9000153807
Level:
A
Find the number of the positive integers with three digits which can be written using
the digits \(2\),
\(3\),
\(4\) and
\(5\). Each
digit can be used at most once.
\(24\)
\(64\)
\(256\)
\(81\)
A
9000168702
Level:
A
Given the ellipse \(4x^{2} + 9y^{2} + 16x - 18y - 11 = 0\),
find the coordinates of the vertex on the minor axis.
\([-2;3]\)
\([-2;4]\)
\([0;1]\)
\([1;1]\)
9000168703
Level:
A
Given the ellipse \(25x^{2} + 9y^{2} - 150x + 18y + 9 = 0\),
find the vertex on the major axis.
\([3;4]\)
\([3;2]\)
\([8;-1]\)
\([6;-1]\)
9000168704
Level:
A
Given the ellipse \(9x^{2} + 4y^{2} + 36x - 24y + 36 = 0\),
find the vertex on the minor axis.
\([-4;3]\)
\([-5;3]\)
\([-2;0]\)
\([-2;1]\)
9000168705
Level:
A
Given the ellipse \(16x^{2} + 9y^{2} + 32x - 36y - 92 = 0\),
find the vertex on the major axis.
\([-1;-2]\)
\([-1;-1]\)
\([3;2]\)
\([2;2]\)
9000150404
Level:
A
Evaluate the following definite integral.
\[
\int _{2}^{6} \frac{2}
{x}\, \text{d}x
\]
\(\ln 9\)
\(\ln 2\)
\(\ln 3\)
\(2\ln 6\)
9000150407
Level:
A
Evaluate the following definite integral.
\[
\int _{1}^{2}7^{x}\, \text{d}x
\]
\(\frac{42}
{\ln 7} \)
\(49\ln 7\)
\(42\)
\(42\ln 7\)
9000150408
Level:
A
Evaluate the following definite integral.
\[
\int _{0}^{ \frac{\pi }{
4} } \frac{2}
{\cos ^{2}x}\, \text{d}x
\]
\(2\)
\(0\)
\(4\)
\(\pi \)
9000151310
Level:
A
Find the value of the real parameter
\(a\) which ensures, that
the following two lines \(p\)
and \(q\)
are perpendicular.
\[
p\colon ax + y - 4 = 0,\qquad q\colon x + 2y + 4 = 0.
\]
\(- 2\)
\(2\)
\(1\)
\(- 1\)
9000150105
Level:
A
Evaluate the following integral on \(\mathbb{R}\).
\[
\int \left (6^{x} - 6x^{6}\right )\, \mathrm{d}x
\]
\(\frac{6^{x}}
{\ln 6} -\frac{6x^{7}}
{7} + c,\ c\in \mathbb{R}\)
\(6^{x}\ln 6 - 6x^{7} + c,\ c\in \mathbb{R}\)
\(6^{x}\ln 6 -\frac{6x^{7}}
{7} + c,\ c\in \mathbb{R}\)
\(\frac{6^{x}}
{\ln 6} - 6x^{7} + c,\ c\in \mathbb{R}\)