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

The values of $\theta, \lambda$ for which the following equations $\sin \theta x – cos\theta y + (\lambda +1)z = 0$; $\cos\theta x + \sin\theta\, y – \lambda z = 0$;$ \lambda x +(\lambda + 1)y + \cos\theta z = 0$ have non trivial solution, is

normal

The values of $x,y,z$ in order of the system of equations $3x + y + 2z = 3,$ $2x – 3y – z = – 3$, $x + 2y + z = 4,$ are

medium

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medium

If $px^4 + qx^3 + rx^2 + sx + t$ $\equiv$ $\left| {\begin{array}{*{20}{c}}{{x^2}\, + \,\,3x}&{x\, – \,1}&{x\, + \,3}\\{x\, + \,1}&{2\, – \,x}&{x\, – \,3}\\{x\, – \,3}&{x\, + \,4}&{3x}\end{array}} \right|$ then $t =$

normal

Let $a,b,c$ be positive real numbers. The following system of equations in $x, y$ and $ z $ $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} – \frac{{{z^2}}}{{{c^2}}} = 1$, $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1, – \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$ has

hard

(IIT-1995)

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