Roots of equation left| {,begin{array}{*{20}{c}}{1 + x}&1&11&{1 + x}&11&1&{1

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3 and 4 .Determinants and Matrices

easy

The roots of the equation $\left| {\,\begin{array}{*{20}{c}}{1 + x}&1&1\\1&{1 + x}&1\\1&1&{1 + x}\end{array}\,} \right| = 0$ are

$0, -3$

$0, 0, -3$

$0, 0, 0, -3$

None of these

Solution

(b) $\left| {\,\begin{array}{*{20}{c}}{1 + x}&1&1\\1&{1 + x}&1\\1&1&{1 + x}\end{array}\,} \right|\, = \,0$

$ \Rightarrow $$\left| {\,\begin{array}{*{20}{c}}
{3 + x}&0&1\\
{3 + x}&x&1\\
{3 + x}&{ – x}&{1 + x}
\end{array}\,} \right|\, = \,0$, $\left( \begin{array}{l}{C_1} \to {C_1} + {C_2} + {C_3}\\{C_2} \to {C_2} – {C_3}\end{array} \right)$

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

$ \Rightarrow $ $(x + 3)\,\left| {\,\begin{array}{*{20}{c}}1&0&1\\0&x&0\\0&{ – x}&x\end{array}\,} \right|\, = 0$, $\left( \begin{array}{l}{R_2} \to {R_2} – {R_1}\\{R_3} \to {R_3} – {R_1}\end{array} \right)$

$ \Rightarrow $ $(x + 3){x^2} = 0 \Rightarrow x = 0,\,0,\, – 3$.

Trick : Obviously the equation is of degree three, therefore it must have three solutions.

So check for option $ (b).$

Standard 12

Mathematics

Let $d \in R$, and $A = \left[ {\begin{array}{*{20}{c}} { – 2}&{4 + d}&{\left( {\sin \,\theta } \right) – 2}\\ 1&{\left( {\sin \,\theta } \right) + 2}&d\\ 5&{\left( {2\sin \,\theta } \right) – d}&{\left( { – \sin \,\theta } \right) + 2 + 2d} \end{array}} \right]$, $\theta \in \left[ {0,2\pi } \right]$. If the minimum value of det $(A)$ is $8$, then a value of $d$ is

hard

(JEE MAIN-2019)

The set of all values of $\lambda $ for which the system of linear equations $x – 2y – 2z = \lambda x$ ; $x + 2y + z = \lambda y$ ; $-x – y = \lambda z$ has non zero solutions.

hard

(JEE MAIN-2019)

The ordered pair $(a, b)$, for which the system of linear equations $3 x-2 y+z=b$ ; $5 x-8 y+9 z=3$ ; $2 x+y+a z=-1$ has no solution, is

medium

(JEE MAIN-2022)

The value of the determinant$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{1 – x}&1\\1&1&{1 + y}\end{array}\,} \right|$is

medium

The value of $'a'$ for which the system of equation $a^3x + (a + 1)^3y + (a + 2)^3 z = 0$ ; $ax + (a + 1)y + (a + 2)z = 0$ ; $x + y + z = 0$ has a non-zero solution is :-

normal