9000063607
Level:
B
Find the following limit.
\[
\lim _{n\to \infty } \frac{1}
{\log 10^{n}}
\]
\(0\)
\(1\)
\(10\)
\(\infty \)
B
9000063608
Level:
B
Find the following limit.
\[
\lim _{n\to \infty }\frac{2^{n} + 3^{n}}
{3^{n}}
\]
\(1\)
\(2\)
\(3\)
\(\infty \)
9000063806
Level:
B
Consider the sequence \(a_{n+1} = a_{n} - 2a_{n-1}\)
with \(a_{3} = 0\)
and \(a_{4} = -16\).
Find \(a_{2} - a_{1}\).
\(4\)
\(16\)
\(- 4\)
\(8\)
9000063302
Level:
B
Differentiate the following function.
\[
f(x)= (3x^{2} + 2)^{3}
\]
\(f'(x) = 18x(3x^{2} + 2)^{2},\ x\in \mathbb{R}\)
\(f'(x) = 18x(3x^{2} + 2),\ x\in \mathbb{R}\)
\(f'(x) = 18x^{2}(3x + 2)^{2},\ x\in \mathbb{R}\)
\(f'(x) = 108x^{2},\ x\in \mathbb{R}\)
9000064101
Level:
B
Find the slope of the tangent to the graph of
\(f(x) = x^{2} + 3x - 2\) at the
point \([1;2]\).
\(-\frac{1}
{5}\)
\(5\)
\(- 5\)
\(\frac{1}
{5}\)
9000060603
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} = -x\),
\(a_{2} = -5\) and
\(a_{3} = 0\) are
three consecutive terms of an arithmetic sequence.
\(x = 10\)
\(x = 0\)
\(x = -5\)
\(x = 5\)
\(x = -10\)
9000046608
Level:
B
Identify an inequality which is true for both
\(x = \frac{\pi } {6}\) and
\(x = -\frac{\pi }{6}\).
\(\cos x > 0\)
\(\sin x > \frac{1}
{2}\)
\(|\mathop{\mathrm{tg}}\nolimits x| < \frac{1}
{2}\)
\(\mathop{\mathrm{cotg}}\nolimits x\leq - 1\)
9000060606
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} = 5x + 1\),
\(a_{2} = x\) and
\(a_{3} = 7x + 3\) are
three consecutive terms of an arithmetic sequence.
\(x = -0.4\)
\(x = 0.4\)
\(x = 2.5\)
\(x = -2.5\)
\(x = 5\)
9000046609
Level:
B
Identify an inequality which is true for every
\(x\) from the
interval \(\left ( \frac{\pi }{4}; \frac{3\pi }
{4}\right )\).
\(\sin x\geq \frac{\sqrt{2}}
{2} \)
\(\mathop{\mathrm{tg}}\nolimits x > 1\)
\(\cos x > 0\)
\(\mathop{\mathrm{cotg}}\nolimits x\geq \frac{\sqrt{3}}
{3} \)
9000060610
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} =\log x\),
\(a_{2} = 2\) and
\(a_{3} =\log x^{3}\) are
three consecutive terms of an arithmetic sequence.
\(x = 10\)
\(x = 0\)
\(x = 0.1\)
\(x = 1\)
\(x = -1\)