Value of a for which system of equations {a^3}x + {(a + 1)^3}y + {(a + 2)^3}z = 0, ax + (a + 1)y + (

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3 and 4 .Determinants and Matrices

hard

The value of a for which the system of equations ${a^3}x + {(a + 1)^3}y + {(a + 2)^3}z = 0,$ $ax + (a + 1)y + (a + 2)z = 0,$ $x + y + z = 0,$ has a non zero solution is

$-1$

$0$

$1$

None of these

Solution

(a) The system will have a non-zero solution, if
$\Delta \equiv \left| {\,\begin{array}{*{20}{c}}{{a^3}}&{{{(a + 1)}^3}}&{{{(a + 2)}^3}}\\a&{a + 1}&{a + 2}\\1&1&1\end{array}\,} \right| = 0$

$ \Rightarrow \left| {\,\begin{array}{*{20}{c}}{{a^3}}&{3{a^2} + 3a + 1}&{3{{(a + 1)}^2} + 3(a + 1) + 1}\\{{a^2}}&1&1\\1&0&0\end{array}\,} \right| = 0$

by $\begin{array}{l}{C_2} \to {C_2} – {C_1}\\{C_3} \to {C_3} – {C_2}\end{array}$

==> $3{a^2} + 3a + 1 – \{ 3{(a + 1)^2} + 3(a + 1) + 1\} $

(expanding along ${R_3}$)

==> $ – 6(a + 1) = 0 \Rightarrow a = – 1$.

Standard 12

Mathematics

If the system of equations

$x-2 y+3 z=9$

$2 x+y+z=b$

$x-7 y+a z=24$

has infinitely many solutions, then $a – b$ is equal to

medium

(JEE MAIN-2020)

The number of solutions of the equations $x + 4y – z = 0,$ $3x – 4y – z = 0,\,x – 3y + z = 0$ is

medium

Let $x, y, z > 0$ are respectively $2^{nd}, 3^{rd}, 4^{th}$ term of $G.P.$ and $\Delta = \left| {\begin{array}{*{20}{c}}
{{X^k}}&{{X^{k + 1}}}&{{X^{k + 2}}}\\
{{Y^k}}&{{Y^{k + 1}}}&{{Y^{k + 2}}}\\
{{Z^k}}&{{Z^{k + 1}}}&{{Z^{k + 2}}}
\end{array}} \right| = {\left( {r – 1} \right)^2}\left( {1 – \frac{1}{{{r^2}}}} \right)$ , (where $r$ is common ratio), then $k=$ …….

normal

Let $A = \left[ {\begin{array}{*{20}{c}}
2&b&1 \\
b&{{b^2} + 1}&b \\
1&b&2
\end{array}} \right]$ where $b > 0$. Then the minimum value of $\frac{{\det \left( A \right)}}{b}$ is

hard

(JEE MAIN-2019)

The system of linear equation $x + y + z = 2, 2x + 3y + 2z = 5$, $2x + 3y + (a^2 -1)\,z = a + 1$ then

hard

(JEE MAIN-2019)

The value of a for which the system of equations ${a^3}x + {(a + 1)^3}y + {(a + 2)^3}z = 0,$ $ax + (a + 1)y + (a + 2)z = 0,$ $x + y + z = 0,$ has a non zero solution is

Solution

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If the system of equations

$x-2 y+3 z=9$

$2 x+y+z=b$

$x-7 y+a z=24$

has infinitely many solutions, then $a – b$ is equal to

Let $A = \left[ {\begin{array}{*{20}{c}}
2&b&1 \\
b&{{b^2} + 1}&b \\
1&b&2
\end{array}} \right]$ where $b > 0$. Then the minimum value of $\frac{{\det \left( A \right)}}{b}$ is