = 3?
• What is the nature of the solutions for the equations x + y + z = 3, 2x + 2y + 2z = 7, and x − y + 3z = 3?
• Consider the matrix M = [0 1 a; 1 2 3; 3 b 1] and its adjugate adj M = [-1 1 -1; 8 -6 2; -5 3 -1], where a and b are real numbers. Which of the following statements is/are true?
• Given that the 3x3 determinant |x, 2, x; x², x, 6; x, 1, x| equals Ax⁴ + Bx³ + Cx² + Dx + E, find the sum of the digits of the square of (5A + 4B + 3C + 2D + E).
• The determinant with rows (x², (y+z)², yz), (y², (x+z)², zx), (z², (x+y)², xy) is divisible by which of the following?
• Consider the determinant equation: det([[a1 + b1*x, a1*x + b1, c1], [a2 + b2*x, a2*x + b2, c2], [a3 + b3*x, a3*x + b3, c3]]) = 0. Which of the following are possible conditions that guarantee this holds for all choices of ai, bi, ci?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →