For all values of A, B, C and P, Q, R,the val | vedclass

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

Question

(A) The given determinant is $\Delta = \left| \begin{array}{ccc} \cos(A-P) & \cos(A-Q) & \cos(A-R) \ \cos(B-P) & \cos(B-Q) & \cos(B-R) \ \cos(C-P) &

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(A) The given determinant is $\Delta = \left| \begin{array}{ccc} \cos(A-P) & \cos(A-Q) & \cos(A-R) \ \cos(B-P) & \cos(B-Q) & \cos(B-R) \ \cos(C-P) & \cos(C-Q) & \cos(C-R) \end{array} ight|$.Using the formula $\cos(x-y) = \cos x \cos y + \sin x \sin y$,we can write each element as a sum of two terms.This allows us to express the determinant as the product of two matrices:$\Delta = \left| \begin{array}{ccc} \cos A & \sin A & 0 \ \cos B & \sin B & 0 \ \cos C & \sin C & 0 \end{array} ight| 	imes \left| \begin{array}{ccc} \cos P & \cos Q & \cos R \ \sin P & \sin Q & \sin R \ 0 & 0 & 0 \end{array} ight|$.Since the third column of the first matrix is all zeros and the third row of the second matrix is all zeros,the product of these two matrices is zero.Alternatively,the determinant can be split into $8$ determinants,each of which has at least two identical columns (or rows),making the value of each determinant $0$.Thus,the value of the determinant is $0$.

For all values of $A, B, C$ and $P, Q, R$,the value of $\left| \begin{array}{ccc} \cos(A-P) & \cos(A-Q) & \cos(A-R) \\ \cos(B-P) & \cos(B-Q) & \cos(B-R) \\ \cos(C-P) & \cos(C-Q) & \cos(C-R) \end{array} \right|$ is

Similar Questions

If $\omega$ is a complex cube root of unity,then the value of the determinant $\left| \begin{array}{ccc} 2 & 2\omega & -\omega^2 \\ 1 & 1 & 1 \\ 1 & -1 & 0 \end{array} \right|$ is:

Let $S$ be the set of all values of $\theta \in [-\pi, \pi]$ for which the system of linear equations
$x + y + \sqrt{3} z = 0$
$-x + (\tan \theta) y + \sqrt{7} z = 0$
$x + y + (\tan \theta) z = 0$
has a non-trivial solution. Then $\frac{120}{\pi} \sum_{\theta \in S} \theta$ is equal to

The number of distinct real roots of $\left| {\begin{array}{*{20}{c}}{\sin x}&{\cos x}&{\cos x}\\{\cos x}&{\sin x}&{\cos x}\\{\cos x}&{\cos x}&{\sin x}\end{array}} \right| = 0$ in the interval $-\frac{\pi}{4} \le x \le \frac{\pi}{4}$ is

If $\left| \begin{array}{ccc} a & b & a\alpha - b \\ b & c & b\alpha - c \\ 2 & 1 & 0 \end{array} \right| = 0$ and $\alpha \neq \frac{1}{2}$,then

If $A=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4\end{array}\right]$,then show that $|3 A|=27|A|$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module