If left| {begin{array}{*{20}{c}}1&a&{{a^2}}1&x&{{x^2}}{{b^2}}&{ab}&{{a^2}} e

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3 and 4 .Determinants and Matrices

normal

If $\left| {\begin{array}{*{20}{c}}1&a&{{a^2}}\\1&x&{{x^2}}\\{{b^2}}&{ab}&{{a^2}} \end{array}} \right|$ $= 0$ , then :

$x = a$

$x = b$

$x = \frac{a}{b}$

both $(A)$ and $(C)$

Solution

$R_2 \rightarrow R_2 -R_1$ and $R_3 \rightarrow R_3 – R_1$ gives
$(x – a) (b – 1)$ $\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2}}\\0&1&{x + a}\\ {b + 1}&a&0\end{array}\,} \right|$ open by $c_1$ and get the value of $x = a/b, x = a$

Standard 12

Mathematics

Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.

Statement $-1$ : If the graphs of two linear equations in two variables are neither parallel nor the same, then there is a unique solution to the system. Statement $-2$ : If the system of equations $ax + by = 0, cx + dy = 0$ has a non-zero solution, then it has infinitely many solutions.

Statement $-3$ : The system $x + y + z = 1, x = y, y = 1 + z$ is inconsistent. Statement $-4$ : If two of the equations in a system of three linear equations are inconsistent, then the whole system is inconsistent.

normal

If $a, b, c$ are all different and $\left| {\begin{array}{*{20}{c}}a&{{a^3}}&{{a^4}\, – \,1}\\b&{{b^3}}&{{b^4}\, – \,1}\\c&{{c^3}}&{{c^4}\, – \,1}\end{array}} \right|$ $= 0$ , then :

normal

$\left| {\,\begin{array}{*{20}{c}}{a + b}&{a + 2b}&{a + 3b}\\{a + 2b}&{a + 3b}&{a + 4b}\\{a + 4b}&{a + 5b}&{a + 6b}\end{array}\,} \right| = $

medium

(IIT-1986)

Which of the following matrices can $NOT$ be obtained from the matrix $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$ by a single elementary row operation?

easy

(JEE MAIN-2022)

$\left| {\,\begin{array}{*{20}{c}}{1 + x}&1&1\\1&{1 + y}&1\\1&1&{1 + z}\end{array}\,} \right| = $

hard

If $\left| {\begin{array}{*{20}{c}}1&a&{{a^2}}\\1&x&{{x^2}}\\{{b^2}}&{ab}&{{a^2}} \end{array}} \right|$ $= 0$ , then :

Solution

Similar Questions

If $a, b, c$ are all different and $\left| {\begin{array}{*{20}{c}}a&{{a^3}}&{{a^4}\, – \,1}\\b&{{b^3}}&{{b^4}\, – \,1}\\c&{{c^3}}&{{c^4}\, – \,1}\end{array}} \right|$ $= 0$ , then :

$\left| {\,\begin{array}{*{20}{c}}{a + b}&{a + 2b}&{a + 3b}\\{a + 2b}&{a + 3b}&{a + 4b}\\{a + 4b}&{a + 5b}&{a + 6b}\end{array}\,} \right| = $

Which of the following matrices can $NOT$ be obtained from the matrix $\left[\begin{array}{cc}-1 & 2 \\ 1 & -1\end{array}\right]$ by a single elementary row operation?

$\left| {\,\begin{array}{*{20}{c}}{1 + x}&1&1\\1&{1 + y}&1\\1&1&{1 + z}\end{array}\,} \right| = $

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