12. f (x+y) + f (x-y) = 2f(x)f (y) and f (alpha) = -1 . Find the period of f

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The minimum area bounded by the function $y = f (x)$ and $y = α x + 9 (α ϵ R)$ where f satisfies the relation $f (x + y) = f (x) + f (y) + y \sqrt{f (x)} \forall x, y ϵ R$ and $f^{'} (0) = 0 & f (0) = 0$ is 9A, value of A is ___

Q. For all non - negative integers x and y, f(x,y) is defined as below.
f(0, y) = y + 1
f(x + 1, 0) = f(x, 1)
f(x + 1, y + 1) = f(x, f(x + 1, y)) find f(1, 2)?

Q. If

f (x) = \frac{2^{x} + 2^{- x}}{2}

, then f(x + y) f(x − y) is equal to
(a)

\frac{1}{2} [f (2 x) + f (2 y)]

(b)

\frac{1}{2} [f (2 x) - f (2 y)]

(c)

\frac{1}{4} [f (2 x) + f (2 y)]

(d)

\frac{1}{4} [f (2 x) - f (2 y)]

Q. The natural number

α

, for which

n \sum k = 1 f (α + k) = 2 n (31 + n)

where the function f satisfies the relation f(x+y)=f(x)+f(y) for all natural numbers x, y and further f(1)=4 is

$If f (x) = \sqrt{1 + x^{2}}, then$ ____

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