If p{λ ^4} + q{λ ^3} + r{λ ^2} + sλ + t = left| {,begin{array}{*{20}{c}}{{λ ^2} + 3λ }&{λ

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3 and 4 .Determinants and Matrices

hard

If $p{\lambda ^4} + q{\lambda ^3} + r{\lambda ^2} + s\lambda + t = $ $\left| {\,\begin{array}{*{20}{c}}{{\lambda ^2} + 3\lambda }&{\lambda - 1}&{\lambda + 3}\\{\lambda + 1}&{2 - \lambda }&{\lambda - 4}\\{\lambda - 3}&{\lambda + 4}&{3\lambda }\end{array}\,} \right|,$ the value of $t$ is

$16$

$18$

$17$

$19$

(IIT-1981)

Solution

(b) Since it is an identity in $\lambda $ so satisfied by every value of $\lambda $.

Now put $\lambda = 0 $ in the given equation, we have $t = \left| {\,\begin{array}{*{20}{c}}0&{ – 1}&3\\1&2&{ – 4}\\{ – 3}&4&0\end{array}\,} \right|\, = – 12 + 30 = 18$.

Standard 12

Mathematics

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

hard

(JEE MAIN-2014)

Let $A_1, A_2, A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A +1, A +2$, respectively. Let $a , b , c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $A_1, A_2, A_3$, respectively such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$ If $a=29$, then the sum of first $20$ terms of an $AP$ whose first term is $c – a – b$ and common difference is $\frac{ d }{12}$, is equal to $……..$.

hard

(JEE MAIN-2023)

If $S$ is the set of distinct values of $'b'$ for which the following system of linear equations $x + y + z = 1;x + ay + z = 1;ax + by + z = 0$ has no solution , then $S$ is :

hard

(JEE MAIN-2017)

If the following system of linear equations

$2 x+y+z=5$

$x-y+z=3$

$x+y+a z=b$

has no solution, then :

hard

(JEE MAIN-2021)

$\left| {\,\begin{array}{*{20}{c}}{1/a}&1&{bc}\\{1/b}&1&{ca}\\{1/c}&1&{ab}\end{array}\,} \right| = $

easy

If $p{\lambda ^4} + q{\lambda ^3} + r{\lambda ^2} + s\lambda + t = $ $\left| {\,\begin{array}{*{20}{c}}{{\lambda ^2} + 3\lambda }&{\lambda - 1}&{\lambda + 3}\\{\lambda + 1}&{2 - \lambda }&{\lambda - 4}\\{\lambda - 3}&{\lambda + 4}&{3\lambda }\end{array}\,} \right|,$ the value of $t$ is

Solution

Similar Questions

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then det. $[( I+ B)^{50} -50B]$ is equal to

If $S$ is the set of distinct values of $'b'$ for which the following system of linear equations $x + y + z = 1;x + ay + z = 1;ax + by + z = 0$ has no solution , then $S$ is :

If the following system of linear equations $2 x+y+z=5$ $x-y+z=3$ $x+y+a z=b$ has no solution, then :

$\left| {\,\begin{array}{*{20}{c}}{1/a}&1&{bc}\\{1/b}&1&{ca}\\{1/c}&1&{ab}\end{array}\,} \right| = $

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If the following system of linear equations

$2 x+y+z=5$

$x-y+z=3$

$x+y+a z=b$

has no solution, then :