Let a ,b ,c be such that b + c ne 0 if left| {begin{array}{*{20}{c}}a&{a + 1}&{a

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3 and 4 .Determinants and Matrices

medium

Let $a ,b ,c $ be such that $b + c \ne 0$ if

$\left| {\begin{array}{{20}{c}}a&{a + 1}&{a - 1}\\{ - b}&{b + 1}&{b - 1}\\c&{c - 1}&{c + 1}\end{array}} \right| + \left| {\begin{array}{{20}{c}}{a + 1}&{b + 1}&{c - 1}\\{a - 1}&{b - 1}&{c + 1}\\{{{\left( { - 1} \right)}^{n + 2}} \cdot a}&{{{\left( { - 1} \right)}^{n + 1}} \cdot b}&{{{\left( { - 1} \right)}^n} \cdot c}\end{array}} \right| = 0$ then $n$ equals to

Zero

any even integer

any odd integer

any integer

(AIEEE-2009)

Solution

$\left| {\begin{array}{*{20}{c}}
a&{a + 1}&{a – 1}\\
{ – b}&{b + 1}&{b – 1}\\
c&{c – 1}&{c + 1}
\end{array}} \right| + \left| {\begin{array}{*{20}{c}}
{{{\left( { – 1} \right)}^{n + 2}}a}&{a + 1}&{a – 1}\\
{{{\left( { – 1} \right)}^{n + 1}}b}&{b + 1}&{b – 1}\\
{{{\left( { – 1} \right)}^n}c}&{c – 1}&{c + 1}
\end{array}} \right|$

$ = \left| {\begin{array}{*{20}{c}}
{a + {{\left( { – 1} \right)}^{n + 2}}a}&{a + 1}&{a – 1}\\
{ – b + {{\left( { – 1} \right)}^{n + 1}}b}&{b + 1}&{b – 1}\\
{c + {{\left( { – 1} \right)}^n}c}&{c – 1}&{c + 1}
\end{array}} \right|$

$=0$ if $n$ is an obb integer.

Standard 12

Mathematics

For how many diff erent values of $a$ does the following system have at least two distinct solutions?

$a x+y=0$

$x+(a+10) y=0$

medium

(KVPY-2017)

If the lines $x + 2ay + a = 0, x + 3by + b = 0$ and $x + 4cy + c = 0$ are concurrent, then $a, b$ and $c$ are in :-

normal

Let $A = \left[ {\begin{array}{*{20}{c}}5&{5\alpha }&\alpha \\0&\alpha &{5\alpha }\\0&0&5\end{array}} \right]$, If ${\left| A \right|^2} = 25$, then $\left| \alpha \right|$ equals

medium

(AIEEE-2007)

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

normal

Let $\omega = – \frac{1}{2} + i\frac{{\sqrt 3 }}{2}$. Then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{ – 1 – {\omega ^2}}&{{\omega ^2}}\\1&{{\omega ^2}}&{{\omega ^4}}\end{array}\,} \right|$ is

easy

(IIT-2002)

Solution

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For how many diff erent values of $a$ does the following system have at least two distinct solutions?

$a x+y=0$

$x+(a+10) y=0$