Evaluate the determinant: left| begin{array}{ | vedclass

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

Question

(A) Let $\Delta = \left| \begin{array}{ccc} \sin^2 x & \cos^2 x & 1 \ \cos^2 x & \sin^2 x & 1 \ -10 & 12 & 2 \end{array} ight|$.Applying the colum

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) Let $\Delta = \left| \begin{array}{ccc} \sin^2 x & \cos^2 x & 1 \ \cos^2 x & \sin^2 x & 1 \ -10 & 12 & 2 \end{array} ight|$.Applying the column operation $C_1 	o C_1 + C_2$:$\Delta = \left| \begin{array}{ccc} \sin^2 x + \cos^2 x & \cos^2 x & 1 \ \cos^2 x + \sin^2 x & \sin^2 x & 1 \ -10 + 12 & 12 & 2 \end{array} ight|$Since $\sin^2 x + \cos^2 x = 1$,we have:$\Delta = \left| \begin{array}{ccc} 1 & \cos^2 x & 1 \ 1 & \sin^2 x & 1 \ 2 & 12 & 2 \end{array} ight|$In this determinant,column $C_1$ and column $C_3$ are identical (both are $\begin{bmatrix} 1 \ 1 \ 2 \end{bmatrix}$).According to the properties of determinants,if any two columns are identical,the value of the determinant is $0$.Therefore,$\Delta = 0$.

Evaluate the determinant: $\left| \begin{array}{ccc} \sin^2 x & \cos^2 x & 1 \\ \cos^2 x & \sin^2 x & 1 \\ -10 & 12 & 2 \end{array} \right|$

Similar Questions

Evaluate the determinant: $\left| \begin{array}{ccc} x & 4 & y + z \\ y & 4 & z + x \\ z & 4 & x + y \end{array} \right|$

$\left| {\begin{array}{*{20}{c}} 1 & 5 & \pi \\ {{\log }_e}e & 5 & {\sqrt 5 } \\ {{\log }_{10}}10 & 5 & e \end{array}} \right| = $

$f(x) = \left| \begin{array}{ccc} 1 & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) \end{array} \right|$, then $f(100)$ is equal to:

The total number of distinct $x \in \mathbb{R}$ for which $\left|\begin{array}{ccc}x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3\end{array}\right|=10$ is

The determinant $\left|\begin{array}{ccc}a^{2}+10 & a b & a c \\ a b & b^{2}+10 & b c \\ a c & b c & c^{2}+10\end{array}\right|$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module