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F4 A Maths Questions
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1. let y = f(x) be a function such that for any real numbers a and b, f(a+b) + f(a-b) = 2[f(a) + f(b)]. Prove that f(x) is an even function. 2. a and b are the real roots of the quadratic equation x^2 - 5x + k = 0 such that |a - b| =3. Find the value of k. Step Please...
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(1) f(a + b) + f(a - b) = 2[f(a) + f(b)] for all real values of a and b Let b = 0 f(a) + f(a) = 2[f(a) + f(0)] => 2f(a) = 2f(a) + 2f(0) => f(0) = 0 Let a = 0 f(b) + f(-b) = 2[f(0) + f(b)] for all real values of b f(b) + f(-b) = 2f(b) for all real values of b f(-b) = f(b) for all real values of b Hence f(x) is an even function (2) x^2 - 5x + k = 0 has roots a and b and | a - b | = 3 Sum of roots a + b = 5 Product of roots ab = k a^2 + 2ab + b^2 = 25 a^2 - 2ab + b^2 + 4ab = 25 (a - b)^2 = 25 - 4ab (a - b)^2 = 25 - 4k 9 = 25 - 4k 4k = 16 k = 4
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