Let f: ℝ → ℝ be defined as\(f(x) = \left\{ \begin{array}{ll} [e^x] & x < 0 \\ [a e^x + [x-1]] & 0 \leq x < 1 \\ [b + [\sin(\pi x)]] & 1 \leq x < 2 \\ [[e^{-x}] - c] & x \geq 2 \\ \end{array} \right.\)Where a, b, c ∈ ℝ and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?
Let the solution curve y = y(x) of the differential equation\([ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ] x \frac{dy}{dx} = x + [ \frac{x}{\sqrt{x² -y²}} + e^\frac{y}{x} ]y\)pass through the points (1, 0) and (2α, α), α> 0. Then α is equal to
Let the eccentricity of the hyperbola\(H : \frac{x²}{a²} - \frac{y²}{b²} = 1\)be √(5/2) and length of its latus rectum be 6√2, If y = 2x + c is a tangent to the hyperbola H. then the value of c2 is equal to
