Number of distinct real roots of left| {,begin{array}{*{20}{c}}{sin x}&{cos x}&{cos x}{cos x

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3 and 4 .Determinants and Matrices

medium

The number of distinct real roots of $\left| {\,\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}\,} \right| = 0$ in the interval $ - \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is

$0$

$2$

$1$

$3$

(IIT-2001)

Solution

(c) Here, $(2\cos x + \sin x)\,\left| {\,\begin{array}{*{20}{c}}1&{\cos x}&{\cos x}\\1&{\sin x}&{\cos x}\\1&{\cos x}&{\sin x}\end{array}\,} \right|\, = 0$
or $(2\cos x + \sin x)\,\left| {\,\begin{array}{*{20}{c}}1&{\cos x}&{\cos x}\\0&{\sin x – \cos x}&0\\0&0&{\sin x – \cos x}\end{array}\,\,} \right|\, = 0$
or $(2\cos x + \sin x){(\sin x – \cos x)^2} = 0$
$\therefore $ $\tan x = – 2,\,1.\,$ But $\tan x \ne – 2$ in $\left[ { – \frac{\pi }{4},\,\,\,\,\frac{\pi }{4}} \right]$
$\therefore $$\tan x = 1$. So, $x = \frac{\pi }{4}$.

Standard 12

Mathematics

If $\mathrm{a}_{\mathrm{r}}=\cos \frac{2 \mathrm{r} \pi}{9}+i \sin \frac{2 \mathrm{r} \pi}{9}, \mathrm{r}=1,2,3, \ldots, i=\sqrt{-1}$ then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to :

medium

(JEE MAIN-2021)

$\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| = $

easy

Let $a ,b ,c $ be such that $b + c \ne 0$ if

$\left| {\begin{array}{{20}{c}}a&{a + 1}&{a – 1}\\{ – b}&{b + 1}&{b – 1}\\c&{c – 1}&{c + 1}\end{array}} \right| + \left| {\begin{array}{{20}{c}}{a + 1}&{b + 1}&{c – 1}\\{a – 1}&{b – 1}&{c + 1}\\{{{\left( { – 1} \right)}^{n + 2}} \cdot a}&{{{\left( { – 1} \right)}^{n + 1}} \cdot b}&{{{\left( { – 1} \right)}^n} \cdot c}\end{array}} \right| = 0$ then $n$ equals to

medium

(AIEEE-2009)

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

hard

(JEE MAIN-2014)

$\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = $

medium

The number of distinct real roots of $\left| {\,\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}\,} \right| = 0$ in the interval $ - \frac{\pi }{4} \le x \le \frac{\pi }{4}$ is

Solution

Similar Questions

If $\mathrm{a}_{\mathrm{r}}=\cos \frac{2 \mathrm{r} \pi}{9}+i \sin \frac{2 \mathrm{r} \pi}{9}, \mathrm{r}=1,2,3, \ldots, i=\sqrt{-1}$ then the determinant $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to :

$\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| = $

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

$\left| {\,\begin{array}{*{20}{c}}a&b&c\\b&c&a\\c&a&b\end{array}\,} \right| = $

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