begin{aligned} & text { If }left|begin{array} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given the determinant $\Delta = \left|\begin{array}{ccc}n^2 & (n+1)^2 & (n+2)^2 \ (n+1)^2 & (n+2)^2 & (n+3)^2 \ (n+2)^2 & (n+3)^2 & (n+4)^2\end{

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(A) Given the determinant $\Delta = \left|\begin{array}{ccc}n^2 & (n+1)^2 & (n+2)^2 \ (n+1)^2 & (n+2)^2 & (n+3)^2 \ (n+2)^2 & (n+3)^2 & (n+4)^2\end{array}ight|$.Applying row operations $R_2 ightarrow R_2 - R_1$ and $R_3 ightarrow R_3 - R_2$,we get:$\Delta = \left|\begin{array}{ccc}n^2 & (n+1)^2 & (n+2)^2 \ 2n+1 & 2n+3 & 2n+5 \ 2 & 2 & 2\end{array}ight|$.Applying $R_2 ightarrow R_2 - R_1$ again,we get:$\Delta = \left|\begin{array}{ccc}n^2 & 2n+1 & 2 \ 2n+1 & 2 & 0 \ 2 & 0 & 0\end{array}ight| = 2(0 - 4) = -8$.Now,given the second determinant equation:$\left|\begin{array}{ccc}1 & -4 & 7 \ -2 & 3 & -5 \ 3 & x & -3\end{array}ight| = 2\Delta + 1 = 2(-8) + 1 = -15$.Expanding the determinant:$1(-9 + 5x) + 4(6 + 15) + 7(-2x - 9) = -15$.$-9 + 5x + 84 - 14x - 63 = -15$.$-9x + 12 = -15$.$-9x = -27$.$x = 3$.

Similar Questions

If $\left| \begin{array}{ccc} x + 1 & x + 2 & x + 3 \\ x + 2 & x + 3 & x + 4 \\ x + a & x + b & x + c \end{array} \right| = 0$,then $a, b, c$ are in

$\left| {\begin{array}{{20}{c}}{{{\log }_3}512}&{{{\log }_4}3}\\{{{\log }_3}8}&{{{\log }_4}9}\end{array}} \right| \times \left| {\begin{array}{{20}{c}}{{{\log }_2}3}&{{{\log }_8}3}\\{{{\log }_3}4}&{{{\log }_3}4}\end{array}} \right| = $

If $f(x)=\left|\begin{array}{ccc}x-3 & 2x^2-18 & 2x^3-81 \\ x-5 & 2x^2-50 & 4x^3-500 \\ 1 & 2 & 3\end{array}\right|$,then the value of $f(1) \cdot f(3)+f(3) \cdot f(5)+f(5) \cdot f(1)$ is:

If $D = \left| \begin{array}{ccc} \frac{1}{z} & \frac{1}{z} & -\frac{(x+y)}{z^2} \\ -\frac{(y+z)}{x^2} & \frac{1}{x} & \frac{1}{x} \\ -\frac{y(y+z)}{x^2z} & \frac{x+2y+z}{xz} & -\frac{y(x+y)}{xz^2} \end{array} \right|$,then the incorrect statement is:

If $\begin{bmatrix} 2 + x & 3 & 4 \\ 1 & -1 & 2 \\ x & 1 & -5 \end{bmatrix}$ is a singular matrix,then $x$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module