Properties of Determinants

Q. If $\omega$ is the cube root of unity, then $\left|\begin{array}{ccc}1& \omega & {\omega }^2\\ \omega & {\omega }^2& 1\\ {\omega }^2& 1& \omega \end{array}\right|=$

1. $1$
2. $0$
3. $\omega$
4. ${\omega }^2$

Q. In a triangle ABC, the value of the determinant $\left|\begin{array}{ccc}sin\frac{A}{2}& sin\frac{B}{2}& sin\frac{c}{2}\\ sin(A+B+C)& sin\frac{B}{2}& cos\frac{A}{2}\\ cos\left(\frac{A+B+C}{2}\right)& tan(A+B+C)& sin\frac{C}{2}\end{array}\right|$ is less than or equal to

1. $\frac{1}{2}$
2. $\frac{1}{4}$
3. $\frac{1}{8}$
4. None of these

Q.

If $Z=\left|\begin{array}{ccc}2& 5-i& 7+i\\ 5+i& 2& 3-i\\ 7-i& 3+i& 7\end{array}\right|$ and arg (z) = $\theta$ then $\theta$ = ___

Q. The determinant $\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}\right|$ is not equal to

1. $\left|\begin{array}{ccc}2& 1& 1\\ 2& 2& 3\\ 2& 3& 6\end{array}\right|$
2. $\left|\begin{array}{ccc}2& 1& 1\\ 3& 2& 3\\ 4& 3& 6\end{array}\right|$
3. $\left|\begin{array}{ccc}1& 2& 1\\ 1& 5& 3\\ 1& 9& 6\end{array}\right|$
4. $\left|\begin{array}{ccc}3& 1& 1\\ 6& 2& 3\\ 10& 3& 6\end{array}\right|$

Q. If $\left|\begin{array}{ccc}x+1& 3& 5\\ 2& x+2& 5\\ 2& 3& x+4\end{array}\right|=0,$ then x =

1. 1, 9
2. -1, -9
3. 1, -9
4. -1, 9

Q. If a+b+c=0, then the solution of the equation $\left|\begin{array}{ccc}a-x& c& b\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$ is

1. $0$
2. $\pm \frac{3}{2}(a^2+b^2+c^2)$
3. $0,\pm \sqrt{\frac{3}{2}(a^2+b^2+c^2)}$
4. $0,\pm \sqrt{(a^2+b^2+c^2)}$

Q. $\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|=$

1. $(a+b+c{)}^2$
2. $(a+b+c{)}^3$
3. $(a+b+c)(ab+bc+ca)$
4. None of these

Q. The value of the determinant $\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$ is

1. a+b+c
2. $(a+b+c{)}^2$
3. 0
4. 1+a+b+c

Q.

1. Circle

2. Ellipse

3. Parabola

4. Straight line

Q.

\( \text { If a point }(x, y) \text { moves on a curve and satisfies the equation }\left|\begin{array}{ccc} a & b & a x+b y \\ b & c & b x+c y \\ a x+b y & b x+a y & 0 \end{array}\right|=0 \) Then,