Let S₁ and S₂ be respectively sets of all a ∈ R -{0} for which system of linear equations a x+

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3 and 4 .Determinants and Matrices

hard

Let $S_1$ and $S_2$ be respectively the sets of all $a \in R -\{0\}$ for which the system of linear equations

$a x+2 a y-3 a z=1$

$(2 a+1) x+(2 a+3) y+(a+1) z=2$

$(3 a+5) x+(a+5) y+(a+2) z=3$

has unique solution and infinitely many solutions. Then

$n \left( S _1\right)=2$ and $S _2$ is an infinite set

$S_1$ is an infinite set an $n\left(S_2\right)=2$

$S _1=\Phi$ and $S _2= R -\{0\}$

$S _1= R -\{0\}$ and $S _2=\Phi$

(JEE MAIN-2023)

Solution

$\begin{array}{l}\Delta=\left|\begin{array}{lll}a & 2 a & -3 a \\ 2 a+1 & 2 a+3 & a+1 \\ 3 a+5 & a+5 & a+2\end{array}\right| \\ =a\left(15 a^2+31 a+36\right)=0 \Rightarrow a=0 \\ \Delta \neq 0 \text { for all } a \in R-\{0\} \\ \text { Hence } S_1=R-\{0\} \quad S_2=\Phi\end{array}$

Standard 12

Mathematics

$l,m,n$ are the ${p^{th}},{q^{th}}$and ${r^{th}}$term of a G.P., all positive, then $\left| {\,\begin{array}{{20}{c}}{\log l}&{p\,\,\,\,\,\begin{array}{{20}{c}}1\end{array}}\\{\log m}&{q\,\,\,\,\,\begin{array}{{20}{c}}1\end{array}}\\{\log n}&{r\,\,\,\,\,\begin{array}{{20}{c}}1\end{array}}\end{array}\,} \right|$ equals

medium

(AIEEE-2002)

Let $\omega $ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt { – 3} $ . If $\left| {\begin{array}{*{20}{c}}1&1&1\\1&{ – {\omega ^2} – 1}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^7}}\end{array}} \right| = 3k$ then $k$ is equal to :

hard

(JEE MAIN-2017)

Find equation of line joining $(1,2)$ and $(3,6)$ using determinates

easy

Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3 x+2 y+5 z=3$ $9 x+4 y+(28+[\lambda]) z=[\lambda]$ has a solution is:

hard

(JEE MAIN-2021)

Evaluate $\left|\begin{array}{rr}2 & 4 \\ -1 & 2\end{array}\right|$

easy

Let $S_1$ and $S_2$ be respectively the sets of all $a \in R -\{0\}$ for which the system of linear equations $a x+2 a y-3 a z=1$ $(2 a+1) x+(2 a+3) y+(a+1) z=2$ $(3 a+5) x+(a+5) y+(a+2) z=3$ has unique solution and infinitely many solutions. Then

Solution

Similar Questions

Let $\omega $ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt { – 3} $ . If $\left| {\begin{array}{*{20}{c}}1&1&1\\1&{ – {\omega ^2} – 1}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^7}}\end{array}} \right| = 3k$ then $k$ is equal to :

Find equation of line joining $(1,2)$ and $(3,6)$ using determinates

Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3 x+2 y+5 z=3$ $9 x+4 y+(28+[\lambda]) z=[\lambda]$ has a solution is:

Evaluate $\left|\begin{array}{rr}2 & 4 \\ -1 & 2\end{array}\right|$

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Let $S_1$ and $S_2$ be respectively the sets of all $a \in R -\{0\}$ for which the system of linear equations

$a x+2 a y-3 a z=1$

$(2 a+1) x+(2 a+3) y+(a+1) z=2$

$(3 a+5) x+(a+5) y+(a+2) z=3$

has unique solution and infinitely many solutions. Then