If A₁B₁C₁,, A₂B₂C₂,, A₃B₃C₃ are three digit number each of which is divisible by k

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3 and 4 .Determinants and Matrices

normal

If $A_1B_1C_1,\, A_2B_2C_2,\, A_3B_3C_3$ are three digit number each of which is divisible by $k$ and $\Delta = \left| {\begin{array}{*{20}{c}}
{{A_1}{\kern 1pt} }&{{B_1}}&{{C_1}} \\
{{A_2}}&{{B_2}}&{{C_2}} \\
{{A_3}}&{{B_3}}&{{C_3}}
\end{array}} \right|$ ; then $\Delta $ is divisible by

$k$

$k^2$

$k^3$

None

Solution

Let $\mathrm{A}_{1} \mathrm{B}_{1} \mathrm{C}_{1}=100 \mathrm{A}_{1}+10 \mathrm{B}_{1}+\mathrm{C}_{1}=\mathrm{pk}$

$\mathrm{A}_{2} \mathrm{B}_{2} \mathrm{C}_{2}=100 \mathrm{A}_{2}+10 \mathrm{B}_{2}+\mathrm{C}_{2}=\mathrm{q} \mathrm{k}$

$\mathrm{A}_{3} \mathrm{B}_{3} \mathrm{C}_{3}=100 \mathrm{A}_{3}+10 \mathrm{B}_{3}+\mathrm{C}_{3}=\mathrm{rk}\left(\text { where } \mathrm{p}_{1} \mathrm{q}_{1} \mathrm{r} \in \mathrm{I}\right)$

so $\Delta=\mathrm{C}_{3} \rightarrow \mathrm{C}_{3}+100 \mathrm{C}_{1}+10 \mathrm{C}_{2}$

$\Delta=\left|\begin{array}{lll}{\mathrm{A}_{1}} & {\mathrm{B}_{1}} & {\mathrm{pk}} \\ {\mathrm{A}_{2}} & {\mathrm{B}_{2}} & {\mathrm{qk}} \\ {\mathrm{A}_{3}} & {\mathrm{B}_{3}} & {\mathrm{rk}}\end{array}\right|=\mathrm{k} \cdot$ Integes (divisible by $k$ )

Standard 12

Mathematics

If $f(\theta ) =\left| {\begin{array}{*{20}{c}}
1&{\cos {\mkern 1mu} \theta }&1\\
{ – \sin {\mkern 1mu} \theta }&1&{ – \cos {\mkern 1mu} \theta }\\
{ – 1}&{\sin {\mkern 1mu} \theta }&1
\end{array}} \right|$ and $A$ and $B$ are respectively the maximum and the minimum values of $f(\theta )$, then $(A , B)$ is equal to

hard

(JEE MAIN-2014)

$\left| {\,\begin{array}{*{20}{c}}1&a&b\\{ – a}&1&c\\{ – b}&{ – c}&1\end{array}\,} \right| = $

easy

Consider the system of linear equations

$-x+y+2 z=0$

$3 x-a y+5 z=1$

$2 x-2 y-a z=7$

Let $S_{1}$ be the set of all $\mathrm{a} \in {R}$ for which the system is inconsistent and $S_{2}$ be the set of all $a \in {R}$ for which the system has infinitely many solutions. If $n\left(S_{1}\right)$ and $n\left(S_{2}\right)$ denote the number of elements in $S_{1}$ and $\mathrm{S}_{2}$ respectively, then

hard

(JEE MAIN-2021)

The maximum value of

$f(x)=\left|\begin{array}{ccc} \sin ^{2} x & 1+\cos ^{2} x & \cos 2 x \\ 1+\sin ^{2} x & \cos ^{2} x & \cos 2 x \\ \sin ^{2} x & \cos ^{2} x & \sin 2 x \end{array}\right|, x \in R \text { is }$

medium

(JEE MAIN-2021)

Given the system of equation $a(x + y + z)=x,b(x + y + z) = y, c(x + y + z) = z$ where $a,b,c$ are non-zero real numbers. If the real numbers $x,y,z$ are such that $xyz \neq 0,$ then $(a + b + c)$ is equal to-

normal

If $A_1B_1C_1,\, A_2B_2C_2,\, A_3B_3C_3$ are three digit number each of which is divisible by $k$ and $\Delta = \left| {\begin{array}{*{20}{c}} {{A_1}{\kern 1pt} }&{{B_1}}&{{C_1}} \\ {{A_2}}&{{B_2}}&{{C_2}} \\ {{A_3}}&{{B_3}}&{{C_3}} \end{array}} \right|$ ; then $\Delta $ is divisible by

Solution

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If $A_1B_1C_1,\, A_2B_2C_2,\, A_3B_3C_3$ are three digit number each of which is divisible by $k$ and $\Delta = \left| {\begin{array}{*{20}{c}}
{{A_1}{\kern 1pt} }&{{B_1}}&{{C_1}} \\
{{A_2}}&{{B_2}}&{{C_2}} \\
{{A_3}}&{{B_3}}&{{C_3}}
\end{array}} \right|$ ; then $\Delta $ is divisible by

The maximum value of

$f(x)=\left|\begin{array}{ccc} \sin ^{2} x & 1+\cos ^{2} x & \cos 2 x \\ 1+\sin ^{2} x & \cos ^{2} x & \cos 2 x \\ \sin ^{2} x & \cos ^{2} x & \sin 2 x \end{array}\right|, x \in R \text { is }$