Exams › JEE Advanced › Maths

If a, b, c are in A.P. and positive reals alpha, beta, gamma are in G.P., then the equation given by the determinant of rows (x + a, x² + log(alpha), k), (x + b, x² + log(beta), k), (x + c, x² + log(gamma), k) = 0 is:

1. is an identity
2. has a root x = 1
3. has a root x = 0
4. has real and identical roots

Correct answer: is an identity

Solution

Both column-1 entries (x+a, x+b, x+c) and column-2 entries (via the logs) are in A.P., so R1 + R3 - 2R2 makes a full row zero for every x, making the determinant identically zero.

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 →