If left|begin{array}{ccc}x+1 & x & x x & x+λ & x x & x & x+λ^2end{array}ri

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

hard

If $\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^2\end{array}\right|=\frac{9}{8}(103 x+81)$, then $\lambda$, $\frac{\lambda}{3}$ are the roots of the equation

A

$4 x ^2+24 x -27=0$

B

$4 x ^2-24 x +27=0$

C

$4 x ^2+24 x +27=0$

D

$4 x ^2-24 x -27=0$

(JEE MAIN-2023)

Solution

Put $x =0$

$\begin{aligned}& \left|\begin{array}{ccc}1 & 0 & 0 \\0 & \lambda & 0 \\0 & 0 & \lambda^2\end{array}\right|\end{aligned}=\frac{9}{8} \times 81$

$\lambda^3=\frac{9^3}{8} \therefore \lambda=\frac{9}{2}$

$\therefore \frac{\lambda}{3}=\frac{3}{2}$

$\therefore$ Required equation is : $x^2-x\left(\frac{9}{2}+\frac{3}{2}\right) x +\frac{27}{4}=0$ $4 x^2-24 x+27=0$

Standard 12

Mathematics

Share

Similar Questions

The values of $\lambda$ and $\mu$ for which the system of linear equations

$x+y+z=2$

$x+2 y+3 z=5$

$x+3 y+\lambda z=\mu$

has infinitely many solutions are, respectively

medium

(JEE MAIN-2020)

${x_1} + 2{x_2} + 3{x_3} = a\ \ ;\ \ 2{x_1} + 3{x_2} + {x_3} =b\ \ ;\ \ $ $3{x_1} + {x_2} + 2{x_3} = c$ this system of equations has

medium

Evaluate the determinant $\Delta=\left|\begin{array}{rrr}1 & 2 & 4 \\ -1 & 3 & 0 \\ 4 & 1 & 0\end{array}\right|$

easy

If $x, y, z$ are in arithmetic progression with common difference $d , x \neq 3 d ,$ and the
determinant of the matrix $\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]$ is zero, then the value of $k ^{2}$ is ….. .

hard

(JEE MAIN-2021)

Let for any three distinct consecutive terms $a, b, c$ of an $A.P,$ the lines $a x+b y+c=0$ be concurrent at the point $\mathrm{P}$ and $\mathrm{Q}(\alpha, \beta)$ be a point such that the system of equations $ x+y+z=6, $ $ 2 x+5 y+\alpha z=\beta$ and $x+2 y+3 z=4$, has infinitely many solutions. Then $(P Q)^2$ is equal to________.

hard

(JEE MAIN-2024)