$\left| {\,\begin{array}{*{20}{c}}{{{\sin }^2}x}&{{{\cos }^2}x}&1\\{{{\cos }^2}x}&{{{\sin }^2}x}&1\\{ - 10}&{12}&2\end{array}\,} \right| = $

  • A

    $0$

  • B

    $12{\cos ^2}x - 10{\sin ^2}x$

  • C

    $12{\sin ^2}x - 10{\cos ^2}x - 2$

  • D

    $10\sin 2x$

Similar Questions

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 $........$.

  • [JEE MAIN 2023]

 If the system of equations $x+2 y-3 z=2$, $2 x+\lambda y+5 z=5$, $14 x+3 y+\mu z=33$ has infinitely many solutions, then $\lambda+\mu$ is equal to:

  • [JEE MAIN 2025]

$2x + 3y + 4z = 9$,$4x + 9y + 3z = 10,$$5x + 10y + 5z = 11$ then the value of $ x$ is

If $\left| {\,\begin{array}{*{20}{c}}{1 + ax}&{1 + bx}&{1 + cx}\\{1 + {a_1}x}&{1 + {b_1}x}&{1 + {c_1}x}\\{1 + {a_2}x}&{1 + {b_2}x}&{1 + {c_2}x}\end{array}\,} \right|,$ $ = {A_0} + {A_1}x + {A_2}{x^2} + {A_3}{x^3}$ then ${A_1}$ is equal to

If the system of equations $ax + y + z = 0 , x + by + z = 0 \, \& \, x + y + cz = 0$ $(a, b, c \ne 1)$ has a non-trivial solution, then the value of $\frac{1}{{1\, - \,a}}\,\, + \,\,\frac{1}{{1\, - \,b}}\,\, + \,\,\frac{1}{{1\, - \,c}}$ is :