Let $∆ = \left|\begin{array}{ccc}\mathrm{Ax}& {\mathrm{x}}^2& 1\\ \mathrm{By}& {\mathrm{y}}^2& 1\\ \mathrm{Cz}& {\mathrm{z}}^2& 1\end{array}\right| \mathrm{and} {∆}_1 = \left|\begin{array}{ccc}\mathrm{A}& \mathrm{B}& \mathrm{C}\\ {\mathrm{x}}^2& {\mathrm{y}}^2& {\mathrm{z}}^2\\ 1& 1& 1\end{array}\right|, \mathrm{then} \left|\begin{array}{ccc}\mathrm{Ax}& \mathrm{By}& \mathrm{Cz}\\ {\mathrm{x}}^2& {\mathrm{y}}^2& {\mathrm{z}}^2\\ 1& 1& 1\end{array}\right|$

• ${∆}_1 = 2∆$

• ${∆}_1 = - ∆$

• ${∆}_1 = ∆$

• ${∆}_1 \ne ∆$

The function f(x) = x² + 2x - 5 is strictly increasing in the interval

• $(- \infty , - 1)$

• $[- 1, \infty )$

• $(- \infty , 1]$

• $(- 1, \infty )$

3.

The point on the curve y² = x where the tangent makes an angle of $\mathrm{\pi }$/4 with X-axis is

• (4, 2)

• $\left(\frac{1}{2}, \frac{1}{4}\right)$

• $\left(\frac{1}{4}, \frac{1}{2}\right)$

• (1, 1)

If $\left|\begin{array}{cc}3& \mathrm{x}\\ \mathrm{x}& 1\end{array}\right| = \left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$, then x is equal to

• 4

• 8

• 2

• $\pm 2\sqrt{2}$

If $2\left|\begin{array}{cc}1& 3\\ 0& \mathrm{x}\end{array}\right| + \left|\begin{array}{cc}\mathrm{y}& 0\\ 1& 2\end{array}\right| = \left|\begin{array}{cc}5& 6\\ 1& 8\end{array}\right|$, then the value of x and y are

• x = 3, y = 3

• x = - 3, y = 3

• x = 3, y = - 3

• x = - 3, y = - 3

The range of ${\sec }^{-1}\left(\mathrm{x}\right)$ is

• $\left[0, \mathrm{\pi }\right] - \left\{\frac{\mathrm{\pi }}{2}\right\}$

• $\left[- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right]$

• $\left\{- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right\}$

• $\left[0, \mathrm{\pi }\right]$

If y = $\left|\begin{array}{ccc}\mathrm{f}\left(\mathrm{x}\right)& \mathrm{g}\left(\mathrm{x}\right)& \mathrm{h}\left(\mathrm{x}\right)\\ \mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right|$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\left|\begin{array}{ccc}\mathrm{f}\text{'}\left(\mathrm{x}\right)& \mathrm{g}\text{'}\left(\mathrm{x}\right)& \mathrm{h}\text{'}\left(\mathrm{x}\right)\\ \mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right|$

• $\left|\begin{array}{ccc}\mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{f}\left(\mathrm{x}\right)& \mathrm{g}\left(\mathrm{x}\right)& \mathrm{h}\left(\mathrm{x}\right)\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\end{array}\right|$

• $\left|\begin{array}{ccc}\mathrm{f}\text{'}\left(\mathrm{x}\right)& \mathrm{l}& \mathrm{a}\\ \mathrm{g}\text{'}\left(\mathrm{x}\right)& \mathrm{m}& \mathrm{b}\\ \mathrm{h}\text{'}\left(\mathrm{x}\right)& \mathrm{n}& \mathrm{c}\end{array}\right|$

• $\left|\begin{array}{ccc}\mathrm{l}& \mathrm{m}& \mathrm{n}\\ \mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{f}\text{'}\left(\mathrm{x}\right)& \mathrm{g}\text{'}\left(\mathrm{x}\right)& \mathrm{h}\text{'}\left(\mathrm{x}\right)\end{array}\right|$

If A is a square matrix of order 3 x 3, then $\left|\mathrm{KA}\right|$ is equal to

• ${\mathrm{k}}^2\left|\mathrm{A}\right|$

• $\mathrm{K}\left|\mathrm{A}\right|$

• $3\mathrm{K}\left|\mathrm{A}\right|$

• ${\mathrm{k}}^3\left|\mathrm{A}\right|$

If A = $\frac{1}{\mathrm{\pi }}\left|\begin{array}{cc}{\sin }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)& {\cot }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right|, \mathrm{B} = \left|\begin{array}{cc}- {\cos }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)& - {\tan }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right|$, then A - B is

• 0

• $\frac{1}{2}\mathrm{I}$

• I

• 2I

If f(x) = $$\begin{gathered} \left\{{\mathrm{kx}}^2 \mathrm{if} \mathrm{x} \le 2 \\ 3 \mathrm{if} \mathrm{x} > 2\right\} \end{gathered}$$is continuous at x = 2, then the value of k is

• 3/4

• 4

• 4/3

• 3