Q3.

Let the set of all values of $p \in R$, for which both the roots of theequation $x^2 - (p + 2)x + (2p+9)= 0$ are negative real numbers, be the interval $(\alpha, \beta)$. Then$\beta - 2\alpha$ is equal to

Q4.

Let the system of equations :

$2x + 3y + 5z = 9$,

$7x+3y-2z = 8$,

$12x + 3y - (4 + \lambda) z = 16 – \mu$,

have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching theline $4x = 3y$ is

Q5.

Let A be a $3 \times 3$ matrix such that $| \text{adj  (adj A)}| = 81$. If $S=\{ n \in Z : |(\text{adj (adj A)|}^{\frac{(n-1)^2}{2}} = |A|^{(3n^2-5n-4)} \}$, then $\sum_{n \in S} |A|^{(n^2+n)}$ is equal to