9000064103
Level:
B
Find the normal line to the graph of the function
\(f(x) = 2x^{2} - 2x + 1\) at the
point \([2;5]\).
\(x + 6y - 32 = 0\)
\(6x - y - 7 = 0\)
\(x + 6y - 4 = 0\)
\(- 6x + y - 7 = 0\)
B
9000063808
Level:
B
Given the sequence \(\left (2n + 3\right )_{n=1}^{\infty }\),
find the recurrence relation for this sequence.
\(a_{n+1} = a_{n} + 2,\ a_{1} = 5\)
\(a_{n+1} = a_{n} + 3,\ a_{1} = 5\)
\(a_{n+1} = a_{n} + 4,\ a_{1} = 5\)
\(a_{n+1} = a_{n} + 5,\ a_{1} = 5\)
9000064104
Level:
B
Let \(p\) be the tangent to
the graph of the function \(f(x) = x^{2} - x - 6\)
parallel to the line \(y = 3x + 1\).
Find the point \(A\)
where \(p\) touches
the graph of \(f\).
\(A = \left [2;-4\right ]\)
\(A = \left [2;4\right ]\)
\(A = \left [1;6\right ]\)
\(A = \left [-1;-4\right ]\)
9000063809
Level:
B
Given the sequence \(\left ( \frac{1}
{n(n+1)}\right )_{n=1}^{\infty }\),
find the recurrence relation for this sequence.
\(a_{n+1} = \frac{n}
{n+2}a_{n},\ a_{1} = \frac{1}
{2}\)
\(a_{n+1} = \frac{n}
{n+1}a_{n},\ a_{1} = \frac{1}
{2}\)
\(a_{n+1} = \frac{n+1}
{n} a_{n},\ a_{1} = \frac{1}
{2}\)
\(a_{n+1} = \frac{n+1}
{n+2}a_{n},\ a_{1} = \frac{1}
{2}\)
9000064105
Level:
B
Find the tangent to the graph of the function
\(f(x) = x\sin x\) at the
point \(\left [ \frac{\pi }{2}; \frac{\pi } {2}\right ]\).
\(y = x\)
\(y = x + 1\)
\(y = 0\)
\(y =\pi -x\)
9000062906
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
\(2\, \mathrm{cm}\). The
radius of each semicircle in the spiral is one half of the radius of the previous
semicircle. Find the total length of the spiral.
\(4\pi \)
\(\frac{4}
{3}\pi \)
\(- 4\pi \)
\(\infty \)
9000060604
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} = x\),
\(a_{2} = x + 2\) and
\(a_{3} = 2x\) are
three consecutive terms of an arithmetic sequence.
\(x = 4\)
\(x = 0\)
\(x = 2\)
\(x = 6\)
\(x = 8\)
9000062908
Level:
B
A quarter circle is an arc formed by one quarter of the full circle. An infinite spiral is
built from quarter circles with an increasing radius. The radius of the first quarter circle
is \(4\, \mathrm{cm}\).
The radius of each quarter circle in the spiral is one half of the radius of the previous
quarter circle. Find the total length of the spiral.
\(4\pi \)
\(8\)
\(\frac{8}
{3}\)
\(\infty \)
9000060608
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} =\log x\),
\(a_{2} =\log(2x)\) and
\(a_{3} = 1\) are
three consecutive terms of an arithmetic sequence.
\(x = 2.5\)
\(x = 1\)
\(x =\log 2\)
\(x = 2\)
\(x = 1\)
9000060601
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} = 10^{2}\),
\(a_{2} = 10^{3}\) and
\(a_{3} = x\) are
three consecutive terms of an arithmetic sequence.
\(x = 1\: 900\)
\(x = 1\: 000\)
\(x = 10\: 000\)
\(x = 1\: 990\)
\(x = 100\: 000\)