If A = left| {,begin{array}{*{20}{c}}{sin (θ + α )}&{cos (θ + α )}&1{sin (θ + β )}&amp

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3 and 4 .Determinants and Matrices

medium

If $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,then

$A = 0$ for all $\theta $

$A$ is an odd Function of $\theta $

$A = 0$ for $\theta = \alpha + \beta + \gamma $

$A$ is independent of $\theta $

Solution

(d) Given $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$
Operate ${R_2} \to {R_2} – {R_1},\,{R_3} \to {R_3} – {R_1}$
$\therefore $$A = \{ \cos (\theta + \gamma ) – \cos (\theta + \alpha )\} $
$\{ \sin (\theta + \beta ) – \sin (\theta + \alpha )\} – \{ \cos (\theta + \beta )$

= $\sin (\beta – \gamma ) – \sin (\beta – \alpha ) – \sin (\alpha – \gamma )$,
which is independent of $\theta $.

Standard 12

Mathematics

The equation $\left| {\begin{array}{{20}{c}}{{{(1 + x)}^2}}&{{{(1 – x)}^2}}&{ – \,(2 + {x^2})}\\{2x + 1}&{3x}&{1 – 5x}\\{x + 1}&{2x}&{2 – 3x}\end{array}} \right|$ $+$ $\left| {\begin{array}{{20}{c}}{{{(1 + x)}^2}}&{2x + 1}&{x + 1}\\{{{(1 – x)}^2}}&{3x}&{2x}\\{1 – 2x}&{3x – 2}&{2x – 3}\end{array}} \right|$ $= 0$

normal

The greatest value of $c \in R$ for which the system of linear equations

$x – cy – cz = 0 \,\,;\,\, cx – y + cz = 0 \,\,;\,\, cx + cy – z = 0 $ has a non -trivial solution, is

hard

(JEE MAIN-2019)

A root of the equation $\left| {\,\begin{array}{*{20}{c}}{3 – x}&{ – 6}&3\\{ – 6}&{3 – x}&3\\3&3&{ – 6 – x}\end{array}\,} \right| = 0$ is

easy

If the system of equations $x + ay = 0,$ $az + y = 0$ and $ax + z = 0$ has infinite solutions, then the value of $a$ is

easy

(IIT-2003)

If $1,\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\,\begin{array}{*{20}{c}}1&{{\omega ^n}}&{{\omega ^{2n}}}\\{{\omega ^n}}&{{\omega ^{2n}}}&1\\{{\omega ^{2n}}}&1&{{\omega ^n}}\end{array}\,} \right|$ is equal to

medium

(AIEEE-2003)

If $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,then

Solution

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The greatest value of $c \in R$ for which the system of linear equations

$x – cy – cz = 0 \,\,;\,\, cx – y + cz = 0 \,\,;\,\, cx + cy – z = 0 $ has a non -trivial solution, is