Let $\alpha ,\,\beta ,\,\gamma $ be the real roots of the equation, ${x^3} + a{x^2} + bx + c = 0,\,\,(a,\,b,\,c \in R\,\,{\rm{and}}\,\,a,\,b \ne 0)$. If the system of equations $({\rm{in}}\,\,u,\,v,\,w)$ given by $\alpha u + \beta v + \gamma w = 0,\,\,\beta u + \gamma v + \alpha w = 0;\,\,\gamma u + \alpha v + \beta w = 0$ has non-trivial solution, then the value of $\frac{{{a^2}}}{b}$ is

JEE MATHEMATICS MCQ: MATRICES

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Analyzing +4 | -1

Let

\alpha ,\,\beta ,\,\gamma

be the real roots of the equation,

{x^3} + a{x^2} + bx + c = 0,\,\,(a,\,b,\,c \in R\,\,\mathrm{and}\,\,a,\,b \ne 0)

. If the system of equations

(\mathrm{in}\,\,u,\,v,\,w)

given by

\alpha u + \beta v + \gamma w = 0,\,\,\beta u + \gamma v + \alpha w = 0;\,\,\gamma u + \alpha v + \beta w = 0

has non-trivial solution, then the value of

\frac{{{a^2}}}{b}

View answer and explanation

Correct option: B

Option B (3) is correct. The condition for non-trivial solution implies that the determinant of the coefficient matrix is zero, leading to the relation

\alpha = \beta = \gamma

, and thus

\frac{a^2}{b} = 3

More Questions from MATRICES

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\left|\begin{array}{ccc}a & b-c & b+c \\ c+a & b & c-a \\ a-b & a+b & c\end{array}\right|=0

and

a, b, c

are real numbers then the line

a x+b y+c=0

passes through the fixed point is The number of solutions of the system of equations

{x_2} - {x_3} = 1,\; - {x_1} + 2{x_3} = - 2,\;{x_1} - 2,\;{x_1} - 2{x_2} = 3

is If the equations

2x + 3y + 1 = 0,3x + y - 2 = 0

and

ax + 2y - b = 0

are consistent, then The value of the determinant

\left|\begin{array}{ccc}1 & \left(\alpha^{2 x}-\alpha^{-2 x}\right)^2 & \left(\alpha^{2 x}+\alpha^{-2 x}\right)^2 \\ 1 & \left(\beta^{2 x}-\beta^{-2 x}\right)^2 & \left(\beta^{2 x}+\beta^{-2 x}\right)^2 \\ 1 & \left(\gamma^{2 x}-\gamma^{-2 x}\right)^2 & \left(\gamma^{2 x}+\gamma^{-2 x}\right)^2\end{array}\right|

, is If

\omega

is the cube root of unity, then

\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{{\omega ^2}}\\ \omega &{{\omega ^2}}&1\\ {{\omega ^2}}&1&\omega \end{array}} \right|

is equal to For any matrix

A

of order

2 \times 2

, if

A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]

, then

|A|

is equal to