Evaluate the determinants

$\left|\begin{array}{ccc}
3 & -4 & 5 \\
1 & 1 & -2 \\
2 & 3 & 1
\end{array}\right|$

If the system of equations $\alpha x+y+z=5, x+2 y+$ $3 z=4, x+3 y+5 z=\beta$ has infinitely many solutions, then the ordered pair $(\alpha, \beta)$ is equal to:

The system of equations : $2x\, \cos^2\theta + y\, \sin2\theta – 2\sin\theta = 0$ $x\, \sin2\theta + 2y\, \sin^2\theta = – 2\, \cos\theta$ $x\, \sin\theta – y \cos\theta = 0$ , for all values of $\theta$ , can

Let $A_1, A_2, A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A +1, A +2$, respectively. Let $a , b , c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $A_1, A_2, A_3$, respectively such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$ If $a=29$, then the sum of first $20$ terms of an $AP$ whose first term is $c – a – b$ and common difference is $\frac{ d }{12}$, is equal to $……..$.

Number of triplets of $a, b \, \& \,c$ for which the system of equations,$ax – by = 2a – b$ and $(c + 1) x + cy = 10 – a + 3 b$ has infinitely many solutions and $x = 1, y = 3$ is one of the solutions, is :