Exams › JEE Main › Maths

Find the smallest integer value of a for which the inequality 1 + log₅(1 + x²) <= log₅(a*x² + 4x + a) holds for all real x.

1. 5
2. 6
3. 7
4. 8

Correct answer: 7

Solution

Rewriting gives 5(1+x²) <= a*x²+4x+a, i.e., (a-5)*x²+4x+(a-5) >= 0 for all x; requiring a-5 > 0 and discriminant <= 0 yields a >= 7, so the smallest integer is 7.

Related JEE Main Maths questions

• Match the maximum number of integral values of x satisfying each equation or inequality in Column I with the entries in Column II. Column I: (A) log₃(log₂(−log_(1/2) x)) < 0 (B) (x−5)(x−9) / (x⁴ − 1) < 0 (C) ||x + 2| − 1| <= 1 (D) ||x − 3| − |x − 5|| = 2|x − 4| Column II: p = 4, q = 3, r = 1, s = 5 Which option correctly matches each entry in Column I to the corresponding maximum number of integral values?
• Find the complete solution set of the inequality (log₁₀(x))² >= log₁₀(x) + 2. If the solution set is (0, a] union [b, infinity), find the value of the product a*b.
• Given log₅(N) = I1 + f1 and log₃(N) = I2 + f2, where I1 and I2 are integers and f1, f2 are in [0, 1). If I1 = 2 and I2 = 3, what is the maximum integer value of N?
• If log_(0.3)(x - 1) < log_(0.09)(x - 1), then x lies in which interval?
• Let A be the set {(n, 2n): n ∈ N} and let B be the set {(2n, 3n): n ∈ N}. What is the intersection A ∩ B?
• Let set A contain 3 elements and set B contain 6 elements. Then the cardinality of their union must satisfy

⚔️ Practice JEE Main Maths free + battle 1v1 →