Consider the following statements

(i) If any two rows or columns of a determinant are identical, then the value of the determinant is zero.
(ii) Ifthe corresponding rows and columns of a determinant are interchanged, then the value of determmant does not change.
(iii) If any two rows (or columns) of a determinant are mterchanged, then the value of the determinant changes in sign.

Which of these are correct?

• (i) and (iii)

• (i) and (ii)

• (i), (ii) and (iii)

• (ii) and (iii)

If P = $\left|\begin{array}{cc}\mathrm{x}& 1\\ 1& \mathrm{x}\end{array}\right|$ and Q = $\left|\begin{array}{ccc}\mathrm{x}& 1& 1\\ 1& \mathrm{x}& 1\\ 1& 1& \mathrm{x}\end{array}\right|$, then $\frac{\mathrm{dQ}}{\mathrm{dX}}$ is equal to

• - 3P

• 3P + 1

• 3P

• 1 - 3P

If $\left|\mathrm{A}\right| \ne 0$ and A is of order n, then adj(adj A) is equal to

• ${\left|\mathrm{A}\right|}^{\mathrm{n}}$

• ${\left|\mathrm{A}\right|}^2$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 1} . \mathrm{I}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 2} . \mathrm{A}$

If A is any non-zero square matrix of order n, then A (adj A) is equal to

• I

• $\left|\mathrm{A}\right|$I

• ${\left|\mathrm{A}\right|}^{\mathrm{n}}$I

• None of these

If A = $\left[\begin{array}{cc}1& \tan \left(\mathrm{x}\right)\\ - \tan \left(\mathrm{x}\right)& 1\end{array}\right]$, then the value of $\left|{\mathrm{A}}^{\mathrm{T}}{\mathrm{A}}^{- 1}\right|$ is

• $\cos \left(4\mathrm{x}\right)$

• ${\sec }^2\left(\mathrm{x}\right)$

• $- \cos \left(4\mathrm{x}\right)$

• 1

If A = $\left[\begin{array}{cc}4& 11\\ 2& 6\end{array}\right]$, then A^-1 is equal to

• $\left[\begin{array}{cc}- 1& \frac{- 11}{2}\\ 3& 2\end{array}\right]$

• $\left[\begin{array}{cc}3& \frac{- 11}{2}\\ - 1& 2\end{array}\right]$

• $\left[\begin{array}{cc}3& 2\\ - 1& \frac{- 11}{2}\end{array}\right]$

• None of these

The area of the triangle formed by the points (a, b + c), (b, c + a) and (c, a + b) will be

• abc

• a² + b² + c²

• ab + bc + ca

• 0

$\mathrm{If} \mathrm{A} = \left[\begin{array}{ccc}1& 0& 1\\ 2& 1& 0\\ 3& 2& 1\end{array}\right], \mathrm{then} \det \mathrm{A} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 2

• 5

• 3

• 4

$$\begin{gathered} \mathrm{If} {\mathrm{x}}^2 + {\mathrm{y}}^2 + {\mathrm{z}}^2 \ne 0, \mathrm{x} = \mathrm{cy} + \mathrm{bz}, \mathrm{y}= \mathrm{az} + \mathrm{cx} \mathrm{and} \mathrm{z} = \mathrm{bx} +\mathrm{ay}, \\ \mathrm{then} {\mathrm{a}}^2 + {\mathrm{b}}^2 +{\mathrm{c}}^2 +2\mathrm{abc} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• 1

• 2

• a + b + c

• ab + bc + ca

$\mathrm{adj}\left[\begin{array}{ccc}1& 0& 2\\ - 1& 1& - 2\\ 0& 2& 1\end{array}\right] = \left[\begin{array}{ccc}5& \mathrm{a}& - 2\\ 1& 1& 0\\ - 2& - 2& \mathrm{b}\end{array}\right], \mathrm{then} \left[\mathrm{a} \mathrm{b}\right] \mathrm{is} \mathrm{equal} \mathrm{to}$

• [- 4 1]

• [- 4 - 1]

• [4 1]

• [4 - 1]