Let $$P = \left[ {\matrix{ 1 & 0 & 0 \cr 4 & 1 & 0 \cr {16} & 4 & 1 \cr } } \right]$$ and I be the identity matrix of order 3. If $$Q = [{q_{ij}}]$$ is a matrix such that $${P^{50}} - Q = I$$ and $${{{q_{31}} + {q_{32}}} \over {{q_{21}}}}$$ equals

Let b_i > 1 for I = 1, 2, ......, 101. Suppose log_eb₁, log_eb₂, ......., log_eb₁₀₁ are in Arithmetic Progression (A.P.) with the common difference log_e2. Suppose a₁, a₂, ......, a₁₀₁ are in A.P. such that a₁ = b₁ and a₅₁ = b₅₁. If t = b₁ + b₂ + .... + b₅₁ and s = a₁ + a₂ + ..... + a₅₁, then

Let a, b $$\in$$ R and f : R $$\to$$ R be defined by $$f(x) = a\cos (|{x^3} - x|) + b|x|\sin (|{x^3} + x|)$$. Then f is