Number of distinct real roots of left|begin{array}{lll}sin x & cos x & cos x cos x & sin

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3 and 4 .Determinants and Matrices

hard

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

$1$

$2$

$3$

$4$

(JEE MAIN-2021)

Solution

$\left|\begin{array}{lll} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{array}\right|=0$, $\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$

Apply: $R_{1} \rightarrow R_{1}-R_{2}$ and $R_{2} \rightarrow R_{2}-R_{3}$

$\left|\begin{array}{ccc} \sin x-\cos x & \cos x-\sin x & 0 \\ 0 & \sin x-\cos x & \cos x-\sin x \\ \cos x & \cos x & \sin x \end{array}\right|=0$

$(\sin x-\cos x)^{2}\left|\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ \cos x & \cos x & \sin x\end{array}\right|=0$

$(\sin x-\cos x)^{2}(\sin x+2 \cos x)=0$

$\sin x=\cos x \quad$ or $\quad \sin x=-2 \cos x$

$\tan x=1$ or $\quad \tan x=-2$ (Not valid)

$\therefore x=\frac{\pi}{4}$

Standard 12

Mathematics

The values of $x,y,z$ in order of the system of equations $3x + y + 2z = 3,$ $2x – 3y – z = – 3$, $x + 2y + z = 4,$ are

medium

If $A$, $B$ and $C$ are square matrices of order $3$ such that $A = \left[ {\begin{array}{*{20}{c}} x&0&1 \\ 0&y&0 \\ 0&0&z \end{array}} \right]$ and $\left| B \right| = 36$, $\left| C \right| = 4$, $\left( {x,y,z \in N} \right)$ and $\left| {ABC} \right| = 1152$ then the minimum value of $x + y + z$ is

normal

The value of $x,$ if $\left| {\,\begin{array}{*{20}{c}}{ – x}&1&0\\1&{ – x}&1\\0&1&{ – x}\end{array}\,} \right| = 0 $ is equal to

easy

If $\left| {\begin{array}{*{20}{c}}
{^9{C_4}}&{^9{C_5}}&{^{10}{C_r}} \\
{^{10}{C_6}}&{^{10}{C_7}}&{^{11}{C_{r + 2}}} \\
{^{11}{C_8}}&{^{11}{C_9}}&{^{12}{C_{r + 4}}}
\end{array}} \right| = 0$ then $r$ is equal to

normal

$\Delta = \left| {\,\begin{array}{*{20}{c}}{a + x}&b&c\\b&{x + c}&a\\c&a&{x + b}\end{array}\,} \right|$,which of the following is a factor for the above determinant