For any two real numbers a and b, we define a R b if and only if sin²(a) + cos²(b) = 1. The relation R is

• reflexive but not symmetric

• symmetric but not transitive

• transitive but not reflexive

• an equivalence relation

For the curve x² + 4xy + 8y = 64 the tangents are parallel to the x-axis only at the points

• $\left(0, 2\sqrt{2}\right) \mathrm{and} \left(0, - 2\sqrt{2}\right)$

• (8, - 4) and (- 8, 4)

• $\left(8\sqrt{2}, - 2\sqrt{2}\right) \mathrm{and} \left(- 8\sqrt{2}, 2\sqrt{2}\right)$

• (9, 0) and (- 8, 0)

If f(x) = $$\begin{gathered} \left\{{\mathrm{x}}^3 - 3\mathrm{x} + 2, \mathrm{x} < 2, \\ {\mathrm{x}}^3 - 6{\mathrm{x}}^2 + 9\mathrm{x} + 2, \mathrm{x} \ge 2\right\} \end{gathered}$$

then

• $\underset{\mathrm{x}\to 2}{\lim }\mathrm{f}\left(\mathrm{x}\right)$ does not exist

• f is not continuous at x = 2

• f is continuous but not differentiable at x = 2

• f is continuous and differentiable at x = 2

Let exp (x) denote the exponential function e^x. If f (x) = $\exp \left({\mathrm{x}}^{\frac{1}{\mathrm{x}}}\right)$, x > 0, then the minimum value off in the interval [2, 5] is

• $\exp \left({\mathrm{e}}^{\frac{1}{\mathrm{e}}}\right)$

• $\exp \left(2^{\frac{1}{2}}\right)$

• $\exp \left(5^{\frac{1}{5}}\right)$

• $\exp \left(3^{\frac{1}{3}}\right)$

The minimum value of the function f (x) = $2\left|\mathrm{x} - 1\right| + \left|\mathrm{x} - 2\right|$ is

• 0

• 1

• 2

• 3

If P = $\left(\begin{array}{ccc}2& - 2& - 4\\ - 1& 3& 4\\ 1& - 2& - 3\end{array}\right)$, then P⁵ is equal to

• P

• 2P

• - P

• - 2P

67.

Consider the system of equations x + y + z = 0, $\mathrm{\alpha x} + \mathrm{\beta y} + \mathrm{\gamma z} = 0$ and ${\mathrm{\alpha }}^2\mathrm{x} + {\mathrm{\beta }}^2\mathrm{y} + {\mathrm{\gamma }}^2\mathrm{z} = 0$. Then, the system of equations has

• a unique solution for all values of $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$

• infinite number of solutions, if any two of $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ are equal.

• a unique solution, if $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$ are distinct

• more than one, but finite number of solutions depending on values of $\mathrm{\alpha }, \mathrm{\beta } \mathrm{and} \mathrm{\gamma }$

The value of the integral

${\int }_{- 1}^1\left\{\frac{{\mathrm{x}}^{2013}}{{\mathrm{e}}^{\left|\mathrm{x}\right|}\left({\mathrm{x}}^2 +\hspace{0.17em}\cos \left(\mathrm{x}\right)\right)} + \frac{1}{{\mathrm{e}}^{\left|\mathrm{x}\right|}}\right\}d\mathrm{x}$ is equal to

• 0

• $1 - {\mathrm{e}}^{- 1}$

• $2{\mathrm{e}}^{- 1}$

• $2\left(1 - {\mathrm{e}}^{- 1}\right)$

The value of I = ${\int }_0^{\mathrm{\pi }/4}\left({\tan }^{\mathrm{n} + 1}\left(\mathrm{x}\right)\right)d\mathrm{x} + \frac{1}{2}{\int }_0^{\mathrm{\pi }/2}\left({\tan }^{\mathrm{n} + 1}\left(\frac{\mathrm{x}}{2}\right)\right)d\mathrm{x}$ is

• $\frac{1}{\mathrm{n}}$

• $\frac{\mathrm{n} + 2}{2\mathrm{n} + 1}$

• $\frac{2\mathrm{n} - 1}{\mathrm{n}}$

• $\frac{2\mathrm{n} - 3}{3\mathrm{n} - 2}$

The value of the integral

${\int }_1^2{\mathrm{e}}^{\mathrm{x}}\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right) + \frac{\mathrm{x} + 1}{\mathrm{x}}\right)d\mathrm{x}$

• ${\mathrm{e}}^2\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$

• ${\mathrm{e}}^2 - \mathrm{e}$

• ${\mathrm{e}}^2\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right) - \mathrm{e}$

• ${\mathrm{e}}^2 - \mathrm{e}\left(1 + {\log }_{\mathrm{e}}\left(2\right)\right)$