Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main20268 April 2026Evening ShiftMathematicsFunctionsActual

Let f be a polynomial function such that ₂(f(x)) = ( ₂ (2+ 2 3 + 2 9 + ) ) ₃ (1+ f(x) f(1/x) ) , x>0 and f(6)=37 . Then _ n=1 ¹⁰f(n) is equal to ________.

Correct answer

0

Step-by-step solution

The sum of the infinite geometric progression is given by: S = 2 + 2 3 + 2 9 + = 2 1 - 1 3 = 3 Substituting this into the given equation: ₂(f(x)) = ₂(3) ₃ (1 + f(x) f(1/x) ) Using the base change property _a(b) _b(c) = _a(c) : ₂(f(x)) = ₂ (1 + f(x) f(1/x) ) Equating the arguments: f(x) = 1 + f(x) f(1/x) f(x)f(1/x) - f(1/x) = f(x) f(x)f(1/x) = f(x) + f(1/x) The only polynomial functions satisfying this relation are of the form f(x) = 1 x^n . Given f(6) = 37 : 1 6^n = 37 Taking the positive sign, 6^n = 36 n = 2 . Thu

Practice Functions on Quantrex Academy →

More from Functions

Let N denote the set of all positive integers. Consider the sets A = 1, 2, 3, 4, 5 and B = 1, 2, 3, 4, 5, 6, 7 . Let S be the set of all functions f: A B such that f(2) 2 and f(4) 2026Let f:(1, ) R be a function defined as f(x) = x-1 x+1 . Let f^ i+1 (x) = f(f^i(x)) , i=1, 2, , 25 , where f^1(x)=f(x) . If g(x) + f²⁶(x) = 0 , x (1, ) , then the area of the region 2026Let f: R R be defined as f(x) = 2x^2 - 3x + 2 3x^2 + x + 3 . Then f is : 2026Let e be the base of natural logarithm and let f: 1, 2, 3, 4 1, e, e^2, e^3 and g: 1, e, e^2, e^3 1, 1 2 , 1 3 , 1 4 be two bijective functions such that f is strictly decreasing a 2026Let f: R R be a function such that f(x) + 3f ( 2 - x ) = x , x R . Let the maximum value of f on R be . If the area of the region bounded by the curves g(x) = x^2 and h(x) = x^3 , 2026The sum of all the integral values of p such that the equation 3 ^2 x + 12 x - 3 = p , x R , has at least one solution, is: 2026Let A = 1, 2, 3, 4, 5, 6 . The number of one-one functions f: A A such that f(1) 3 , f(3) 4 and f(2) + f(3) = 5 , is __________. 2026For the function f:[1, ) [1, ) defined by f(x)=(x-1)^4+1 , among the two statements: (I) The set S= x [1, ): f(x)=f⁻¹(x) contains exactly two elements, and (II) The set S= x [1, ): 2026 Full Functions list All JEE Main PYQs