9000063602
Level:
B
Find the following limit.
\[
\lim _{n\to \infty }(-1)^{n} \frac{3}
{2n + 1}
\]
\(0\)
\(-\frac{3}
{2}\)
\(\frac{3}
{2}\)
\(- 1\)
B
9000063604
Level:
B
Find the following limit.
\[
\lim _{n\to \infty }\sin 2\pi n
\]
\(0\)
\(1\)
\(- 1\)
\(\infty \)
9000063605
Level:
B
Find the following limit.
\[
\lim _{n\to \infty }\log 3^{n}
\]
\(\infty \)
\(- 1\)
\(0\)
\(3\)
9000063607
Level:
B
Find the following limit.
\[
\lim _{n\to \infty } \frac{1}
{\log 10^{n}}
\]
\(0\)
\(1\)
\(10\)
\(\infty \)
9000063608
Level:
B
Find the following limit.
\[
\lim _{n\to \infty }\frac{2^{n} + 3^{n}}
{3^{n}}
\]
\(1\)
\(2\)
\(3\)
\(\infty \)
9000063107
Level:
B
Differentiate the following function.
\[
f(x) =\cos x(1 +\sin x)
\]
\(f'(x) =\cos ^{2}x -\sin ^{2}x -\sin x,\ x\in \mathbb{R}\)
\(f'(x) = -\sin x\cos x,\ x\in \mathbb{R}\)
\(f'(x) =\cos x,\ x\in \mathbb{R}\)
\(f'(x) =\sin x +\sin ^{2}x -\cos ^{2}x,\ x\in \mathbb{R}\)
9000046606
Level:
B
Identify an inequality which is true for
\(x = \frac{3\pi }
{4}\).
\(\mathop{\mathrm{cotg}}\nolimits (-x) > 0\)
\(\sin 2x > 0\)
\(\mathop{\mathrm{tg}}\nolimits x\cdot \mathop{\mathrm{cotg}}\nolimits x < \frac{\sqrt{2}}
{2} \)
\(\cos ^{2}x < 0\)
9000062904
Level:
B
A semicircle is a half of the full circle. An infinite spiral is built from
semicircles with a decreasing radius. The radius of the first semicircle is
\(3\, \mathrm{cm}\). The
radius of each semicircle in the spiral is smaller by one third of the radius of the
previous semicircle. Find the total length of the spiral.
\(9\pi \)
\(9\)
\(\frac{9}
{5}\pi \)
\(\infty \)
9000063109
Level:
B
Differentiate the following function.
\[
f(x) = 3^{x}\cdot x^{3}
\]
\(f'(x) = 3^{x}x^{2}(x\ln 3 + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x+1}x^{2}\ln 3,\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x + 3),\ x\in \mathbb{R}\)
\(f'(x) = 3^{x}x^{2}(x\ln x + 3),\ x\in \mathbb{R}^{+}\)
9000046607
Level:
B
Identify an inequality which is true for both
\(x = \frac{4\pi }
{3}\) and
\(x = \frac{5\pi }
{3}\).
\(\sin x < \frac{1}
{2}\)
\(\mathop{\mathrm{tg}}\nolimits x > 0\)
\(\cos x < 0\)
\(\mathop{\mathrm{cotg}}\nolimits x\leq - 1\)