If U = [2 - 3 4], X = [0 2 3], $\mathrm{V} = \left[\begin{array}{c}3\\ 2\\ 1\end{array}\right] \mathrm{and} \mathrm{Y} = \left[\begin{array}{c}2\\ 2\\ 4\end{array}\right]$, then UV + XY is equal to

• 20

• [- 20]

• - 20

• [20]

If matrix A = $\left[\begin{array}{cc}1& - 1\\ 1& 1\end{array}\right]$, then

• A' = $\left[\begin{array}{cc}1& 1\\ 1& - 1\end{array}\right]$

• ${\mathrm{A}}^{-1} = \left[\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right]$

• $\mathrm{A} . \left[\begin{array}{cc}1& 1\\ - 1& 1\end{array}\right] = 2\mathrm{I}$

• $\mathrm{\lambda A} = \left[\begin{array}{cc}\mathrm{\lambda }& - \mathrm{\lambda }\\ - 1& 1\end{array}\right]$

If $\left|\begin{array}{ccc}- {\mathrm{a}}^2& \mathrm{ab}& \mathrm{ac}\\ \mathrm{ab}& - {\mathrm{b}}^2& \mathrm{bc}\\ \mathrm{ac}& \mathrm{bc}& - {\mathrm{c}}^2\end{array}\right|$ = ka²b²c², then k is equal to

• 2

• 4

• - 4

• 8

If $\left|\begin{array}{ccc}6\mathrm{i}& - 3\mathrm{i}& 1\\ 4& 3\mathrm{i}& - 1\\ 20& 3& \mathrm{i}\end{array}\right|$ = x + iy, then (x, y) is

• (3, 1)

• (1, 3)

• (0, 3)

• (0, 0)

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

• $\frac{\mathrm{y}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) - \mathrm{y}\right)}{\mathrm{x}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) - \mathrm{x}\right)}$

• $\frac{\mathrm{x}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) + \mathrm{y}\right)}{\mathrm{y}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) + \mathrm{x}\right)}$

• $\frac{\mathrm{x}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) - \mathrm{y}\right)}{\mathrm{y}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) - \mathrm{x}\right)}$

• $\frac{\mathrm{y}\left({\mathrm{xlog}}_{\mathrm{e}}\left(\mathrm{y}\right) + \mathrm{y}\right)}{\mathrm{x}\left({\mathrm{ylog}}_{\mathrm{e}}\left(\mathrm{x}\right) + \mathrm{x}\right)}$

If y = ${\mathrm{e}}^{\left(1 + {\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• e

• 1

• 0

• ${\log }_{\mathrm{e}}\left(\mathrm{x}\right){\mathrm{e}}^{{\log }_{\mathrm{e}}\left(\mathrm{ex}\right)}$

The value of m for which the function f(x) = $$\begin{gathered} \left\{{\mathrm{mx}}^2, \mathrm{x} \le 1 \\ 2\mathrm{x}, \mathrm{x} > 1\right\} \end{gathered}$$ is differentiable at x = 1, is

• 0

• 1

• 2

• does not exist

18.

The function x⁵ - 5x⁴ + 5x³ - 10 has a maximum, when x is equal to

• 3

• 2

• 1

• 0

If u = $2\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - \hat{\mathrm{k}}$ and v = $6\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$, then the unit vector perpendicular to u and v is

• $\hat{\mathrm{i}} - 10\hat{\mathrm{j}} - 18\hat{\mathrm{k}}$

• $\frac{1}{\sqrt{17}}\left(\frac{1}{5}\hat{\mathrm{i}} - 2\hat{\mathrm{j}} - \frac{18}{5}\hat{\mathrm{k}}\right)$

• $\frac{1}{\sqrt{473}}\left(7\hat{\mathrm{i}} - 10\hat{\mathrm{j}} - 18\hat{\mathrm{k}}\right)$

• None of the above

The ratio in which the xy-plane divides the join of (a, b, c) and (- a, - c, - b), is

• a : b

• b : c

• c : a

• c : b