If A = $\left[\begin{array}{ccc}1& 1& 0\\ 2& 1& 5\\ 1& 2& 1\end{array}\right]$, then a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ is equal to

• 1

• 0

• - 1

• 2

If Rolle's theorem for f(x) = ${\mathrm{e}}^{\mathrm{x}}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right)$ is verified on $\left[\frac{\mathrm{\pi }}{4}, \frac{5\mathrm{\pi }}{4}\right]$, then the value of c is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

If $2{\tan }^{-1}\left(\cos \left(\mathrm{x}\right)\right) = {\tan }^{-1}\left(2\csc \left(\mathrm{x}\right)\right)$, then $\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)$ is equal to

• $2\sqrt{2}$

• $\sqrt{2}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

The approximate value of f(x) = x³ + 5x² - 7x + 9 at x = 1.1 is

• 8.6

• 8.5

• 8.4

• 8.3

5.

The point on the curve 6y = x³ + 2 at which y-coordinate is changing 8 times as fast as x-coordinate is

• (4, 11)

• (4, - 11)

• (- 4, 11)

• (- 4, - 11)

If the function f(x) defined by

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right), \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{for} \mathrm{x} = 0\right\} \end{gathered}$$

is continuous at x = 0, then k is equal to

• 0

• 1

• - 1

• $\frac{1}{2}$

If $\mathrm{y} = {\mathrm{e}}^{\mathrm{m}{\sin }^{-1}\left(\mathrm{x}\right)} \mathrm{and} \left(1 - {\mathrm{x}}^2\right){\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2$ = Ay², then A is equal to

• m

• - m

• m²

• - m²

$\frac{{\tan }^{-1}\left(\sqrt{3}\right) - {\sec }^{-1}\left(- 2\right)}{{\csc }^{-1}\left(- \sqrt{2}\right) + {\cos }^{-1}\left(- {\displaystyle \frac{1}{2}}\right)}$ is equal to

• $\frac{4}{5}$

• $- \frac{4}{5}$

• $\frac{3}{5}$

• 0

For what value of k, the function defined by

f(x) = $$\begin{gathered} \left\{\frac{\log \left(1 + 2\mathrm{x}\right)\sin \left(\mathrm{x}°\right)}{{\mathrm{x}}^2}, \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{k} , \mathrm{for} \mathrm{x} = 0\right\} \end{gathered}$$

is continuous at x = 0 ?

• 2

• $\frac{1}{2}$

• $\frac{\mathrm{\pi }}{90}$

• $\frac{90}{\mathrm{\pi }}$

If ${\log }_{10}\left(\frac{{\mathrm{x}}^2 - {\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right) = 2$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $- \frac{99\mathrm{x}}{101\mathrm{y}}$

• $\frac{99\mathrm{x}}{101\mathrm{y}}$

• $- \frac{99\mathrm{y}}{101\mathrm{x}}$

• $\frac{99\mathrm{y}}{101\mathrm{x}}$