Special Determinants

Q. If $A+B+C=\pi$, then $\left|\begin{array}{ccc}sin(A+B+C)& sinB& cosC\\ -sinB& 0& tanA\\ cos(A+B)& -tnaA& 0\end{array}\right|$ is equal to

1. 1
2. \N
3. -1
4. 2

Q. $\left|\begin{array}{ccc}0& a& -b\\ -a& 0& c\\ b& -c& 0\end{array}\right|=$

1. -2abc
2. abc
3. \N
4. $a^2+b^2+c^2$

Q.

$\text{Given f (x)=}\left|\begin{array}{ccc}1& x& x^2\\ 1& a& a^2\\ 1& b& b^2\end{array}\right|\text{Then, is (x - a) a factor to this determinant}$

1. True

2. False

Q.

$\text{Given f (x)=}\left|\begin{array}{ccc}1& x& x^2\\ 1& a& a^2\\ 1& b& b^2\end{array}\right|\text{Then, is (x - a) a factor to this determinant}$

1. True

2. False

Q.

$\text{Express the det}A=\left|\begin{array}{ccc}a& bc& abc\\ b& ca& abc\\ c& ab& abc\end{array}\right|\text{as product of factors}$

1. $abc(a+b)(b+c)(c+a)$

2. $abc(a-b)(b-c)(c-a)$

3. $ab(a-b)(b-c)$

4. $abc(a+b)(b-c)(c-a)$

Q. $\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=$

1. $a^2+b^2+c^2$
2. (a+b) (b+c) (c+a)
3. (a-b) (b-c) (c-a)
4. None of these