C

2010016806

Level: 
C
The domain of the expression \( \frac{\cos⁡^2 x}{1+\sin ⁡x} \) is the set:
\( \left\{x\in\mathbb{R}\colon x\neq\frac{3\pi}2 + 2k\pi,\ k\in\mathbb{Z} \right\} \)
\( \mathbb{R}\)
\( \left\{x\in\mathbb{R}\colon x\neq\frac{\pi}2 + 2k\pi,\ k\in\mathbb{Z} \right\} \)
\( \left\{x\in\mathbb{R}\colon x\neq \pi + 2k\pi,\ k\in\mathbb{Z} \right\} \)

2010016404

Level: 
C
Function \( f \) is given completely by the next graph. Identify which of the following statements is true.
\( f(x)=|-\cos x|;\ x\in [ -2\pi;2\pi ]\)
\( f(x)=-|\cos x|;\ x\in [ -2\pi;2\pi ]\)
\( f(x)=|\sin x|;\ x\in [ -2\pi;2\pi ]\)
\( f(x)=-|\sin x|;\ x\in [ -2\pi;2\pi ]\)

2000016301

Level: 
C
Investors Thomas and Paul invested the same amount. After the first year, the value of Thomas's investments decreased by \(5{\small{{}^\text{o}\mkern-5mu/\mkern-3mu_\text{o}}}\), but after the following year, their value increased by \(5{\small{{}^\text{o}\mkern-5mu/\mkern-3mu_\text{o}}}\). Paul's investments were more stable. After the first year, the value of his investments increased by \(2{\small{{}^\text{o}\mkern-5mu/\mkern-3mu_\text{oo}}}\), but after the second year, it decreased again by \(2{\small{{}^\text{o}\mkern-5mu/\mkern-3mu_\text{oo}}}\). Determine the true statement about the value of Thomas and Paul's investments two years after the investment.
Paul's investments will have a higher value.
Thomas's investments will have a higher value.
The values of both investments will again be the same.
It is not possible to determine the ratio of the values of Paul and Thomas's investments from the given data.

2010016114

Level: 
C
Let a point \(B\) be the intersection point of the sphere \(x^2 + y^2 + z^2 + 4x + 2y - 4z - 8 = 0\) and \(y\)-axis. Find the equations of all the tangent planes to the given sphere at the point \(B\).
\(2x -3y -2z -12 = 0\), \(2x + 3y - 2z -6 = 0\)
\(2x + 3y - 2z +12 = 0\), \(2x -3 y -2z +6 = 0\)
\(2x -3y -2z -12 = 0\), \(2x -3 y -2z +6 = 0\)
\(2x + 3y - 2z +12 = 0\), \(2x + 3y - 2z -6 = 0\)

2010016113

Level: 
C
Let a point \(A\) be the intersection point of the sphere \(x^2 + y^2 + z^2 - 4x - 2y + 4z - 5 = 0\) and \(z\)-axis. Find the equations of all the tangent planes to the given sphere at the point \(A\).
\(2x + y + 3z + 15 = 0\), \(2x + y - 3z + 3 = 0\)
\(2x + y - 3z -15 = 0\), \(2x + y + 3z - 3 = 0\)
\(2x + y + 3z + 15 = 0\), \(2x + y + 3z - 3 = 0\)
\(2x + y - 3z - 15 = 0\), \(2x + y - 3z + 3 = 0\)