Let the point P of the focal chord PQ of the parabola $y^2=16 x$ be $(1,-4)$. If the focus of the parabola divides the chord $P Q$ in the ratio $m: n, \operatorname{gcd}(m, n)=1$, then $m^2+n^2$ is equal to :

If the system of equations

$$ \begin{aligned} & 2 x+\lambda y+3 z=5 \\ & 3 x+2 y-z=7 \\ & 4 x+5 y+\mu z=9 \end{aligned} $$

has infinitely many solutions, then $\left(\lambda^2+\mu^2\right)$ is equal to :

Let $A$ be a $3 \times 3$ real matrix such that $A^2(A-2 I)-4(A-I)=O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha, \beta$, and $\gamma$ are real constants, then $\alpha+\beta+\gamma$ is equal to :

$$If\,\mathop {\lim }\limits_{x \to 0} {{\cos (2x) + a\cos (4x) - b} \over {{x^4}}}is\,finite,\,then\,(a + b)\,is\,equal\,to:$$