The value of $\left|\begin{array}{ccc}\mathrm{y} +\hspace{0.17em}\mathrm{z}& \mathrm{x}& \mathrm{x}\\ \mathrm{y}& \mathrm{z} + \mathrm{x}& \mathrm{y}\\ \mathrm{z}& \mathrm{z}& \mathrm{x} +\hspace{0.17em}\mathrm{y}\end{array}\right|$ is

• xyz

• x²y²z²

• 4xyz

• 4x²y²z²

If A and B are square matrices of order 3 such that $\left|\mathrm{A}\right|$ = - 1, $\left|\mathrm{B}\right|$ = 3, then $\left|3\mathrm{AB}\right|$ is equal to

• - 9

• - 81

• - 27

• 81

Let A = $\left[\begin{array}{ccc}0& 0& - 1\\ 0& - 1& 0\\ - 1& 0& 0\end{array}\right]$, the only correct statement about the matrix A, is

• A is a zero matrix

• A = (- 1)I, where I is a unit matrix

• A^{- 1} does not exist

• A² = I

If A = $\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$, then A² is equal to

• 5A

• 10A

• 16A

• 32A

The trigonometric equation ${\sin }^{-1}\left(\mathrm{x}\right) = 2{\sin }^{-1}\left(\mathrm{a}\right)$, has a solution for

• $\frac{1}{2} < \left|\mathrm{a}\right| \le \frac{1}{\sqrt{2}}$

• all real values of a

• $\left|\mathrm{a}\right| \le \frac{1}{\sqrt{2}}$

• $\left|\mathrm{a}\right| \ge \frac{1}{\sqrt{2}}$

${\tan }^{-1}\left(\frac{1}{2}\right) + {\tan }^{-1}\left(\frac{1}{3}\right)$ is equal to

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

The equation of normal to the circle 2x² + 2y² - 2 - 5y + 3 = 0 at (1, 1) is

• 2x + y = 3

• x - 2y = 3

• x + 2y = 3

• None of these

A function f from the set of natural numbers to integers defined by f(n) = $$\begin{gathered} \left\{\frac{\mathrm{n} - 1}{2}, \mathrm{when} \mathrm{n} \mathrm{is} \mathrm{odd} \\ - \frac{\mathrm{n}}{2}, \mathrm{when} \mathrm{n} \mathrm{is} \mathrm{even}\right\} \end{gathered}$$, is

• one - one but not onto

• onto but not one - one

• one - one and onto both

• neither one - one nor onto

9.

If x^y = e^{x - y}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is

• $\frac{1 + \mathrm{x}}{1 + \log \left(\mathrm{x}\right)}$

• $\frac{1 - \log \left(\mathrm{x}\right)}{1 + \log \left(\mathrm{x}\right)}$

• not defined

• $\frac{\log \left(\mathrm{x}\right)}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}$

If f(x) = xⁿ, then the value of $\mathrm{f}\left(1\right) - \frac{\mathrm{f}\text{'}\left(1\right)}{1!} + \frac{\mathrm{f}\text{'}\text{'}\left(1\right)}{2!} - \frac{\mathrm{f}\text{'}\text{'}\text{'}\left(1\right)}{3!} + ... + \frac{{\left(- 1\right)}^{\mathrm{n}}\mathrm{f}{\text{'}}^{\mathrm{n}}\left(1\right)}{\mathrm{n}!}$ is

• 2ⁿ

• 2^{n - 1}

• 0

• 1