If ${\Delta _1} = \left| {\begin{array}{*{20}{c}}
x&{\sin \,\theta }&{\cos \,\theta } \\
{\sin \,\theta }&{ - x}&1 \\
{\cos \,\theta }&1&x
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}}
x&{\sin \,2\theta }&{\cos \,\,2\theta } \\
{\sin \,2\theta }&{ - x}&1 \\
{\cos \,\,2\theta }&1&x
\end{array}} \right|$, $x \ne 0$ ; then for all $\theta \in \left( {0,\frac{\pi }{2}} \right)$
If ${\Delta _1} = \left| {\begin{array}{*{20}{c}}
x&{\sin \,\theta }&{\cos \,\theta } \\
{\sin \,\theta }&{ - x}&1 \\
{\cos \,\theta }&1&x
\end{array}} \right|$ and ${\Delta _2} = \left| {\begin{array}{*{20}{c}}
x&{\sin \,2\theta }&{\cos \,\,2\theta } \\
{\sin \,2\theta }&{ - x}&1 \\
{\cos \,\,2\theta }&1&x
\end{array}} \right|$, $x \ne 0$ ; then for all $\theta \in \left( {0,\frac{\pi }{2}} \right)$
- [JEE MAIN 2019]
- A
${\Delta _1} - {\Delta _2} = - 2{x^3}$
- B
${\Delta _1} + {\Delta _2} = - 2({x^3} + x - 1)$
- C
${\Delta _1} - {\Delta _2} = x\left( {\cos \,2\theta - \cos \,4\theta } \right)$
- D
${\Delta _1} + {\Delta _2} = - 2{x^3}$
Similar Questions
Let $\omega = - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}$. Then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{ - 1 - {\omega ^2}}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^4}}\end{array}\,} \right|$ is
Let $\omega = - \frac{1}{2} + i\frac{{\sqrt 3 }}{2}$. Then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{ - 1 - {\omega ^2}}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^4}}\end{array}\,} \right|$ is
- [IIT 2002]
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{\left( {{b^2} + {c^2}} \right)}&{ab}&{ac}\\
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{ac}&{bc}&{\left( {{a^2} + {b^2}} \right)}
\end{array}} \right| = K{a^2}{b^2}{c^2}$ then value of $K$ is
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{\left( {{b^2} + {c^2}} \right)}&{ab}&{ac}\\
{ab}&{\left( {{c^2} + {a^2}} \right)}&{bc}\\
{ac}&{bc}&{\left( {{a^2} + {b^2}} \right)}
\end{array}} \right| = K{a^2}{b^2}{c^2}$ then value of $K$ is
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The value of $\left| {\,\begin{array}{*{20}{c}}{41}&{42}&{43}\\{44}&{45}&{46}\\{47}&{48}&{49}\end{array}\,} \right| = $
Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N=N M$. Further, if $M \neq N^2$ and $M^2=N^4$, then
$(A)$ determinant of $\left( M ^2+ MN ^2\right)$ is $0$
$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $\left( M ^2+ MN ^2\right) U$ is the zero matrix
$(C)$ determinant of $\left( M ^2+ MN ^2\right) \geq 1$
$(D)$ for a $3 \times 3$ matrix $U$, if $\left( M ^2+ MN ^2\right) U$ equals the zero matrix then $U$ is the zero matrix
Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N=N M$. Further, if $M \neq N^2$ and $M^2=N^4$, then
$(A)$ determinant of $\left( M ^2+ MN ^2\right)$ is $0$
$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $\left( M ^2+ MN ^2\right) U$ is the zero matrix
$(C)$ determinant of $\left( M ^2+ MN ^2\right) \geq 1$
$(D)$ for a $3 \times 3$ matrix $U$, if $\left( M ^2+ MN ^2\right) U$ equals the zero matrix then $U$ is the zero matrix
- [IIT 2014]