If $\left| {\begin{array}{*{20}{c}}{a\, + \,1}&{a\, + \,2}&{a\, + \,p}\\{a\, + \,2}&{a\, +\,3}&{a\, + \,q}\\{a\, + \,3}&{a\, + \,4}&{a\, + \,r}\end{array}} \right|$ $= 0$ , then $p, q, r$ are in :

The number of distinct real roots of $\left|\begin{array}{lll}\sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is

Let $\mathrm{A}(-1,1)$ and $\mathrm{B}(2,3)$ be two points and $\mathrm{P}$ be a variable point above the line $A B$ such that the area of $\triangle \mathrm{PAB}$ is $10$ . If the locus of $\mathrm{P}$ is $\mathrm{ax}+\mathrm{by}=15$, then $5 a+2 b$ is :

Given the system of equation $a(x + y + z)=x,b(x + y + z) = y, c(x + y + z) = z$ where $a,b,c$  are non-zero real numbers. If the real numbers $x,y,z$ are such that $xyz \neq 0,$ then  $(a + b + c)$ is equal to-

The determinant $\,\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&2&3\\1&3&6\end{array}\,} \right|$ is not equal to

Let $\alpha, \beta$ and $\gamma$ be real numbers such that the system of linear equations

$x+2 y+3 z=\alpha$

$4 x+5 y+6 z=\beta$

$7 x+8 y+9 z=\gamma-$

is consistent. Let $| M |$ represent the determinant of the matrix

$M=\left[\begin{array}{ccc}\alpha & 2 & \gamma \\ \beta & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$

Let $P$ be the plane containing all those $(\alpha, \beta, \gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.

($1$) The value of $| M |$ is

($2$) The value of $D$ is