C

1003083108

Level: 
C
The parabolas of the functions \( f \) and \( g \) have the same vertex \( V \) and \( f(x)=ax^2+c \), where \( a \) and \( c \) are nonzero real numbers. Find \( g(x) \) such that the graphs of \( f \) and \( g \) are symmetric about the vertex \( V \) and that \( y \)-axis is their line of symmetry.
\( g(x)=-ax^2+c\), i.e. the equations of \( f \) and \( g \) differ in the sign of the coefficient at the quadratic term only
\( g(x)=ax^2-c\), i.e. the equations of \( f \) and \( g \) differ in the sign of the coefficient at the linear term only
\( g(x)=-ax^2-c \), i.e. \( g(x)=-f(x) \)
None of the statements above is true.

1003109404

Level: 
C
One of the roots of the quadratic equation \( x^2 + px + 1 - 3\,\mathrm{i} = 0 \) with a complex parameter \( p \) is \( x_1 = -\mathrm{i} \). Choose the equivalent form of the given equation.
\( (x + \mathrm{i})(x -3 - \mathrm{i}) = 0 \)
\( (x + \mathrm{i})(x - 3 +\mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x- 3-\mathrm{i}) = 0 \)
\( (x +\mathrm{i})(x + 3 + \mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x- 3 + \mathrm{i}) = 0 \)
\( (x -\mathrm{i})(x + 3 +\mathrm{i}) = 0 \)

1003109403

Level: 
C
One of the following equations has the solutions \( x_1=\frac12-\mathrm{i} \) and \( x_2=-\frac12+2\,\mathrm{i} \). Find this equation.
\( 4x^2-4\,\mathrm{i}\,x+7+6\,\mathrm{i}=0 \)
\( 4x^2-4\,\mathrm{i}\,x-9+3\,\mathrm{i}=0 \)
\( 4x^2+4\,\mathrm{i}\,x+7+6\,\mathrm{i}\,=0 \)
\( 4x^2+4\,\mathrm{i}\,x-9+3\,\mathrm{i}=0 \)

1003109402

Level: 
C
Which of the given quadratic equations has the roots \( x_1 = 1 +\mathrm{i} \) and \( x_2 = (1 +\mathrm{i})^2 \)?
\( x^2 - (1 + 3\,\mathrm{i})x - 2 + 2\,\mathrm{i} = 0 \)
\( x^2 - (1 + 3\,\mathrm{i})x - 2 - 2\,\mathrm{i} = 0 \)
\( x^2 - (1 + 3\,\mathrm{i})x + 2 + 2\,\mathrm{i} = 0 \)
\( x^2 - (1 + 3\,\mathrm{i})x + 2 - 2\,\mathrm{i} = 0 \)

1003109401

Level: 
C
Which of the given quadratic equations has the roots \( x_1 = 2 + \mathrm{i} \) and \( x_2 = 1 - 3\,\mathrm{i} \)?
\( x^2 - (3 - 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)
\(x^2 + (3 - 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)
\( x^2 - (3 + 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)
\( x^2 + (3 + 2\,\mathrm{i})x + 5 - 5\,\mathrm{i} = 0 \)

1003082308

Level: 
C
Let \( [x;y]\in\mathbb{N}\times\mathbb{N} \). Find all \( [x;y] \), that satisfy \[ x(8 + 4\,\mathrm{i}) + y(1 - 4\,\mathrm{i}) + 5 = x(3 +\mathrm{i}) + 6(y - 2\,\mathrm{i}) + 9\,\mathrm{i}. \]
There is no \( [x;y] \). (There is no solution.)
\( [1;0] \)
\( [0;1] \)
\( [-1; 0] \)
\( [0;-1] \)

1003082307

Level: 
C
Let \( z_1 = x^2 + 9y\,\mathrm{i}-20\,\mathrm{i} \) and \( z_2 = 7x-12+ y^2\,\mathrm{i} \). Find all \( [x;y] \in \mathbb{R}\times\mathbb{R} \) such that \( z_1= z_2 \).
\( [x;y]\in\left\{[3;4], [3;5], [4;4], [4;5]\right\} \)
\( [x;y]\in\left\{[4;3], [4;4], [5;3], [5;4]\right\} \)
\( [x;y]\in\left\{[-3;-4], [-3;-5], [-4;-4], [-4;-5]\right\} \)
\( [x;y]\in\left\{[-4;-3], [-4;-4], [-5;-3], [-5;-4]\right\} \)

1003082306

Level: 
C
Let \( [x;y]\in\mathbb{R}\times\mathbb{R} \). Find all \( [x;y] \) that satisfy \[ (3x + 2y\,\mathrm{i})\cdot(3x - 2y\,\mathrm{i}) + y^2\,\mathrm{i} = 97 + 4\,\mathrm{i}. \]
\( [x;y]\in\left\{[3;2], [-3;2], [3;-2], [-3;-2]\right\} \)
\( [x;y]\in\left\{[3;2], [-3;2]\right\} \)
\( [x;y]\in\left\{[3;2], [3;-2]\right\}\)
\( [x;y]\in\left\{[3;2], [-3;-2]\right\} \)

1003107605

Level: 
C
Evaluate the following integral on \( (0;\infty) \). \[ \int\frac{7x+2}{15x^2+3x}\mathrm{d}x \]
\( \frac23\ln⁡ x-\frac15\ln\left(x+\frac15\right)+c\text{, }c\in\mathbb{R} \)
\( \ln\frac{\sqrt[3]{x^2}}{\sqrt[5]{x+1}}+c\text{, }c\in\mathbb{R} \)
\( \frac23\ln x-\ln(5x+1)+c\text{, }c\in\mathbb{R} \)
\(\ln\frac{\sqrt[3]{x^2}}{\sqrt[5]{x+5}}+c\text{, }c\in\mathbb{R} \)