If xleft[begin{array}{l}2 3end{array}right]+yleft[begin{array}{l}-1 1end{array}right]=left[begin{arr

Hindi

3 and 4 .Determinants and Matrices

easy

If $x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{l}-1 \\ 1\end{array}\right]=\left[\begin{array}{l}10 \\ 5\end{array}\right],$ find values of $\mathrm{x}$ and $\mathrm{y}$.

$x=3$ and $y=-4$

Solution

$x\left[\begin{array}{l}2 \\ 3\end{array}\right]+y\left[\begin{array}{l}-1 \\ 1\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}2 x \\ 3 x\end{array}\right]+\left[\begin{array}{c}-y \\ y\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$

$\Rightarrow\left[\begin{array}{l}2 x-y \\ 3 x+y\end{array}\right]=\left[\begin{array}{c}10 \\ 5\end{array}\right]$

Comparing the corresponding elements of these two matrices, we get:

$2 x-y=10$ and $3 x+y=5$

Adding these two equations, we have:

$5 x=15 \Rightarrow x=3$

Now, $3 x+y=5$ $\Rightarrow y=5-3 x$

$\Rightarrow y=5-9=-4$

$\therefore x=3$ and $y=-4$

Standard 12

Mathematics

If $A = \left[ {\begin{array}{*{20}{c}}{\,\,\,\cos \alpha }&{\sin \alpha }\\{ – \sin \alpha }&{\cos \alpha }\end{array}} \right]$, then ${A^2} = $

easy

If $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right]$ and $B=\left[\begin{array}{rrr}3 & -1 & 3 \\ -1 & 0 & 2\end{array}\right],$ then find $2 A-B$.

easy

The value of a for which the matrix $A = \left( {\begin{array}{*{20}{c}}a&2\\2&4\end{array}} \right)$is singular if

easy

Let $S=\left\{n \in N \mid\left(\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right)^{n}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \forall a, b, c, d \in R\right\}$, where $i=\sqrt{-1} .$ Then the number of $2 -$ digit numbers in the set $\mathrm{S}$ is $……$

medium

(JEE MAIN-2021)

A manufacturer produces three products $x,\, y,\, z$ which he sells in two markets. Annual sales are indicated below:

Market $x$ $y$ $z$

$I$ $10,000$ $2,000$ $18,000$

$II$ $6,000$ $20,000$ $8,000$

If unit sale prices of $x, \,y$ and $z$ are Rs. $2.50$, Rs. $1.50$ and Rs. $1.00,$ respectively, find the total revenue in each market with the help of matrix algebra.

medium

Market	$x$	$y$	$z$
$I$	$10,000$	$2,000$	$18,000$
$II$	$6,000$	$20,000$	$8,000$