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IAT IISER2026MathematicsTrigonometric Ratios & Identities

Consider the function f : R R defined by f(x) = ^2(7x) - ^2(5x) . Which of the following statements is NOT TRUE?

Options

  1. Af is increasing on ( 3 2 , 2 ) .
  2. Bf(x) > 0 , for all x (0, 48 ) .
  3. Cf (x + 2 ) + f(x) = 0 , for all x R .
  4. Df ( 12 ) = 0

Correct answer

A. f is increasing on ( 3 2 , 2 ) .

Step-by-step solution

f(x) = ^2(7x) - ^2(5x) Using the identity ^2 A - ^2 B = (A+B) (A-B) , we get: f(x) = (12x) (2x) Checking the given options: For option (b): If x (0, 48 ) , then 12x (0, 4 ) and 2x (0, 24 ) . Since both 12x and 2x lie in the first quadrant, (12x) > 0 and (2x) > 0 . Thus, f(x) > 0 . This statement is true. For option (c): f (x + 2 ) = (12 (x + 2 ) ) (2 (x + 2 ) ) f (x + 2 ) = (12x + 6 ) (2x + ) f (x + 2 ) = (12x)(- (2x)) = -f(x) Therefore, f (x + 2 ) + f(x) = 0 . This statement is true. For option (d): f ( 12 ) = (12

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